%A king gave all his castles to his 7 sons before he died.

The youngest son received some

number of castles, the second youngest received twice as many, the third three times as
many, etc. so that the oldest son received seven times as many castles as the youngest.
The queen, however, did not find this fair and ordered each son to give two of his castles
to every other son that was younger than himself. After this all seven princes had the
same number of castles. How many castles did the old king originally have?
(a) 98 (b) 126 (c) 105 (d) 84 (e) 112
Four children guess the height of their teacher to be 196cm, 163cm, 178cm, and 185cm.
They all missed the exact height and they were off (in some order) by 1 cm, 6cm, 16cm
and 17cm. What is the sum of the digits in the teachers real height?
(a) 15 (b) 16 (c) 17 (d) 18 (e) 19
The gas mileage of a car is
16 miles per gallon at 80 miles per hour,
18 miles per gallon at 70 miles per hour,
20 miles per gallon at 60 miles per hour.
If that car were driven at 80 mph for 2 hours, at 70 mph for 3 hours, and at 60 mph for 5
hours, about how many gallons of gas would be used?
(a) 35 (b) 37 (c) 39 (d) 41 (e) 43
Which of the following are always true for all nonzero real numbers x, y and z.
i. (x y) z = x (y z)
ii. (x y) z = x (y z)
iii. (x y) z = x (y z)
iv. (x y) z = x (y z)
v. x (y z) = x (y z)
(a) only i, iii (b) only i, iv (c) only ii, iii, iv (d) only i, iii, iv (e) all of the above
%The square below can be filled in so that each row and each column contains each of
the numbers 1, 2, 3, and 4 exactly once. What does x equal?

Four numbers, a, b, c and d, are either 0 or 1. It is known that,

(i)
if b = 0, then c = 1
(ii)
if c = 0, then a = d
(iii)
if d = 0, then a = 1
Suppose c = 0, what is the value of a + b + d?

%Each of the letters A, B, D, and M represents a decimal digit. Two have the value 9, one
has the value 8, and the other has the value 1. The largest possible sum of the 3-digit
numbers BAD, DAM, and MAD is
(A) 2159 (B) 2655 (C) 2656 (D) 2657 (E) 2958
I have a list of 5 positive integers whose median is 3 and whose mode is 4. Name the 5
numbers in my list.
You fold a 1-inch square piece of paper along its diagonal, then in half and then in half
again. What are the lengths of the sides of this new shape?
A woman and her two children need to cross a river. They have a rowboat, but this boat
can only carry a maximum of 150 lbs. If the mother weighs 130 lbs and each child
weighs 75 lbs, what is the least number of river crossings required to get the entire family
across the river?
It takes me 10 minutes to eat a big mac. However, if I ever consume any amount over 2
big macs within an hour, I will get sick. What is the fastest I can eat 6 big macs without
getting sick? 140 minutes
How many positive multiples of 11 less than 117 are there which are not multiples of 2, 3, 5, or
7? (1)

Anna has two unmarked water bottles. One holds 12 ounces of fluid and the other holds
42 ounces. Anna has access to running water, but has no other water containers. What is
the smallest nonzero amount of water that Anna can exactly measure out?
If a, b, and c are non-zero integers, what is the smallest possible integer value of
a b2 c2 ?
Lights are evenly spaced on a circular path. If the 10th and the 32nd light are directly
opposite each other (endpoints of a diameter), how many lights are there?
A. 54

B. 48

C. 44

D. 22

%Let m be the smallest of 10 distinct positive integers, each of which is less than
100. If the average (arithmetic mean) of these numbers is 88, the smallest m could be is
a. 1 b. 8 c. 15 d. 25 e. 77
%Of 100 people in a room, 60 play guitar and 50 play piano. At most, how many play
one instrument but not the other?
a. 10 b. 40 c. 50 d. 55 e. 90
Of 100 kids in a room, 70 play tennis and 50 play golf. At most how many kids play one
of these sports but not the other? (80)

Suppose you are writing positive integers in a row, without blank spaces, like this:
123456789101112 . . .
What will be the 1000th digit?
%Let us play the following game. You have $1. With every move, you can either double
your money or add $1 to it. What is the smallest number of moves you have to make to
get to $200?
(A) 6 (B) 7 (C) 8 (D) 9 (E) It is impossible to get to $200
Suppose y > 0, x > y and z 0, the inequality which is always correct is
(a) x +z > y z
(b) xy > yz
(c) x / z > y / z
(d) xz2 > yz2
An abundant number, N, is a whole number whose factors have a sum which is greater
than 2N. For example, 12 is an abundant number, because 12 has a sum of its factors: 1 +
2 + 3 + 4 + 6 + 12 = 28 which is greater than 24 ( 2 12 ). How many abundant numbers
which are greater than 50 and less than 60?
The number in an unshaded square is obtained by adding the numbers
connected to it from the row above. (The 11 is one such number that equals
5 + 6.) The value of x is
(A) 4 (B) 6 (C) 9 (D) 15 (E) 10

%On a rectangular table with 5 units long and 2 units wide, a ball is rolled
from point P at an angle of 45to PQ and bounces off SR. The ball continues
to bounce off the sides at 45until it reaches S. How many bounces of the
ball are required?
(A) 9 (B) 8 (C) 7 (D) 5 (E) 4

The sequence appears as: A, B, B, C, C, C, D, D, D, D, E, E, E, E, E, F, F

. .
The 100th letter in the sequence is

Using only digits 1, 2, 3, 4, and 5, a sequence is created as follows: one

1, two 2s, three 3s, four 4s, five 5s, six 1s, seven 2s, and so on.
The sequence appears as: 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 1, 1,
1, 1, 1, 1, 2, 2, ... .
The 100th digit in the sequence is
(A) 1 (B) 2 (C) 3 (D) 4 (E) 5
A deck of 100 cards is numbered from 1 to 100. Each card has the
same number printed on both sides. One side of each card is red and
the other side is yellow. Barsby places all the cards, red side up, on a
table. He first turns over every card that has a number divisible by 2.
He then examines all the cards, and turns over every card that has a
number divisible by 3. How many cards have the red side up when
Barsby is finished?
(A) 83 (B) 17 (C) 66 (D) 50 (E) 49
The numbers 123,456,789 and 999,999,999 are multiplied. How many
of the digits in the final result are 9s?
(A) 0 (B) 1 (C) 2 (D) 3 (E) 17
Twelve balloons are arranged in a circle as shown. Counting clockwise,
every third balloon is popped. C is the first one popped. This continues
around the circle until two unpopped balloons remain. The last two
remaining balloons are
(A) B, H (B) B, G (C) A, E (D) E, J (E) F, K

