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Singapore Math Olympiad Questions
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Algebraic Expressions and Equations
< 1 >
1. S96 Q6
Suppose 41
23222
22
ba
ba. What is the ratio a : b?
2. S96 Q7
In the figure below, ABCD is a rectangle and ABGE, HFGC, DEFH are squares. What is
the ratio of AB to BC?
D H C
EF
G
A B
3. S97 Q2
Which of the following is the largest?
(A) 317.9 682.1 (B) 317.3 682.7 (C) 271.5 728.5 (D) 610.8 389.2
(E) 493.7 506.3
4. S98 Q1
Ann sells an item at $10 less than the list price and receives 10% of her selling price as
her commission. Bill sells the same item at $20 less than the list price and receives 20%
of his selling price as his commission. If they both get the same commission, what is the
list price in dollars?
5. S98 Q23
Consider the system of equations in x, y, z, w:
8x + 4y + 6z + 2w = –32
x + 7y + 3z + 5w = 32
5x + 3y + 7z + w = –32
2x + 6y + 4z + 8w = 32
Find the value of 100 + x.
6. S98 Q24
Suppose a, b, c, d are four numbers such that a + b + c + d = 0 and
abc + bcd + cda + dab = 3. Find the value of 3333 dcba .
7. S00 Q8
If x and y are two real number such that xy = 24 and x + y = 11, find the value of 22 yx .
8. S00 Q13
Let x, y, z be three positive real numbers such that x + y + z = 6. Find the largest possible
value of xy + yz + zx.
Circles
< 2 >
1. S96 Q17
In the diagram, AB and CB are diameters of semicircles ADB and CDB respectively. The
line AD is a tangent of the semicircle CPB at P. Suppose PBC = 27. Find PAC.
C BA
P
D
2. S97 Q10
In the diagram, the two circles meet each other at C. The diameter AB of the bigger circle
is tangent to the smaller circle at D. If DE bisects ADC and BAC = 27, find BCD.
BA
E
D
C
3. S97 Q14
In the diagram, TPA and TQB are tangents to the circle at P and Q respectively. If
PQ = PR, which of the following must be true?
I. APR and QRP II. QTP and QPR III. QPR and APR
T AP
R
Q
B
Circles
< 3 >
4. S98 Q21
In the figure below, ABCD is a convex quadrilateral, BAC = 20, BCA = 35,
BDC = 40, BDA = 70 and AC intersects BD at M. Find the size of BMC in
degrees.
BA
DC
M
5. S99 Q16
In a square ABCD of area 666 cm2, O is its centre, P is a point inside ABCD such that
OPB = 45. Suppose that PA : PB = 5 : 7. Find the length of PB in cm.
6. S00 Q7
Let ABCD be a convex quadrilateral such that AB = BC = AC = AD. Suppose
BDC = x, find the value of x.
More Challenging Problems
7. S98 Q29
In the figure below, there are 1999 circles ,1C ,2C , 1999C between two lines 1l and 2l
such that for i = 1, 2, , 1998,
(i) iC touches 1iC ,
(ii) iC is tangent to both 1l and 2l and
(iii) the radius ir of iC is less than the radius 1ir of 1iC .
Suppose ir = 1 cm and 1000r = 13 cm, find the value of 1999r in cm.
Circles
< 4 >
8. S99 Q23
Let A, B, C, D be four points on a circle in that order and E the intersection of AC and
BD. Suppose that BC = CD = 4, AE = 6 and the lengths of BE and DE are integers. Find
the length of BD.
Divisibility and Prime Factorisation
< 5 >
1. S95 Q5
How many pairs of positive integers a, b are there such that 28922 ba ?
2. S95 Q18
In how many ways can the fraction 35
1 be written as the sum of the reciprocals of two
natural numbers?
3. S96 Q15
What of the following number is prime?
(A) 21147 53 (B) 1242 (C) 24 121147 (D) 11134
(E) 1262
4. S96 Q19
What is the number of divisors of 6710 532 ? Here the divisors of 6710 532 include
1 and itself.
5. S97 Q20
Let x and y be positive integers. Find y if x is a perfect square and the difference of x y
and x + y is 1000.
6. S97 Q21
The expenses of a party numbering 43 were $229. If each man $10, each woman paid $5
and each child paid $2, what was the largest possible number of men in the party?
7. S97 Q26
Lim’s stamp collection consists of three albums. Two tenths of his stamps are in the first
album, several sevenths in the second album, and there are 303 stamps in the third album.
