Math Olympiad Challenges
45 Problems — Free Response
(+1)
1
arithmetic
easy
Find the number A such that the following statement is true: 5 × 10 + 6 × 10 = 11 × A.
Show answer
Answer: 10
5 × 10 + 6 × 10 = 50 + 60 = 110, and 110 = 11 × A, so A = 110 ÷ 11 = 10.
2
algebra
easy
What is the missing number B which satisfies the given equation? 2 × B + 5 × 4 − 14 = B + 3 × 2 + 6.
Show answer
Answer: 6
Left side: 2B + 20 − 14 = 2B + 6. Right side: B + 6 + 6 = B + 12. So 2B + 6 = B + 12, giving B = 6.
3
number-theory
medium
Suppose N is a whole number. For how many values of N less than 150 is N/10 + N/20 also a whole number?
Show answer
Answer: 8
N/10 + N/20 = 2N/20 + N/20 = 3N/20. This is a whole number only when 20 divides 3N; since 3 and 20 share no factors, 20 must divide N. Since 0 is a whole number, the values below 150 are 0, 20, 40, 60, 80, 100, 120, 140 — that is 8 values.
4
number-theory
medium
For how many values of N is 144/N² also a whole number?
Show answer
Answer: 6
144 = 2⁴ × 3². We need N² to divide 144, so N can be 1, 2, 3, 4, 6 or 12 (since 1, 4, 9, 16, 36 and 144 all divide 144). That is 6 values.
5
logic
medium
A bag contains a total of 15 balls of 5 different colours. Each colour has a unique number of balls in the bag. What is the least number of balls to be picked (without looking) to ensure three different coloured balls are picked?
Show answer
Answer: 10
Five different positive counts summing to 15 must be 1, 2, 3, 4, 5. In the worst case you pick every ball of the two most numerous colours: 5 + 4 = 9 balls of only two colours. One more pick must be a third colour, so 9 + 1 = 10 balls are needed.
6
money
medium
Seven 50¢ coins and eight $1 coins are equally distributed (randomly) among five children. What is the difference between the maximum amount and the minimum amount a child can receive? (Give your answer in dollars.)
Show answer
Answer: $1.50
There are 15 coins, so each child gets 3 coins. The most a child can get is three $1 coins = $3.00 (there are 8 such coins). The least is three 50¢ coins = $1.50 (there are 7 such coins). The difference is $3.00 − $1.50 = $1.50.
7
counting
easy
A palindromic number is a number that remains the same when its digits are reversed. E.g. 4554 is a palindromic number. How many palindromic numbers are there between 100 and 999, inclusive?
Show answer
Answer: 90
A 3-digit palindrome has the form ABA. The first digit A can be 1–9 (9 choices) and the middle digit B can be 0–9 (10 choices); the last digit is forced. So there are 9 × 10 = 90.
8
number-theory
medium
If two different natural numbers have the same digits but in reverse order, each number is called the palimage of the other. For example, 12 and 21 are palimages of each other. What is the maximum difference between two-digit palimages?
Show answer
Answer: 72
For a two-digit number and its reverse, the difference is 9 × (difference of the two digits). Both numbers must be two-digit, so both digits are 1–9; the largest digit gap is 9 − 1 = 8. Maximum difference = 9 × 8 = 72 (from 91 and 19).
9
rate
easy
Alan runs at a rate of 100 meters per 15 seconds. At this rate, how much distance will he cover in 15 minutes? Give your answer in kilometers.
Show answer
Answer: 6 km
15 minutes = 900 seconds. That is 900 ÷ 15 = 60 intervals of 15 seconds, so 60 × 100 m = 6000 m = 6 km.
10
rate
medium
In a construction company, 10 workers complete one-tenth of their total in 10 hours. How long will 50 workers take to complete their entire work?
Show answer
Answer: 20 hours
10 workers do 1/10 of the job in 10 hours, so the whole job takes 10 workers 100 hours = 1000 worker-hours. With 50 workers it takes 1000 ÷ 50 = 20 hours.