Janet has 10 coins consisting of nickels, dimes, and quarters. Seven of

the coins are either dimes or quarters, and eight of the coins are either
dimes or nickels. How many dimes does Janet have?
(A) 2 (B) 3 (C) 4 (D) 5 (E) 6
In the diagram, all triangles are equilateral. What fraction of ABC is
colored black?
3
1
9
4
27
(A)
(B)
(C)
(D)
(E)
4
2
16
9
64

The digits 1, 2, 3, 4 can be arranged to form twenty-four different fourdigit numbers. If these twenty-four numbers are then listed from the
smallest to largest, in what position is 3142?
(A) 13th (B) 14th (C) 15th (D) 16th (E) 17th
In the diagram adjacent edges are at right angles. The four longer
edges are equal in length, and all of the shorter edges are also equal in
length. The area of the shape is 528. What is the perimeter?
(A) 132 (B) 264 (C) 144 (D) 72 (E) 92

How many unshaded squares are in the tenth figure of the pattern?
(A) 38 (B) 40 (C) 42 (D) 44 (E) 46

At the beginning of the game Clock 7, the arrow points to one of the
seven numbers. On each turn, the arrow is rotated clockwise by the
number of spaces indicated by the arrow at the beginning of the turn.
For example, if Clock 7 starts with the arrow pointing at 4, then on
the first turn, the arrow is rotated clockwise 4 spaces so that it now

points at 1. The arrow will then move 1 space on the next turn, and so
on.
1. If the arrow points at 6 on the beginning, which number would the
arrow point after the 21st turn?
2. If the arrow points at 6 after the 21st turn, at which number did the
arrow point after the first turn?
(A) 3 (B) 6 (C) 5 (D) 2 (E) 7

In the diagram, the numbers 1, 2, 4, 5, 6, and 8 are substituted, in some order, for the
letters A, B, C, D, E, and F, so that the number between and below two numbers is the
positive difference between those two numbers. For example, the 7 in the third row is the
positive difference between D and 9. Thus D = 2 because 9 2 = 7. The value of A+ C is
(A) 7 (B) 12 (C) 13 (D) 10 (E) 14

A straight one-way city street has 8 consecutive traffic lights. Every light remains green
for 1.5 minutes, yellow for 3 seconds, and red for 1.5 minutes. The lights are
synchronized so that each light turns red 10 seconds after the preceding one turns red.
What is the longest interval of time, in seconds, during which all 8 lights are green?
(A) 10 (B) 15 (C) 20 (D) 25 (E) 30
Toothpicks are used to form squares in the pattern shown:

Four toothpicks are used to form one square, seven to form two squares, and so on. If this
pattern continues, how many toothpicks will be used to form 10 squares in a row?
(A) 39 (B) 40 (C) 31 (D) 35 (E) 28
The natural numbers from 1 to 2100 are entered sequentially in 7 columns, with the first
3 rows as shown. The number 2002 occurs in column m and row n. The value of m+ n is

(A) 290 (B) 291 (C) 292 (D) 293 (E) 294

Pierre celebrated his birthday on March 2, 2005. On that day, his age equals the sum of
the digits in the year in which he was born. In what year was Pierre born?
(A) 1987 (B) 1980 (C) 1979 (D) 1977 (E) 1971
The largest four-digit number which has the sum of digits as 17 is 9800. The 5th largest
four-digit number whose digits have a sum of 17 is
(A) 9521 (B) 9620 (C) 9611 (D) 9602 (E) 9530
In the diagram, every number beginning at third column, 30, equals
twice the sum of the two numbers to its immediate left. The value of c
is
(A) 50 (B) 70 (C) 80 (D) 100 (E) 200

The digits 1, 1, 2, 2, 3, and 3 are arranged to form an odd six digit

integer. The 1s are separated by one digit, the 2s by two digits, and
the 3s by three digits. What are the last three digits of this integer?
(A) 3 1 2 (B) 1 2 3 (C) 1 3 1 (D) 1 2 1 (E) 2 1 3
The left most digit of an integer of length 2000 digits is 3. In this
integer, any two consecutive digits must be divisible by 17 or 23. The
2000th digit may be either a or b. What is the value of a+ b ?
(A) 3 (B) 7 (C) 4 (D) 10 (E) 17
If p, q, r, s , and t are numbers such that r s , t q , q p, and t r,
which of these numbers is greatest?
(A) t (B) s (C) r (D) q (E) p

In the chart, the products of the numbers represented by the letters in

each of the rows and columns are given. For example, xy 6 and xz
12. If x, y, z, and w are integers,
what is the value of xw?
(A) 150 (B) 300 (C) 31 (D) 75 (E) 30

Given the set 1, 2, 3, 5, 8,13, 21, 34, 55 , how many integers

between 3 and 89 cannot be written as the sum of exactly two
elements of the set?
(A) 51 (B) 57 (C) 55 (D) 34 (E) 43
The first four triangular numbers 1, 3, 6, and 10 are illustrated in the
diagram. What is the tenth triangular number?
(A) 55 (B) 45 (C) 66 (D) 78 (E) 50

A positive integer is to be placed in each box. The product of any four

adjacent integers is always 120. What is the value of x?
(A) 1 (B) 2 (C) 3 (D) 4 (E) 5

Eight squares with the same centre have parallel sides and are one
unit apart. The two largest squares are shown. If the largest square has
a perimeter of 96, what is the perimeter of the smallest square?
(A) 40 (B) 68 (C) 32 (D) 64 (E) 89

Faruq subtracted 5 from a number and then divided by 4. Next, he subtracted 4 from the
original number and then divided by 5. He got the same final answer both times. The
original number was
(A) 4 (B) 15 (C) 9 (D) 20 (E) 9
In the grid shown, it is only possible to travel along an edge in the direction indicated by
the arrow. The number of different paths from A to F is
(A) 9 (B) 5 (C) 3 (D) 6 (E) 4

The sum of the digits of the integer equal to 777777777777 7772 2222222222222232 is
(A) 148 (B) 84 (C) 74 (D) 69 (E) 79
In how many ways can 75 be expressed as the sum of two or more consecutive positive
integers?
(A) 1 (B) 2 (C) 3 (D) 4 (E) 5
In the sequence of five numbers x, _____, 3, _____, 18, each number after the second is
obtained by multiplying the two previous numbers. The value of x is
2
3
(A)
(B)
(C) 1 (D) 9 (E) 1
3
2
In the magic square, the sum of the three numbers in any row, column or diagonal is the
same. The sum of the three numbers in any row is
(A) 0 (B) 1 (C) 3 (D) 7 (E) 9

In a pack of construction paper, the numbers of blue and red sheets

are originally in the ratio 2:7. Each day, Laura uses 1 blue sheet and 3
red sheets. One day, she uses 3 red sheets and the last blue sheet,
leaving her with 15 red sheets. How many sheets of construction paper
were in the pack originally?
(A) 144 (B) 252 (C) 135 (D) 270 (E) 105
Zan has created this rule for generating sequences of whole numbers.
If a number is 25 or less, double the number.