How many stamps does Lim have?
8. S98 Q19 (repeated in a different form in J03 Q28)
What is the largest integer n such that 2n is divisible by every even number 2 and 20
inclusively?
Divisibility and Prime Factorisation
< 6 >
9. S98 Q28
A point (x, y) in the xy-plane is called a lattice point if both x and y are integers. For any
integer n, let f(n) be the number of lattice points on the line segment joining (0, 0) and
(n, n + 5). For instance, we have f(0) = 6 and f(1) = 2. Find the value of
)1998()3()2()1( ffff .
10. S00 Q1
Find the number of positive prime numbers P less than 100 and such that one of the digits
of P is 3 or 7.
11. S00 Q3
Find the number of positive 3-digit integers which are divisible by both 7 and 13.
12. S00 Q15
Find the smallest positive integer N such that 2
N is the square of an integer and
5
N is the
cube of an integer.
13. S00 Q20
Let n be a positive 2-digit integer. Suppose n2 is a four-digit number whose last two digits
are the same as those of n. What is n?
14. S00 Q21
Find the positive integer x such that x
x
740
2
is a prime number.
15. S00 Q27
Let 321 ,, ppp be three positive prime numbers such that
320147141129321 ppp .
Find the value of 321 ppp .
Indices and Logarithm
< 7 >
1. S95 Q1
Which of the following numbers is the largest?
(A) 3 65 (B) 3 56 (C) 3 65 (D) 3 65 (E) 3 56
2. S95 Q3
If n222
666666
333
444455
555555
555
5555
, what is the value of n?
3. S95 Q15
If 0)](log[loglog)](log[loglog)](log[loglog 5533225
1
3
1
2
1 zyx , then
(A) yxz (B) zyx (C) xzy (D) xyz
(E) yzx
4. S96 Q5
What does 81
32log
9
5log2
16
25log equal to?
5. S96 Q12
Let 6a , 32 b and 3)2(c . Which of the following is true?
(A) cba (B) abc (C) bac (D) cab
(E) bca
6. S97 Q8
Suppose a2log10 and b3log10 , what is the value of 12log5 ?
7. S98 Q2
Find the value of
2
6
4
2727100
.
Indices and Logarithm
< 8 >
8. S98 Q10
Let x, y, z be three number all bigger than 1 and let w be a positive number such that
24log wx , 40log wy , 12log wxyz .
Find the value of wzlog .
9. S99 Q17
If 360 a and 560 b , find the value of
)1(2
1
12b
ba
.
10. S00 Q2
Simplify 5725
5
352
11
.
11. S00 Q4
Find the real number x satisfying 17
1
8loglog 77
xx .
12. S00 Q6
Let x and y be two real number such that 330 x and 530 y . Find the value of 1230 yx in simplest form.
More Challenging Problems
13. S95 Q12
Suppose xy = 144 and 3
10loglog yx xy with x, y > 0. Find the value of
2
yx .
14. S96 Q21
Solve the equation xxx
111
255151699 .
15. S99 Q14
Let 2000
1
2000
1
)1999()1999(2
1 x . Find the value of 2000
21
xx .
Modular Arithmetic
< 9 >
1. S95 Q14
Find the number of pairs (m, n) of integers which satisfy the equation
1992756 2323 nnnmmm .
2. S95 Q22
Find the last digit of the number )12)(12)(12)(12)(12)(12)(12( 643216842 .
3. S96 Q2
Find the smallest positive integer x such that when x is divided by 4, the remainder is 1;
when 4
1x is divided by 3, the remainder is 1; and when
3
14
1 x
is divided by 2, the
remainder is 1.
4. S97 Q3
Which of the following is divisible by 8?
(A) 69678478 (B) 765434 (C) 483210 (D) 7184632
(E) none of the above
5. S97 Q9
What is the unit digit of 1997)1997( ?
6. S97 Q12
Which of the following is not a perfect square?
(A) 3196944 (B) 6431296 (C) 14326225 (D) 28313041
(E) 431490910
7. S99 Q10
What is the remainder of 4123456789 when it is divided by 8?
8. S99 Q11
What is the smallest positive integer such that it has remainders 1, 2, 3, 4, 5 when divided
by 3, 4, 5, 6, 7 respectively?
9. S99 Q15
Let n be a positive integer. Suppose that the tens digit of 2n is 7. Find the units digit of 2n .
