2018 Junior Indivi Relay Team Problems
2018 Junior Indivi Relay Team Problems
2018 Junior Indivi Relay Team Problems
1. Find 1 + 2 – 3 + 4 + 5 – 6 + 7 +8 – 9 + … + 97 + 98 – 99.
1 p
2. If 1 where p and q are relatively prime find p + q.
1 q
1
1
1
1
1
2
A) 7 B) 6 C) 5 D) 4 E) 3
3. Compute
(11 + 10 + ... + 1) + (10 + 9 + ... + 2) + (9 + 8 + ... + 3) + (8 + ... + 4) + (7 + 6 + 5) + 6.
28 14 11 6 27
A) B) C) D) E)
25 12 10 5 25
x y
5. If A( x, y ) , find A(1,2) + A(1,3) + A(1,4) + A(2,3) + A(2,4) + A(3,4).
y x
1 5 5
A) 12 B) 13 C) 13 D) 16 E) 15
6 6 6
A) 24 33 52 7 B) 22 33 52 7 C) 22 32 52 7 2
D) 23 32 53 7 E) 23 32 52 7
1 1 1 1
7. Find 1 1 1 ...1 .
2 3 4 2018
1 1 2 2018 2017
A) B) C) D) E)
2017 2018 2017 2017 2018
13 13 3 5 9
A) B) C) D) E)
21 18 4 8 14
2 33 404 5005
9. Find the value of .
6 77 808 9009
10 5 5 15 3
A) B) C) D) E)
63 126 84 196 216
A) 2 B) 4 C) 6 D) 8 E) 9
A) 66 B) 242 C) 36 D) 6 E) 44
123
12. Find 2018-th digit after decimal point of the fraction .
7
A) 5 B) 7 C) 1 D) 4 E) 2
a c
15. If a, b, c and d are 3, 4, 5, 6 in some order find the least value of .
b d
1 4 19 6 9
A) B) C) D) E)
2 3 15 5 5
1. Car with 4 wheels and 1 spare tire travelled for 1000 kilometers. Each of two of
the tires has been used for 500 kilometers. The other three tires have been used
equal number of kilometers. Find this number.
3. Six integers x, x + 1, x + 2, x + 3, x + 4 and x + 5 are such that the sum of any two
of them equals to one of the remaining numbers or can be expressed as a sum of
two or three from the remaining numbers. How many such x exist?
4. Find the number of three digit numbers such that the difference between any
two neighboring digits is not 1.
a c e
7. Find the number of solutions of where a, b, c, d, e, and f are 1, 2,
b d f
3, 4, 5 and 6 in some order.
8. A person is born in year abcd . A year mnpq is good for that person if after his
birthday his age in this year equals a+b+c+d. For example, 2017 is good for a
person born in 2012. How many years are there between year 2000 and year 2018
inclusive that are not good for anyone?
3 6 3
1. Find the number of 3 digit numbers without 0 in their decimal representation with
sum of all digits equals to 23.
2. In a pile of 9 coins one is a fake. Suppose we can find out whether the fake coin
is among them in any move where we pick four coins. Find the minimum number
of moves that guarantee finding the fake coin.
3. The average weight of a group of people is 35.2 kilograms. Albert, who weighs
45.6 kilograms, then joins the group. This raises the average weight of the group
to 36 kilograms. How many children were in the original group?
6. How many numbers in the interval [1, 2018] can be expressed as a sum of k
consecutive positive integers for each k = 2, k = 3 and k = 5?
8. Each cell of a 4 4 table is colored either white or black. Any white cell has
exactly 3 black neighboring cells and any black cell has exactly one white
neighboring cell. Find the number of white cells. (Two cells are neighbors if they
share a common side.)
9. If the number a 2018b is divisible by 12 but not divisible by 9 find the largest
value of a + b.