If a number is more than 25, subtract 12 from it.

For example, if Zan starts with 10, she gets the sequence 10, 20,
40, 28, 16, . If the third number in Zans sequence is 36, what
is the sum of the four distinct numbers that could have been the
first number in her sequence?
John, Mike and Chantel will divide a pile of pennies amongst
themselves using the following process: The number of pennies in the
pile is counted.

If the number of pennies in the pile is even, Mike will get

half of the pile.
If the number of pennies in the pile is odd, one penny will
be given to Chantel, and John will get half the pennies
remaining in the pile.

This process is then repeated until the pile is empty. How many pennies will Mike have at
the end if the original pile contains 2005 pennies?
How many combinations of pennies, nickels and/or dimes are there
with a total value of 25?
The sequence of integers in the row of squares and in each of the two
columns of squares form three distinct arithmetic sequences. What is
the value of N?

If I add up 9 of the first 10 positive integers, the result is a perfect square. What is the
number I am missing from the first 10 positive integers?
The largest prime number less than 30 that can be written as the sum of two primes is?

How many whole numbers between 50 and 60 are the products of 2 prime numbers?
Five students are seated at a round table, facing the table.
(1) Howard is next to Tina and on her right.
(2) Jeff is next to Beth and on her right.
(3) Melinda is not sitting next to Beth.
What is the seating arrangement, starting with Melinda and going to her left?
A. Melinda-Tina-Howard-Beth-Jeff
B. Melinda-Howard-Tina-Beth-Jeff
C. Melinda-Tina-Howard-Jeff-Beth
D. Melinda-Jeff-Beth-Howard-Tina
E. Melinda-Jeff-Beth-Tina-Howard
In the Pascal family, each child has at least 2 brothers and at least 1
sister. What is the smallest possible number of children in this
family?
(A) 3
(E) 7

(B) 4

(C) 5

(D) 6

The average of 3 different positive integers is 20. What is the greatest number of them?

Five persons, A, B, C, D, E, sit on 2 chairs and 3 stools, each seating exactly one person.
If A and B sit on the same type of seat, B and D sit on a different type of seat, D and E sit
on different type of seat, then who sits on the chairs?

There are 3 kinds of marbles in a jar. Each costs 20 cents, 30 cents and 40 cents. The total
cost of marbles in the jar is $15. The number of marbles cost 30 cents in the jar cannot be,
(A) 4

(B) 5

(C) 6

(D) 8

(E) 10

In the incomplete table above, the sum of the three integers in each row, column, and
diagonal is the same. If the numerical values in four of the blocks are as shown, what is
the value of w ?
(A) -6
(B) -5
(C) 2
(D) 5
(E) 8
There are five employees--G, H, I, J, and K--in an office. Rumors spread through the
office according to the following rules; Rumors can be passed in either direction between
J and K. Rumors can be passed from G to H, from H to I, and from I to K. A rumor begun
by H that reaches J will be known by all the following employees EXCEPT ?
A. I
B. G
C. K
D. H
E. J
Half the people on a bus get off at each stop after the first, and no one gets on after the
first stop. If only one person gets off at stop number 7, how many people got on at the
first stop?
(A) 128

(B) 64

(C) 32

(D) 16

(E) 8

There were 9 pieces of paper. Some of them were cut into three pieces. As a result, there
are 15 pieces of paper now. How many pieces of paper were cut?
A) 2 B) 3 C) 4 D) 5 E) 6
Greg needs to bring four full sacks of sand from the river to a house located at the other
end of the village. Unfortunately, on his way through the village, half of the sand spills
out of the sack through a hole. How many trips does Greg need to make from the river to
the house in order to bring the required amount of sand?
A) 4 B) 5 C) 6 D) 7 E) 8
Using 6 matches, only one rectangle with a perimeter of 6 matches can be made (see the
picture). How many different rectangles with a perimeter of 14 matches can be made
using 14 matches?
A) 2 B) 3 C) 4 D) 6 E) 12

A picture frame was constructed using pieces of wood which all have the same width.
What is the width of the frame if the inside perimeter of the frame is 8 decimeters less
than its outside perimeter?
A) 1 dm B) 2 dm C) 4 dm D) 8 dm E) It depends on the size of the picture

In a trunk there are 5 chests, in each chest there are 3 boxes, and in each box there are 10
gold coins. The trunk, the chests, and the boxes are locked. At least how many locks need
to be opened in order to take out 50 coins?
A) 5 B) 6 C) 7 D) 8 E) 9
The figure shows a rectangular garden with dimensions of 16 m by 20 m. The gardener
has planted six identical flowerbeds (they are gray in the diagram). What is the perimeter
of each of the flowerbeds?
A) 20 m B) 22 m C) 24 m D) 26 m E) 28 m

Mike chose a three-digit number and a two-digit number. The difference of these
numbers is 989. What is their sum?
A) 1001 B) 1010 C) 2005 D) 1000 E) 1009
Five cards are laying on the table in the order: 5, 1, 4, 3, 2 as shown in the top row of the
picture. They need to be placed in the order shown in the bottom row. In each move, any
two cards may be switched. What is the least number of moves that need to be made?
A) 2 B) 3 C) 4 D) 5 E) 6

Which of the cubes has the plan shown in the picture to the right?

Eva cut a paper napkin into 10 pieces. She then also cut one of the pieces into 10 pieces.
She repeated this process two more times. Into how many pieces did she cut the napkin?
A) 27 B) 30 C) 37 D) 40 E) 47
A rectangular garden with an area of 30 m2 was divided into three rectangular sections of
flowers, vegetables, and strawberries (some of the dimensions are shown in the diagram).
What is the area of the vegetable section, if the flower part has an area of 10 m2?
A) 4 m2 B) 6 m2 C) 8 m2 D) 10 m2 E) 12 m2

How many two digit numbers are there, which can be expressed only by using different
odd digits?
A) 15 B) 20 C) 25 D) 30 E) 50
What number should replace x, if we know that the number in the circle in the upper row
is the sum of the numbers from the two circles right below it.
A) 32 B) 50 C) 55 D) 82 E) 100

In a two-digit number, a is the tens digit and b is the ones digit. Which of the conditions
below ensures that the number will be divisible by 6?
A) a + b = 6 B) b = 6a C) b = 5a D) b = 2a E) a = 2b

What is the mean measure of these angles at the center of this circle?