Modular Arithmetic
< 10 >
10. S00 Q19
Suppose 1ababababab is an 11-digit integer which is divisible by 99. Find the value of
a + b.
11. S00 Q22
Find the largest possible remainder when the square of a prime number is divided by 24.
12. S00 Q26
For any positive integer n, let na be the remainder when n7 is divided by 100. Find the
value of 100321 aaaa .
13. S00 Q29
Find the largest 3-digit integer abc satisfying 14a + 49b + 2c = 263.
Patterns, Sequences and Sums
< 11 >
1. S95 Q2
Given that the sum of the first 100 odd positive integers is p. find the sum of the first 100
even positive integers.
2. S96 Q13
Evaluate the sum 115
1
13
1
12
1222
.
3. S97 Q5
Suppose three numbers 1, a and b are three consecutive terms of both an Arithmetic
Progression (AP) and a Geometric Progression (GP). How many possible pairs (a, b) are
there?
4. S98 Q4
The first four terms of an arithmetic progression are x, y, 1998, 2y. Find the value of x.
5. S98 Q13
The even numbers 2, 4, 6, 8, … are put into groups G1, G2, G3, … in the following way:
G1 = {2, 4}, G2 = {6, 8, 10, 12}, G3 = {14, 16, 18, 20, 22, 24}, …
so that the group G2 = contains 2n numbers. Find the value of x if 1998 appears in Gx.
6. S98 Q16
Find the positive square root of 444444888889.
7. S99 Q2
Suppose that the three dimensions of a certain rectangular solid are in geometric
progression, and total surface area is equal to the sum of all the edges. Find the volume of
this solid.
8. S99 Q5
If f(n) = 2 f(n – 1) + 1 for all positive integral values of n, and f(1) = 1, find a formula for
f(n) in terms of n when n is a positive integer.
9. S99 Q8
Suppose that ,pa qa and ra are the p-th, q-th and r-th terms of an arithmetic
progression. Find the sum rqp aqpaprarq )()()( .
Patterns, Sequences and Sums
< 12 >
10. S99 Q19
Find the number of possible integers n such that there are exactly two positive integers
between nn
1988 and
1988 .
More Challenging Problems
11. S99 Q24
Find the sum !99!33!22!11 .
12. S00 Q30
Let A be the set of all integers in the form )()1( knnn , where n, k are
positive integers. Suppose the elements of A are arranged in ascending order. Find the
2000th
number.
Properties of numbers and Inequality
< 13 >
1. S95 Q17
If the sum of positive numbers a and b is equal to 1, then the smallest possible value
ba
11 is
(A) 1 (B) 2 (C) 3 (D) 4 (E) None of the above
2. S96 Q1
Suppose x and y are positive real numbers. Which of the following expressions must be
larger than x and y?
(A) xy (B) 2)( yx (C) yx 2 (D) 3)( yx
(E) 2)1( yx
3. S96 Q10
Find the maximum value of a such that for any positive real x, y, if x y = 10, then
ax y .
(A) 0 (B) 1 (C) 10 (D) 410 (E) 810
4. S96 Q11
Which of the following is true?
(A) 12345 12347 > 2)12346( (B) 2)1001( + 2)1097( < (1001)(1097)(1.99)
(C) 2)1001( + 2)1097( < (1001)(1097)(1.9999) (D) 12340 12352 > 2)12346(
(E) None of the above
5. S96 Q23
Suppose a, b, c, d, e satisfy the following system of equations: a + b + c = 1,
b + c + d = 2, c + d + e = 3, d + e + a = 4 and e + a + b = 5. Write down the order of
a, b, c, d, e.
6. S97 Q1
Suppose a given circle and a given square have equal area. If the perimemter of the circle
and the square are 1P and 2P respectively, then
(A) 1P = 2P (B) 1P < 2P (C) 1P > 2P (D) 1P = 221 P
(E) None of the above.
Properties of numbers and Inequality
< 14 >
7. S97 Q15
Let M = 54
76
98
99999998 . Which of the following is true?
(A) 2M = 0.0004 (B) 2M < 0.0004 (C) 2M > 0.0004 (D) 2M = 0.04
(E) 2M > 0.04
Trigonometrical Identities
< 15 >
1. S95 Q24
Suppose 20 and
0coscoscos = 0sinsinsin = 0.
Find the value of .
2. S96 Q18
Find x2sin )1(sin 2 x )2(sin 2 x )179(sin 2 x .
3. S97 Q6
What is the value of
4
1sin 1
4
1cos 1 ?