11
calendar
medium
If January 1st, 2005 was a Monday, on what date will the 55th Monday from January 1st, 2005 be?
Show answer
Answer: 14 January 2006
The 1st Monday is January 1, 2005. The 55th Monday is 54 weeks later = 54 × 7 = 378 days after January 1, 2005. 2005 has 365 days, so 378 days later falls 378 − 365 = 13 days into 2006, i.e. January 14, 2006.
12
averages
easy
The average of 5 numbers is 7.6. What number should be added to increase the average of the numbers to 8?
Show answer
Answer: 10
The sum of the 5 numbers is 5 × 7.6 = 38. For 6 numbers to average 8, their sum must be 6 × 8 = 48. The number to add is 48 − 38 = 10.
13
patterns
medium
In the following pattern, how many black squares are there in the 19th image?
Show answer
Answer: 41
Each image is a (k+2)×(k+2) grid with both diagonals shaded black — not a full checkerboard. The 1st image is 3×3 (5 black), the 2nd is 4×4 (8 black), the 3rd is 5×5 (9 black), the 4th is 6×6 (12 black), and so on: 5, 8, 9, 12, 13, 16, 17, 20, 21, 24, 25, 28, 29, 32, 33, 36, 37, 40, 41, ... For an n×n grid, the two diagonals give 2n black squares if n is even, or 2n−1 if n is odd (they cross at one shared center cell). The 19th image is 21×21 (n=21, odd), so it has 2(21)−1 = 41 black squares.
14
logic
medium
A certain number of children stand in a line in ascending height order. The 10th child from the front is also the 16th child from the back. How many children must be added at the back such that the 5th child from the front becomes the 36th child from the back?
Show answer
Answer: 15
The 10th from front = 16th from back means there are 10 + 16 − 1 = 25 children. For the 5th child from the front to be 36th from the back, the line must have 5 + 36 − 1 = 40 children. So 40 − 25 = 15 children must be added.
15
sequences
medium
Among a certain group of kids, the first kid gets 1 sweet, the second kid gets 2 more sweets than the first kid, the third kid gets 3 more sweets than the second kid and so on. How many more sweets does the 12th kid get in comparison to the 6th kid?
Show answer
Answer: 57
The n-th kid gets 1 + 2 + 3 + … + n = n(n+1)/2 sweets. The 12th kid gets 12 × 13 ÷ 2 = 78 and the 6th kid gets 6 × 7 ÷ 2 = 21. The difference is 78 − 21 = 57.
16
rate
medium
Rico walks from point A to point B on a straight path at a speed of 2 km/h. Alice walks from point B to point A on the same path at a speed of 5 km/h. If Rico starts walking from point A at the same time as Alice starts walking from point B, and the distance between points A and B is 14 km, how far would Alice have walked before she crosses Rico?
Show answer
Answer: 10 km
They move toward each other, closing the 14 km gap at 2 + 5 = 7 km/h. They meet after 14 ÷ 7 = 2 hours. In 2 hours Alice walks 5 × 2 = 10 km.
17
geometry
easy
How many degrees does the minute hand sweep between the time 9:00 AM and 11:50 AM?
Show answer
Answer: 1020°
From 9:00 to 11:50 is 2 hours 50 minutes = 170 minutes. The minute hand turns 360° every 60 minutes, i.e. 6° per minute. So it sweeps 170 × 6 = 1020°.
18
logic
medium
A, B, C are different numbers to be chosen from the set {4, 7, 8, 10}. Find the difference between the largest and smallest values of (A × B) / C.
Show answer
Answer: 17.2
To make (A × B)/C largest, use the smallest C = 4 and the two largest for A, B: 8 × 10 ÷ 4 = 20. To make it smallest, use the largest C = 10 and the two smallest: 4 × 7 ÷ 10 = 2.8. The difference is 20 − 2.8 = 17.2.
19
patterns
hard
Given on the right is a series of numbers following a specific pattern. (1,2) refers to the number in the first row second column. What is the number in (10,2)?