A wooden cube with the length of its side equal to 3 m was painted with 0.25 kg of paint.
The cube was then cut up into unit cubes (side length of 1 m). How much paint is needed
to paint the unpainted sides of the little cubes?
A) 1.25 kg B) 1 kg C) 0.75 kg D) 0.5 kg E) 0.25 kg
Five circles have radii of the same length (see the picture). Four of them are touching the
fifth circle, and their centers are the vertices of a square. The ratio of the area of the
shaded region of the circles to the area of unshaded regions of the circles is:
A) 1 : 3 B) 1 : 4 C) 2 : 5 D) 2 : 3 E) 5 : 4

Mark has 42 cubes with side length of 1 cm. He used them to construct a prism, the base
of which has a perimeter of 18 cm. The height of that prism is:
A) 6 cm B) 5 cm C) 4 cm D) 3 cm E) 2 cm
On the board Peter wrote all the three-digit numbers that have the following properties:
the digits in each of the numbers are different, the first digit is the square of the quotient
of the second digit and the third digit. How many numbers did Peter write?
A) 1 B) 2 C) 3 D) 4 E) 8
Equilateral triangle ABC (all sides congruent) has an area equal to 1. A bigger triangle
was constructed out of 49 of these triangles (see the picture). The area of the shaded
region is equal to:
A) 20 B) 22.5 C) 23.5 D) 25 E) 32

How many white squares need to be shaded so that the number of shaded squares equals
exactly to half of the number of white squares?
A) 2 B) 3
C) 4
D) 6
E) It is impossible to calculate it.

Eva is 52 days older than her girlfriend Ania. Eva had her birthday on Tuesday in March
of this year. On which day of the week will Ania celebrate her birthday this year?
A) Monday

B) Tuesday

C) Wednesday

E) Thursday

E) Friday

Numbers were placed into the squares of diagram so that the sum of the numbers in the
first row is equal to 3, the sum of the numbers in the second row is equal to 8, and the
sum of the numbers in the first column is equal to 4. What is the sum of the numbers in
the second column?
A) 4
B) 6
C) 7
D) 8
E) 11

Four square tiles were arranged in a way shown in the picture. The lengths of the sides of
two tiles are indicated in the picture. What is the length of the side of the largest tile?

A) 24

B) 56

C) 64

D) 81

E) 100

Girls and boys from Marias and Mathews class have formed a line. There are 16
students on Marias right, and Mathew is among them. There are 14 students on
Mathews left, and Maria is among them. There are 7 students between Maria and
Mathew. How many students are in this class?
A) 37

B) 30

C) 23

D) 22

E) 16

The sum of the digits of the 10-digit number is 9.What is the product of the digits of this
number?
A) 0

B) 1

C) 45

D) 9 8 7 .. 2 1

E) 10

Out of 125 small, white and black cubes, the big cube was formed (see the picture).
Every two adjacent cubes have different colors. The vertices of the big cube are black.
How many white cubes does the big cube contain?
A) 62

B) 63

C) 64

D) 65

E) 68

A lottery-ticket was 4 dollars. Three boys: Paul, Peter, and Robert made a contribiution
and bought two tickets. Paul gave 1 dollar, Peter gave 3 dollars, and Robert gave 4
dollars. One of the tickets they bought was worth 1000 dollars. Boys shared the award
fairly, meaning, proportionally to their contributions. How much did Peter receive?
A) 300

B) 375

C) 250

D) 750

E) 425

In three soccer games the Dziobaks team scored three goals and lost one. For every
game won the team gets 3 points, for a tie it gets 1 point, and for the game lost it gets 0
points. For sure, the number of points the team earned in those three games was not
equal to which of the following numbers?
A) 7

B) 6

C) 5

D) 4

E) 3

In every white section of a diagram, the products of two numbers from grey sections
one from above and one from the left was placed (for example: 42 = 7 6 ). Some of
these products are represented by letters. Which two letters represent the same number?
A) L and M

B) T and N

C) R and P

D) K and P

E) M and

The different figures represent different digits. Find the digit corresponding to the square.
A) 9

B) 8

C) 7

D) 6

E) 5

Tomek, Romek, Andrzej, and Michal said the following about a certain number: Tomek:
This number is equal to 9; Romek: This number is prime.; Andrzej: This number is
even.; Michal: This number is equal to 15. Only one statement given either by Romek
or Tomek is true, as well as only one statement given by either Andrzej or Michal is true.
What number is it?
A) 1

B)2

C) 3

D) 9

E) 15

What is the smallest number of the little squares that have to be shaded in order to get at
least one axis of symmetry of the figure below?

A) 1

B) 2

C) 3

D) 4

E) 5

One corner of a cube was cut off. Which of the figure below represents the pattern of the
cube after unfolding it?

Four snails: Fin, Pin, Rin, and Tin are moving along identical rectangular tiles. The shape
and length of each snails trip is shown below. How many decimeters has snail Tin gone?
Snail Fin has gone 25 dm.
Snail Pin has gone 37 dm.
Snail Rin has gone 38 dm.
Snail Tin has gone ? dm
A) 27 dm

B) 30 dm

C) 35 dm

D) 36 dm

E) 40 dm

The Island of Turtles has an unusual weather system: Mondays and Wednesdays are
rainy, Saturdays are foggy and the other days are sunny. A group of tourists would like to
go on a 44-day long vacation to the island. Which day of the week should be the first day
of their vacation in order to enjoy the most of the sunny days?
A) Monday

B) Wednesday C) Thursday D) Friday

E) Tuesday

The sum of two natural numbers is equal to 77. If the first number is multiplied by 8 and
the second by 6, then those products are equal. The larger of these numbers is:
A) 23

B) 33

C) 43

D) 44

E) 54

5
2
Ella and Ola had 70 mushrooms altogether. 9 of Ellas mushrooms are brown and 17 of
Olas mushrooms are white. How many mushrooms did Ella have?

A) 27

B) 36

C) 45

D) 54

E) 10

There are 11 fields in the picture. Number 7 is written in the first field and number 6 in
the ninth field. What number has to be placed in the second field so that the sum of the
numbers from every three consecutive fields is equal to 21?
A) 7
The

B) 8

C) 6

D) 10

E) 21

square below was divided into small squares. What part of the area of the shaded
figure is the area of the figure that is not shaded?
1
1
1
2
2
A) 4
B) 5
C) 6
D) 5
E) 7

In a CD store two CDs have the same price. The price of the first CD was reduced by 5 %
and the price of the other one was increased by 15%. After this change the prices of the
two CDs differ by $6.00. How much is the cheaper CD now?
A) $1.50

B) $6.00

C) $28.50

D) $30.00

E) 34.50

In the little squares of a big square the consecutive natural numbers are placed in a way
shown in the figure. Which of the numbers below cannot be placed in the square with
letter x?

A) 128

B) 256

Ania divided number

A) 670
B) 669

C) 81
111
...
1
2004

D) 121

E) 400

by 3. What is the number of zeros in the quotient?