4. S97 Q13
Suppose
cos1
cos1
cos1
cos1
=
sin
x,
where 1800 . Find the value of x.
5. S98 Q11
If tan x + tan y = 24 and cot x + cot y = 28, find the value of tan(x + y).
6. S98 Q27
Find the value of 10(cos16 2 50cos 2 )80sin40sin 2 .
7. S99 Q29
Let 0 . Find the minimum value of
)cos1)(2
(sin3
100
.
8. S00 Q11
If 5cos4sin3 , what is the value of 2
cos2
sin3
?
Trigonometry
< 16 >
1. S95 Q4
In triangle ABC (see diagram), if A : B : C = 1 : 2 : 3, then a : b: c is
B
A
C
c
a
b
(A) 1 : 2 : 3 (B) 1 : 3 : 2 (C) cos 1 : cos 2 : cos 3
(D) sin 1 : sin 2 : sin 3 (E) None of the above
2. S95 Q8
If and are between 0 and 2 , and cos > sin , then
(A) + < 2 (B) + =
2 (C) + >
2 (D) >
(E) >
3. S96 Q3
Let a = tan 224, b = sin 136, c = cos 310. Then
(A) a < b < c (B) b < c < a (C) c < a < b (D) c < b < a
(E) b < a < c
4. S96 Q4
In triangle ABC, (AC + AB) : (AB + BC) : (BC + AC) = 4 : 5 : 6. What is the ratio of
sin BAC : sin ABC : sin ACB?
(A) 6:5:4 (B) 7:5:3 (C) 11:10:9 (D) 12:11:8 (E) 7:6:4
5. S96 Q14
Let triangle ABC be a right angled triangle with ACB = 90. Suppose CD AB and
BD : DA = 1 : 3. Find CAB.
6. S97 Q4
Which of the following is the largest?
(A) tan 48 + cot 48 (B) tan 48 + cos 48 (c) cot 48 + sin 48
(D) sin 48 + cos 48 (E) (sin 48)2 + (cos 48)
2
Trigonometry
< 17 >
7. S97 Q7
In triangle ABC, A = 30, c = 6. Find all the values of a such that it is possible to draw
two distinct triangles as shown in the figure.
A C
B
CA
B
ca a
c
(A) 0 < a < 3 (B) 0 < a < 6 (C) 3 < a < 6 (D) a > 3
(E) a > 6
8. S97 Q17
In triangle ABC, suppose AC = 2(BC)sin B. Find A.
9. S98 Q3
If sin x = 3 cos x, find the value of 900(sin x)(cos x).
10. S00 Q10
A road is in the shape of a regular hexagon (six-sided figure) with each side of length
4 km. Suppose a car starts at a corner and moves along the road for a distance of 14 km.
Let the distance of the car from its starting point be x km. Find the value of x2.
Quadratic equations
< 18 >
1. S96 Q9
Obtain the conditions satisfied by k if the quadratic equation
013)1( 22 xxxxk
has real roots.
2. S97 Q19
Solve the equation 222 1996)1996)1997(( x .
3. S98 Q6
Find the sum of all the solutions of the equation
4
4
5
6
xx
.
4. S98 Q7
A stone is dropped into a dry well and the sound of the stone striking the bottom is heard
7.7 seconds after it is dropped. Assume that the stone falls 16t2 feet in t seconds and that
the velocity of sound is 1120 feet per second. Find the depth of the well in feet.
5. S98 Q9
If the perimeter of a rectangle is 216 cm, what is the smallest possible value of the
length of one of its diagonals in cm?
6. S99 Q4
Find the number of lines which are tangent to both the parabolas 2xy and
1682 xxy .
7. S99 Q7
Solve the following equation for x: 2
3
5
1
1
5
x
x
x
x
8. S99 Q12
The polynomial rqxpxx 23 has three distinct integral roots. If r is a prime number,
what is q?
Quadratic equations
< 19 >
9. S99 Q22
Let 321 ,, xxx and 4x be roots of the equation 014 xx . Find the value of
8
1x 8
2x 8
3x 8
4x .
10. S00 Q28
Let p(x) be a polynomial such that its leading coefficient (i.e. coefficient of the highest
power of x) is 1, and
27(x – 1)p(x) = (x – 27)p(3x)
for all real number x. Find the value of p(4).
More Challenging Problems
1. S95 Q11
How many distinct roots for x does the following equation have?
43232
xx
2. S95 Q21
For each n, let the roots of the quadratic equation 0)12( 22 nxnx be n and n .
