SMO Junior 2017
SMO Junior 2017
SMO Junior 2017
Instructions to contestants
3. For the multble chairc questxons, enter yoLr ansuer an the ansuer sheet bU shading
bLbbLe containins the letter (A, B, C, D or E) cal-respan(tiw to the caryect ansuer.
.4. Far the other shad questions, ur-ite your aneuer un the i".r,"" ,t"rt and shad.e the
propl-iate bLbble behw your ansuer.
6. ThmushoLt this paper, Iet lrl d.,nate the grEatest 'irLteser l$s than or equal ta x. For
exampLe, l2.Il: 2, 3.9 = 3.
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<!*7tct.or,
Multiple Choice Questions
23 24 25 26
1. -{mong ihe flve umbe$
tl' E' 46' +7
ulla
fi, *ncr, o,,e t * the smallesi vatue?
.2324
(A) (Il ) (c)
25
@)
2; (o)
,14 45 46 asl
3- How many ways can th letters of the word "IGt OO" be arradged?
(A) 4 (B) 5 (c) 30 (D) 60 (E) 120
4. Jenny and Mary received identica.l fruit baskets, ea-ch containing 3 apphs, 4 oranges and
2 bananas. Assuming that both Jenny and Mary randomly picked a liuit fiom their own
basket, wha.t is the probability that they both picked a,r apple?
(A)
; (B)
; (c) ; 1l) ; (E) None or the above
5. A cylinder has base radius r and height ?r. If a sphere has the same surface arca as the
cylinder, find the ratio of the volume of the tylinder to ihe votume of the sphere.
1dt
fA, "4J2r32
rB, ' rc, a ,o, "t
Let ABC D be a rcctangular sheet of paper with ,48 = 6 and BC : 8. We can fold the
paper aloag the crease line t-P so that point C coincides with point ,4. Find the lengih
of the resulting line segment ,4I -
AED
(A)
25 ,*, ]! l(-27
)t (D)'7 (E) None of the abowe
42
7l Given tlree consecuti\ positi\ iateg;rs, whlch of the follorring is a pbssible ralue for the
ditrrence of iLe squares oI Lihe larycsl :r,nd the smallesi of ihese three iriegers?
9. Let a arld 6 be positive integels. If the highest co]I1mon facror of a and 6is 6 and the
lowst common multiple of a and b is 233455, how many possible values a.re there for a?
Short Questions
1l_ An n sided polygon has two interior a.ngles of sizs 94" and b1". The remainins interio,
angles are all cqudl ixtu". ll 4. a _20 daFrminF r.F lallF o. n.
13. A list of six positi.!.e intege$ has a unique mode of 4, median of 6 and mea.n of 8. Find
the lalgest possible inteser in the list.
14. In the diagram, ,4F is a dianeter of the ctucle aJld ,4BCD is a square with points B and
C on -4F and poinis A and D on the circle. If AB = 17.y/5 find the lensth of rF.
18. Let ABC be a t a.ngle, D be a point on Ad such that .4D = DC and E be a point on BC
such that B-U : 2rd. Let I. be the intersection of BD and AE. If the area of tdangle
,4BC is 100, find the area of triafigle ADi'.
19. Find the laxgest integer from 1 to 100 which has exactly 3 positive integer divisols. For
example, the only positive divisom of 4 arc 1, 2 and 4.
20. Lt d, b and c be positiw integers such that a2 + bc:257 arr,d ab +ln:101. Detemioe
27. In a trapezium - BCD, AD is paralel to BC and poinis -A and -F arc the midpoints of
48 and DC respeo ir ely. Tl
ArFaot AErD rttt
Ar.a ot fB1-F 3 \ 3'
and the a.rca. of tdargle is v/5, frnd the :rea of the irapezirm ,4BCt.
"48,
23. Lt d,b and c be the three solutions ofthe equation x:3 4x2 +5x 6=0. Deiermine th
,,?"I\Le ol d2 + b2 + c2 + 3abc.
24. I-et a be ar intes$ such that both a + 79 and o + 2 are pefect squa-res. Flnd the largest
possible va.lue of a.
25 DFtFrqri'.F rhF nu- bFr o. in,pgFrj r which .a'i.fy thp tolloni' I :nF.llra :L).
26. If every root of the polynomial 12 +4r - 5 is also a root of the pollnomiat2rs +9f +tu+c,
-o ,. . vnl -" ot b2 "2
I
27. Let m be the mirimom value of the quadratic curve g : 72 4an + 5o2 3(1, where the
\.alue m depends on !r. If 0 S a _< 6, find the maximrm possible .""!lue of m-
28. Let a,b,c,d, anC, ebe fve consecutive positiF integE q here e is the largesh. Ifb+c+dis
a pedect square and a + r + c + d + is d perfect cubc, fi d tLc least possiblc \alue of e.
30. Let a and b be positive real numbN satisfying a + 6 = 10. Find the largest possible .value
of
'/rr,a.+ts+'/tort+n.
31- Find ihe vAlrre of
32. Find the la.rgest possible value of ra such that the polynomial 12 1 (2n 1)r + (n 6)
ha.s two rcal rcots ,1 and 12 satisfuing 11 ! -1 and rr ) 1:
If one of the integers is rcmorred from the first N consecutir inteeers 1,2,3, .. .. N, the
rFsu riDg d\eragF ot thp rpmanins ir rpse-s is .'^O n.
?.
34. Amongst the fractions
723 174
175' 1,75' !75" 175',
there a.re some which can be rcduced to a fraction \vith a smaller denominator such as
tfu : *1, and there are some that cannot be rcduced further like r75!. Find the sum of alt
the ftactions vhich cannot be reduced further.
35- The number of seashells collected by 13 boys and n girls is n2 + 10n 18. If each child
collects eiactly the same number of seashells, determine the !?lue of n.