C) 668
D) 667
E) 665

Imagine that you have 108 red balls and 180 green balls. The balls have to be packed in
boxes in such a way that every box contains the same number of balls and there are balls
of only one color in every box. What is the smallest number of boxes that you need?
A) 288

B) 36

C) 18

D) 8

E) 1

During a competition in the Kangaroo Summer Camp in Zakopane students were given
10 problems to solve. For each correct answer a student was given 5 points and for each
incorrect one the student was loosing 3 points. Everybody solved all the problems.
Mathew got 34 points, Philip got 10 points and John got 2 points. How many problems
did they answer correctly all together?
A) 17

B) 18

C) 15

D) 13

E) 21

Peter wrote out consecutive natural numbers starting with 3 until he had written 35 digits.
What was the greatest number that Peter wrote?
A) 12

B) 22

C) 23

D) 28

E) 35

A pattern, the beginning and the end of which is shown in the picture, is made up of
alternating black and white bars. There are 17 bars altogether. The bars on both ends are
black. How many white bars are there in the pattern?

A) 9

B) 16

C) 7

D) 8

E) 15

Peter has 11 pieces of paper. He cut some of them into three parts and now he has 29
pieces of paper. How many pieces of paper did he cut?
A) 3

B) 2

C) 8

D) 11

E) 9

Points A, B, C, D all of which lie on a straight line, are marked in the figure below. The
distance between points A and C is 10 m, between B and D is 15 m, and between A and D
is 22 m. What is the distance between points B and C?

A) 1 m

B) 2 m

C) 3 m

D) 4 m

E) 5 m

Peter bought 3 kinds of cookies: large, medium, and small. The large cookies cost 4
zlotys each, the medium: 2 zlotys each, and the small: 1 zloty each. (A zloty is the Polish
unit of money.) Altogether, Peter bought 10 cookies and paid 16 zlotys. How many large
cookies did he buy?
A) 5

B) 4

C) 3

D) 2

E) 1

Christopher built a rectangular prism using red and blue cubes of identical size. The outer
walls of this prism are red but all the inner cubes are blue. How many blue cubes did
Christopher use in this construction?

A) 12

B) 24

C) 36

D) 40

E) 48

From a square puzzle, two pieces, which made up the shaded region, were cut out (see
the figure). Which two of the pieces below are these?

A) 1 and 3

B) 2 and 4

C) 2 and 3

D) 1 and 4

E) 3 and 4

At the toy store, among other things, you can buy dogs, bears, and kangaroos. Three dogs
and two bears together cost as much as four kangaroos. For the same amount of money
you can buy one dog and three bears. Then:
A) A dog is twice as expensive as a bear.
B) A bear is twice as expensive as a dog.
C) The prices of a dog and of a bear are identical.
D) A bear is three times as expensive as a dog.
E) A dog is three times as expensive as a bear.
Two of the numbers located on the two circles (see the picture) are represented by letters
A and B. The sum of the numbers on each circle is equal to 55. What number is
represented by letter A?
A) 9
B) 10
C) 13
D) 16
E) 17

A square with the length of side equal to x consists of a square with an area of 81 cm2,

two rectangles with areas of 18 cm2 each, and a small square. What is the value of x?
A) 2 cm

B) 7 cm C) 9 cm D) 10 cm E) 11 cm

How many shortest distances along the edges of the cube are there that connect vertex A
with the opposite vertex B?
A) 4

B) 6

C) 3

D) 12

E) 16

We add two different numbers chosen from the numbers: 1, 2, 3, 4, 5. How many
different sums can we get?
A) 5

B) 6

C) 7

D) 8

E) 9

We multiply two different numbers chosen from the numbers: 1, 2, 3, 4, 5. How many
different products can we get?
A) 7

B) 8

C) 9

D) 10

E) 12

The figure in the picture consists of 7 squares. Square A has the greatest area, and square
B - the smallest area. The lengths of two of the squares are given. How many B squares
will it take to fill up square A completely?
A) 16
B) 25
C) 36
D) 49
E) It is impossible.

Ewa has 20 balls of four colors: yellow, green, blue, and black. 17 of them are not green,
5 are black, and 12 are not yellow. How many blue balls does Ewa have?
A) 3
B) 4
C) 6
D) 7
E) 8
The square ABCD consists of a white square and four shaded rectangles. Each of the
rectangles has a perimeter of 40 cm. The area of square ABCD equals:
A)100 cm2 B) 200cm2 C) 160cm2 D) 400cm2 E) 80cm2

We have six segments with lengths: 1, 2, 3, 2001, 2002, 2003. In how many ways can we
select three of these segments to build a triangle?
A) 1

B) 3

C) 5

D) 6

E) 10

Piotrek is writing the numbers from 0 to 109 into a five-column table using a rule which
is easy to understand (see the picture at the right). Which of the pieces below can not be
filled in with numbers to fit Piotrek's table?

In the figure, the beginning part of the path from point A to point B is shown. How long
is the whole path?

In the addition, every square stands for a certain digit, every triangle stands for another
specific digit, and every circle denotes yet another digit. What is the sum of the numbers
represented by the square and the circle?

A) 6

B) 7

C) 8

D) 9

E) 13

The shaded figure at the picture consists of five identical isosceles right triangles (see the
figure at the left). The area of the shaded figure is:
A) 20 cm2

B) 25 cm2

C) 35 cm2

D) 45 cm2

E) 60 cm2

Robert had a certain number of identical cubes. He glued a tunnel using half of his
blocks (see Picture 1). With some of the remaining cubes he formed a pyramid (see
Picture 2). How many blocks were not used to build those structures?
A/ 34

B/ 28

C/ 22

Picture 1

D/ 18

E/ 15

Picture 2

Daughter is 3 years old, and her mother is 28 years older than the daughter. How many
years later will the mother be three times older than her daughter?
A/ 9

B/ 12

C/ 10

D/ 1

E/ 11

Twenty eight students from the fourth grade competed in the math competition. Each
student earned a different number of points. The number of students who received more
points than Tomek is two times smaller than the number of students who had less points
than Tomek. In which position did Tomek finish that competition?
A/ 6th

B/ 7th

C/ 8th

D/ 9th

E/ 10th

An odometer in a car shows the number 187569 of passed kilometers. This number
consists of all different digits. After passing how many kilometers will the odometer
show a number consisting of all different digits again?
A/ After 777 km B/ After 12,431 km
km

C/ After 431 km D/ After 21 km E/ After 11

Rectangle ABCD (see the picture) is built out of 24 little squares with the length of each
side equal to 1. What is the area of triangle ALM?
A) 5

B) 6

C) 7

D) 8

E) Other

How many different three-digit numbers divisible by 25 can be made with the digits 0, 3,
5, 7 if the digits can be repeated?
A) 16

B) 9

C) 81

D) 64

E) 3

In the picture below, the area of triangle ABD is equal to 15, the area of triangle ABC is
equal to 12 and the area of triangle ABE is equal to 4. What is the area of pentagon
ABCED?
A) 19

B) 31

C) 23

D) 27

E) 35

The weight of each possible pair of boys from a group of 5 was recorded. The following
results were obtained: 90 kg, 92 kg, 93 kg, 94 kg, 95 kg, 96 kg, 97 kg, 98 kg, 100 kg and
101 kg. The total weight of all five boys equals:
A) 225 kg

B) 230 kg

C) 239 kg

D) 240 kg

E) 250 kg

There are four congruent squares. In each of them the midpoints of the sides are indicated
and some regions with areas S1, S2, S3 and S4 are shaded. Which expression below is true?