Determine the value of
)1)(1(
1
33
)1)(1(
1
44
)1)(1(
1
2020 .
3. S96 Q22
Let f(x) = (5 – p)x2 – 6x + (p + 5). Find all real values p such that 0)( xf for all
positive values x.
Triangles
< 20 >
1. S95 Q5
In triangle ABC, AB = AC, BAC = 40 and D is a point in triangle ABC. If
BAC = BAC, calculate BDC.
2. S95 Q13
A given triangle has integral lengths and its perimeter is 8 cm. What is the area of this
triangle?
3. S95 Q19
Consider the quadrilateral ABCD whose diagonal intersect at O. If Area (ACB) = 5,
Area (BCD) = 9, Area (CDA) = 10 and Area(DAB) = 6, find the area of triangle AOB.
4. S95 Q20
In a right angled triangle ABC, B = 41. Square PQRS is inscribed as shown. Let AB = c
and the altitude from C to AB be h. If 3
211
ch, find the length of a side of the square.
A
S
P
R
Q
BC
5. S96 Q8
In the diagram, the sides of triangle ABC are produced as shown. If AB = AC, BG = BH
and AK = KG, find BAC.
B
G
C
A
K
H
Triangles
< 21 >
6. S97 Q23
In the diagram, AN = BM = AB, C = 35. Find BAC.
M
A
P
N
C
B
7. S97 Q24
In the diagram, the areas of BDO, ODC, OCE and OFA are 10 cm2, 15 cm
2, 20 cm
2, and
15 cm2 respectively. Find the area of OFB.
A
C
E
DB
F
O
8. S98 Q14
In ABC, AB = 16, BC = 17 and CA = 18. M is the midpoint of AB and H is the foot of the
altitude from A to BC. Find the length of MH.
9. S98 Q17
In ABC, AB = BC, ABC = 20, and M is a point on AB such that BM = AC. Find the size
of AMC in degrees.
10. S99 Q6
In a triangle ABC, the angle BAC is a right angle and AB = 2AC. AD is the perpendicular
from A to BC. Find the numerical value of DC
BD.
11. S99 Q9
In ABC, let A = 60 and B = 30. A line passes through C, divides ABC into two
pieces of equal area, and cuts AB at M. What is the value of CMB?
Triangles
< 22 >
12. S99 Q28
In the figure below, suppose that AB = AD and BD = AC. Find BCD.
13. S00 Q9
Let ABCD be a trapezium such that BC // AD, 2AB = CD, BC < AD and
ABC + ABC = 120. Suppose the smallest angle of the trapezium is x. Find the value
of x.
14. S00 Q18
ABCD is a square-shaped sheet of paper of area 81 cm2. A square of area 1 cm
2 with one
vertex at A and sides parallel to those of ABCD is removed from ABCD. Then the
remaining part is cut into k congruent triangles (without pasting). Find the smallest
possible value of k.
100
A
B
C
D
Pythagoras’ Theorem
< 23 >
1. S95 Q10
Pictured below are two semicircles. AB is tangent to the smaller semicircle and is parallel
to CD. Given AB = 24, find the area of the shaded region.
A
C
B
D
2. S96 Q16
In the diagram, ABCD is a square with each side of length 1. Suppose BPC is an
equilateral triangle. Find the area of the triangle BPD.
D
B
A
C
P
3. S98 Q12
The length of the perimeter of a right-angled triangle is 180 cm and the length of the
altitude perpendicular to the hypotenuse is 36 cm. Find the length of the hypotenuse in
cm.
4. S99 Q26
In the following figure, AB = AC, BAD is a right angle, BD = 36 cm and DC = 14 cm.
Find the length of AB.
A
B CD
Pythagoras’ Theorem
< 24 >
5. S00 Q14
Let ABC be a right-angled triangle with A = 90, and let D be the midpoint of BC.
Suppose AD = 1 cm, the perimeter of ABC is 82 cm, and the area of ABC is x cm2.
Find the value of x.
6. S00 Q16
ABCD is a trapezium with AB // DC. Suppose AB = 3 cm, BC = 5 cm, CD = 6 cm,
DA = 4 cm and the area of the trapezium is x cm2. Find the value of x.
7. S00 Q17
In a convex quadrilateral of area 64 cm2, the sum of the lengths of a diagonal and a pair
of opposite sides is 216 cm. Suppose the length of the other diagonal is x cm, find the
value of x.