A) S3 < S4 < S1 = S2

B) S3 < S1= S2 = S4

D) S3 < S4 < S1< S2

E) S4 < S3 < S1 < S2

C) S3 < S1 = S4 < S2

You count from 1 to 100 and you clap when you say the multiples of number 3 and the
numbers that are not multiples of 3 but have 3 as the last digit. How many times will you
clap your hands?
A) 30

B) 33

C) 36

D) 39

E) 43

The cyclist went up the hill with the speed of 12 km/h and went down the hill with the
speed of 20 km/h. The ride up the hill took him 16 minutes longer than the ride down the
hill. How many minutes did the cyclist take to go down the hill?
A) 24

B) 40

C) 32

D) 16

E) 28

Symbols P, Q, R, S indicate the total weight of the figures drawn above them. It is known
that any two figures of the same shape have the same weight. If P<Q<R, then:
A) P<S<Q

B) Q<S<R

C) S<P

D) R<S

E) R = S

Ada has 14 gray balls, 8 white balls and 6 black balls in a bag. What is the least number
of the balls she has to take out of her bag having her eyes closed to make sure that she
took at least one ball of each color?
A) 23

B) 22

C) 21

D) 15

E) 9

A computer virus destroys computer memory. On the first day it destroyed

memory. On the second day it destroyed
on the third day it destroyed

of this

of the memory remaining after the first day;

of the memory remaining after two days and on the fourth

day it destroyed of the memory remaining after three days. What part of all the
computer memory was left after those four days?

A)

B)

C)

D)

E)

A net with 32 hexagonal spaces in three rows was made out of matches (see the picture.)
How many matches were used to make this net?
A) 123

B) 124

C) 125

D) 120

E) 121

On the AMC-10 math test you earn 6 points for a correct answer, 0 points for a wrong
answer, and 2.5 points for leaving the answer space blank. There are 25 questions. How
many must you get correct with no wrong answers to score 100 or more?
Jim, Tom, Sam, and Joe divided up $720 made by mowing lawns according to how many
lawns each mowed. Jim mowed 2/3 as many as Tom, and Tom mowed twice as many as
Sam. Joe mowed three times as many as Sam. How much money did Jim make?
A pattern is made by drawing a straight line up 1, then right 2, then down 3, then left 4,
then up 5, right 6, etc. This makes many vertical and horizontal line segments. How far
apart are the vertical segments?
At Cheryl's party, everyone shook hands with everyone else just once. There were 15
shakes. How many people were at the party?
Two cubic dice are rolled. Each die has the number 1, 2, 3, 4, 5, or 6 on a side. Your score
is the sum of the two numbers that are face up after the dice stop rolling. How many
different scores are possible?
Pete is given three positive integers and is told to add the first two, and then multiply the
result by the third. Instead, he multiplies the first two and adds the third to that result.
Surprisingly, he still gets the correct answer of 14. How many different values could the
first number have been?
(A) 5 (B) 4 (C) 6 (D) 3 (E) 7

A purse contains a collection of quarters, dimes, nickels, and pennies. The average value
of the coins in the purse is 17 cents. If a penny is removed from the purse, the average
value of the coins becomes 18 cents. How many nickels are in the purse?
(A) 2 (B) 5 (C) 0 (D) 1 (E) 8
The numbers on opposite sides of a die total 7. What is the sum of the numbers on the
unseen faces of the two dice shown?
(A) 14 (B) 20 (C) 21 (D) 24 (E) 30

A game is said to be fair if your chance of winning is equal to your chance of losing.
How many of the following games, involving tossing a regular six-sided die, are fair?
You win if you roll a 2
You win if you roll an even number
You win if you roll a number less than 4
You win if you roll a number divisible by 3
(A) 0 (B) 1 (C) 2 (D) 3 (E) 4
Chris and Pat are playing catch. Standing 1 m apart, Pat first throws the ball to Chris and
then Chris throws the ball back to Pat. Next, standing 2 m apart, Pat throws to Chris and
Chris throws back to Pat. After each pair of throws, Chris moves 1 m farther away from
Pat. They stop playing when one of them misses the ball. If the game ends when the 29th
throw is missed, how far apart are they standing and who misses catching the ball?
(A) 15 m, Chris (B) 15 m, Pat (C) 14m, Chris (D) 14 m, Pat (E) 16 m, Pat
Emily was at a garage sale where the price of every item was reduced by 10% of its
current price every 15 minutes. At 9:00 a.m., the price of a carpet was $10.00. At 9:15
a.m., the price was reduced to $9.00. As soon as the price of the carpet fell below $8.00,
Emily bought it. At what time did Emily buy the carpet?
(A) 9:45 a.m. (B) 9:15 a.m. (C) 9:30 a.m. (D) 10:15 a.m. (E) 10:00 a.m.
Using an equal-armed balance, if balances OO and OOO balances , which
of the following would not balance O?
(A) O (B) (C) OO (D) (E) O
On a circular track, Alphonse is at point A and Beryl is diametrically opposite at point B.
Alphonse runs counterclockwise and Beryl runs clockwise. They run at constant, but
different, speeds. After running for a while they notice that when they pass each other it is
always at the same three places on the track. What is the ratio of their speeds?
(A) 3 : 2 (B) 3 : 1 (C) 4 : 1 (D) 2 : 1 (E) 5 : 2

How many different combinations of pennies, nickels, dimes and quarters use 48 coins to
total $1.00?
(A) 3 (B) 4 (C) 5 (D) 6 (E) 8
In the diagram, all rows, columns and diagonals have the same sum. What is the value of
x?
(A) 12 (B) 13 (C) 16 (D) 17 (E) 18

Two positive integers have a sum of 11. The greatest possible product of these two
positive integers is
(A) 11 (B) 18 (C) 28 (D) 35 (E) 30
If P=300 and Q=0.3, which of the following calculations gives the largest result?
P

(A) P + Q (B) PQ (C) Q (D)

(E) P Q
P
What is the maximum numbers of rectangular wooden blocks with dimensions 20cm30
cm40 cm that could fit into a rectangular box with inner dimensions 40 cm60 cm80
cm?
(A) 2 (B) 4 (C) 10 (D) 8 (E) 6
A rectangular wooden prism is made up of three pieces, each consisting of four cubes of
wood glued together. Which of the pieces below has the same shape as the black piece?

The area of square ABCD is 64 and AX = BW = CZ = DY = 2. What is the area of square

WXYZ?
(A) 58 (B) 52 (C) 48 (D) 40 (E) 36

In the diagram, the rectangular floor plan of the first floor of a house is shown. The living
room and the laundry room are both square. The areas of three of the rooms are shown on
the diagram. The area of the kitchen, in m2, is
(A) 12 (B) 16 (C) 18 (D) 24 (E) 36

The entire contents of a jug can exactly fill 9 small glasses and 4 large glasses of juice.
The entire contents of the jug could instead fill 6 small glasses and 6 large glasses. If the
entire contents of the jug is used to fill only large glasses, the maximum number of large
glasses that can be filled is
(A) 8 (B) 9 (C) 10 (D) 11 (E) 12
In the Gauss 2004 Olympics, there are six competitors and eight events. The top three
competitors in each event receive gold, silver and bronze medals respectively. (There are
no ties at the Gauss Olympics, and no competitor can win more than one medal on the
same event.) Each competitor scores 5 points for each gold medal, 3 points for each silver
medal, and 1 point for each bronze medal. If one of the competitors had a total of 27
points, what is the maximum number of silver medals she could have won?
(A) 6 (B) 2 (C) 3 (D) 4 (E) 5
In a survey, 90 people were asked What is your favourite pet? Their responses were
recorded and then graphed. In the graph, the bar representing favourite pet is dog has
been omitted. How many people selected a dog as their favourite pet?
(A) 20 (B) 55 (C) 40 (D) 45 (E) 35

Spencer was walking home from school when he realized he had forgotten his
homework. He walked back to the school, picked up his homework and then walked
home. The graph shows his distance from home at different times. In total, how far did he
walk?
(A) 2800 m (B) 1000 m (C) 800 m (D) 1200 m (E) 1400 m

A box contains 14 disks, each colored red, blue or green. There are twice as many red
disks as green disks, and half as many blue as green. How many disks are green?
(A) 2 (B) 4 (C) 6 (D) 8 (E) 10
A palindrome is a positive integer whose digits are the same when read forwards or
backwards. For example, 2002 is a palindrome. What is the smallest number which can
be added to 2002 to produce a larger palindrome?
(A) 11 (B) 110 (C) 108 (D) 18 (E) 1001
The first six letters of the alphabet are assigned values A = 1, B = 2, C = 3, D = 4, E = 5,
and F = 6. The value of a word equals the sum of the values of its letters. For example,
the value of BEEF is 2 + 5 + 5 + 6 = 18. Which of the following words has the greatest
value?
(A) BEEF (B) FADE (C) FEED (D) FACE (E) DEAF
In the diagram, AC = 4, BC = 3, and BD = 10. The area of the shaded triangle is
(A) 14 (B) 20 (C) 28 (D) 25 (E) 12

In the following equations, the letters a, b and c represent different numbers.

13 = 1
a3 = 1 + 7
33 = 1 + 7 + b
43 = 1 + 7 + c
The numerical value of a, b, c is
(A) 58 (B) 110 (C) 75 (D) 77 (E) 79
A perfect number is an integer that is equal to the sum of all of its positive divisors,
except itself. For example, 28 is a perfect number because 28 = 1 + 2 + 4 + 7 + 14. Which
of the following is a perfect number?
(A) 10 (B) 13 (C) 6 (D) 8 (E) 9
The word stop starts in the position shown in the diagram to the
right. It is then rotated 180clockwise about the origin, O, and this
result is then reflected in the x-axis. Which of the following represents
the final image?

Mark has a bag that contains 3 black marbles, 6 gold marbles, 2 purple
marbles, and 6 red marbles. Mark adds a number of white marbles to
the bag and tells Susan if she now draws a marble at random from the

bag, the probability of it being black or gold is

3
. The number of white
7

marbles that Mark adds to the bag is

(A) 5 (B) 2 (C) 6 (D) 4 (E) 3
Each of the integers 226 and 318 have digits whose product is 24. How
many three-digit positive integers have digits whose product is 24?
(A) 4 (B) 18 (C) 24 (D) 12 (E) 21
In the square shown, the numbers in each row, column, and diagonal multiply to give the
same result. The sum of the two missing numbers is
(A) 28 (B) 15 (C) 30 (D) 38 (E) 72

A prime number is called a Superprime if doubling it, and then subtracting 1, results in
another prime number. The number of Superprimes less than 15 is
(A) 2 (B) 3 (C) 4 (D) 5 (E) 6
Rectangular tiles, which measure 6 by 4, are arranged without overlapping, to create a square.
The minimum number of these tiles needed to make a square is
(A) 8 (B) 24 (C) 4 (D) 12 (E) 6 (E)

The point of a dart lands at some point inside the rectangle shown. What is the probability
that the y-coordinate of that point is greater than its x-coordinate?
(The coordinates do not have to be integers.)
(A) 1/3 (B) 3/8 (C) 3/7 (D) 3/5 (E) 3/4

A class picnic has a boy - girl ratio of 5:3 until 3 more girls come, changing the ratio to
10:7. The total number of students at the picnic now is
(A) 17 (B) 34 (C) 51 (D) 68 (E) 85
Anne, Beth and Chris have 10 candies to divide amongst themselves. Anne gets at least 3
candies, while Beth and Chris each get at least 2. If Chris gets at most 3, the number of
candies that Beth could get is

(A) 2 (B) 2 or 3 (C) 3 or 4 (D) 2, 3 or 5 (E) 2, 3, 4, or 5

Enzo has fish in two aquariums. In one aquarium, the ratio of the number of guppies to the
number of goldfish is 2:3. In the other, this ratio is 3:5. If Enzo has 20 guppies in total, the
least number of goldfish that he could have is
(A) 29 (B) 30 (C) 31 (D) 32 (E) 33

In the addition shown, a digit, either the same or different, can be

placed in each of the two boxes. What is the sum of the two missing
digits?
(A) 9 (B) 11 (C) 13 (D) 3 (E) 7

The area of a rectangle is 12 square metres. The lengths of the sides, in metres, are whole
numbers. The greatest possible perimeter (in metres) is
(A) 14 (B) 16 (C) 12 (D) 24 (E) 26
In the diagram, all rows, columns and diagonals have the sum 12. What is the sum of the
four corner numbers?
(A) 14 (B) 15 (C) 16 (D) 17 (E) 12

Paul, Quincy, Rochelle, Surinder, and Tony are sitting around a table. Quincy sits in the
chair between Paul and Surinder. Tony is not beside Surinder. Who is sitting on either
side of Tony?
(A) Paul and Rochelle (B) Quincy and Rochelle (C) Paul and Quincy (D) Surinder and
Quincy (E) Not possible to tell
The month of April, 2000, had five Sundays. Three of them fall on even numbered days.
The eighth day of this month is a
(A) Saturday (B) Sunday (C) Monday (D) Tuesday (E) Friday
A dishonest butcher priced his meat so that meat advertised at $3.79 per kg was actually
sold for $4.00 per kg. He sold 1800 kg of meat before being caught and fined $500. By
how much was he ahead or behind where he would have been had he not cheated?
(A) $478 loss (B) $122 loss (C) Breaks even (D) $122 gain (E) $478 gain
In a basketball shooting competition, each competitor shoots ten balls which are
numbered from 1 to 10. The number of points earned for each successful shot is equal to

the number on the ball. If a competitor misses exactly two shots, which one of the
following scores is not possible?
(A) 52 (B) 44 (C) 41 (D) 38 (E) 35
Sam is walking in a straight line towards a lamp post which is 8 m high. When he is 12 m
away from the lamp post, his shadow is 4 m in length. When he is 8 m from the lamp
post, what is the length of his shadow?
(A) 1

1
1
2
m (B) 2m (C) 2 m (D) 2 m (E) 3m
2
2
3

The total area of a set of different squares, arranged from smallest to largest, is 35 km2 .
The smallest square has a side length of 500 m. The next larger square has a side length
of 1000 m. In the same way, each successive square has its side length increased by 500
m. What is the total number of squares?
(A) 5 (B) 6 (C) 7 (D) 8 (E) 9
Twelve points are marked on a rectangular grid, as shown. How many squares can be
formed by joining four of these points?
(A) 6 (B) 7 (C) 9 (D) 11 (E) 13

Five students named Fred, Gail, Henry, Iggy, and Joan are seated
around a circular table in that order. To decide who goes first in a
game, they play countdown. Henry starts by saying 34, with Iggy
saying 33. If they continue to count down in their circular order, who
will eventually say 1?
(A) Fred (B) Gail (C) Henry (D) Iggy (E) Joan
Which of the following numbers is an odd integer, contains the digit 5, is divisible by 3,
and lies between 122 and 172?
(A) 105 (B) 147 (C) 156 (D) 165 (E) 175
A game is played on the board shown. In this game, a player can move three places in any
direction (up, down, right or left) and then can move two places in a direction

perpendicular to the first move. If a player starts at S, which position on the board (P, Q,
R, T, or W) cannot be reached through any sequence of moves?
(A) P (B) Q (C) R (D) T (E) W

In a softball league, after each team has played every other team 4 times, the total
accumulated points are: Lions 22, Tigers 19, Mounties 14, and Royals 12. If each team
received 3 points for a win, 1 point for a tie and no points for a loss, how many games
ended in a tie?
(A) 3 (B) 4 (C) 5 (D) 7 (E) 10
In the multiplication question, the sum of the digits in the four boxes is
(A) 13 (B) 12 (C) 27 (D) 9 (E) 22

A rectangular field is 80m long and 60m wide. If fence posts are placed at the corners and
are 10m apart along the four sides of the field, how many posts are needed to completely
fence the field?
(A) 24 (B) 26 (C) 28 (D) 30 (E) 32
Kalyn cut rectangle R from a sheet of paper and then cut figure S from R. All the cuts
were made parallel to the sides of the original rectangle. In comparing R to S,

(A) the area and perimeter both decrease

(B) the area decreases and the perimeter increases
(C) the area and perimeter both increase
(D) the area increases and the perimeter decreases
(E) the area decreases and the perimeter stays the same
The diagram shows a magic square in which the sums of the numbers in any row, column
or diagonal are equal. What is the value of n?
(A) 3 (B) 6 (C) 7 (D) 10 (E) 11

Each of the digits 3, 5, 6, 7, and 8 is placed one to a box in the diagram. If the two digit
number is subtracted from the three digit number, what is the smallest difference?
(A) 269 (B) 278 (C) 484 (D) 271 (E) 261

Claire takes a square piece of paper and folds it in half four times without unfolding,
making an isosceles right triangle each time. After unfolding the paper to form a square
again, the creases on the paper would look like

The letters of the word GAUSS and the digits in the number 1998 are each cycled
separately and then numbered as shown.
1. AUSSG 9981
2. USSGA 9819
3. SSGAU 8199
etc.
If the pattern continues in this way, what number will appear in front of GAUSS 1998?
(A) 4 (B) 5 (C) 9 (D) 16 (E) 20
Juan and Mary play a two-person game in which the winner gains 2 points and the loser
loses 1 point. If Juan won exactly 3 games and Mary had a final score of 5 points, how
many games did they play?
(A) 7 (B) 8 (C) 4 (D) 5 (E) 11

Each of the 12 edges of a cube is colored either red or green. Every face of the cube has
at least one red edge. What is the smallest number of red edges?
(A) 2 (B) 3 (C) 4 (D) 5 (E) 6
Ten points are spaced equally around a circle. How many different chords can be formed
by joining any 2 of these points? (A chord is a straight line joining two points on the
circumference of a circle.)
(A) 9 (B) 45 (C) 17 (D) 66 (E) 55

Each time a bar of soap is used, its volume decreases by 10%. What is the minimum
number of times a new bar would have to be used so that less than one-half its volume
remains?
(A) 5 (B) 6 (C) 7 (D) 8 (E) 9
Two natural numbers, p and q, do not end in zero. The product of any pair, p and q, is a
power of 10 (that is, 10, 100, 1000, 10 000 , ...). If p > q, the last digit of pq cannot be
(A) 1 (B) 3 (C) 5 (D) 7 (E) 9
These five counters are overlapped and arranged into a row. The area covered by the five
counters is 31 units.

Two counters are taken away. The area covered by the remaining three counters
is 19 units.

What is the area of one counter

(A) 4 units
(B) 5 units
(C) 6 units
(D) 7 units

When the figure to the right is folded to make a cube, what is the largest sum of the
numbers on a pair of parallel faces?

Zan has created this rule for generating sequences of whole numbers.
If a number is 25 or less, double the number.
If a number is more than 25, subtract 12 from it.
For example, if Zan starts with 10, she gets the sequence 10, 20,
40, 28, 16, . If the third number in Zans sequence is 36, what
is the sum of the four distinct numbers that could have been the
first number in her sequence?
John, Mike and Chantel will divide a pile of pennies amongst
themselves using the following process: The number of pennies in the
pile is counted.
If the number of pennies in the pile is even, Mike will get half of
the pile.
If the number of pennies in the pile is odd, one penny will be
given to Chantel, and John will get half the pennies remaining in
the pile.
This process is then repeated until the pile is empty. How many
pennies will Mike have at the end if the original pile contains 2005
pennies?