Show answer
Answer: 72
In the table the first column reads 2, 9, 17, 24, … increasing by 7 then 8 alternately. So column 1 of row 10 is 2 + (7+8+7+8+7+8+7+8+7) = 69. Within a row the numbers increase by 2 in odd-numbered rows and by 3 in even-numbered rows; row 10 is even, so (10,2) = 69 + 3 = 72.
20
geometry
medium
A circle with centre O is inscribed in a square ABCD as shown in the figure. If the perimeter of square ABCD is 16 cm, find the area of the circle in cm². Give your answer in terms of cm².
Show answer
Answer: 4π cm²
Perimeter 16 cm means each side is 16 ÷ 4 = 4 cm. The inscribed circle has diameter 4 cm, so radius 2 cm. Area = π × 2² = 4π cm².
21
geometry
medium
In the given figure, arc AOD represents a semicircle with centre at O. Length of AB = a where ABCD is a square. Find the area of the given figure.
Show answer
Answer: a² + (πa²)/8
The square has area a². The semicircle is drawn on side AD (length a) as diameter, so its radius is a/2 and its area is ½ × π × (a/2)² = πa²/8. Total area = a² + πa²/8.
22
geometry
medium
Let ABCDEFGH be a cuboid such that AB = 10 cm, BC = 2 cm, BF = 5 cm. How many cubes of length ½ cm would be required to fill the cuboid?
Show answer
Answer: 800
Volume of the cuboid = 10 × 2 × 5 = 100 cm³. Each small cube has volume (½)³ = 1/8 cm³. Number of cubes = 100 ÷ (1/8) = 800.
23
number-theory
hard
Given C > 0, for how many values of B is the given product valid? (The multiplication is AA × CB, with partial products BB and CC0, giving C5B.)
Show answer
Answer: 5
AA × B = BB forces A = 1 (so AA = 11). Then 11 × C shifted one place gives CC0. Adding BB + CC0 = C5B leads to 10C + 10B = 50, i.e. B + C = 5. With C > 0 the pairs (C,B) are (1,4), (2,3), (3,2), (4,1), (5,0), so B can be 0, 1, 2, 3 or 4 — 5 values. (If the letters are required to be different digits, only 3 remain.)
24
geometry
hard
If BD is a diagonal in a rectangle ABCD where AB = 10, AD = 5, AE = 2, FC = 4 as shown in the figure, find the value of: (Area of triangle AED)/(Area of triangle BED) + (Area of triangle BCF)/(Area of triangle BFD).
Show answer
Answer: 11/12
With E on AB (AE = 2, EB = 8) and F on DC (FC = 4, DF = 6): triangle AED and triangle BED share the same height from D, so their areas are in the ratio AE : EB = 2 : 8, i.e. 1/4. Triangle BCF and triangle BFD share the same height from B, so their ratio is FC : DF = 4 : 6 = 2/3. Sum = 1/4 + 2/3 = 3/12 + 8/12 = 11/12.
25
geometry
medium
ABCD is a square park where AB = 100 m. The shaded area represents grass and the solid areas represent congruent squares of concrete. If EF = 5 m, what is the ratio of the area covered by grass to the area covered by concrete?
Show answer
Answer: 99:1
The park area is 100 × 100 = 10000 m². There are 4 concrete squares each 5 × 5 = 25 m², totalling 100 m². Grass = 10000 − 100 = 9900 m². Ratio grass : concrete = 9900 : 100 = 99 : 1.
26
algebra
medium
Andy's current age is 40. Five years from now, his age will be equal to the sum of the ages of his two sons, Mike and Rick. If Mike is 3 years older than Rick, what is Rick's current age?
Show answer
Answer: 16
In 5 years Andy will be 45, so Mike + Rick (in 5 years) = 45. In 5 years Rick = r + 5 and Mike = r + 8 (Mike is 3 older). So (r + 5) + (r + 8) = 45, giving 2r + 13 = 45, r = 16. Rick is 16 now.
27
logic
hard
A 3 × 3 grid contains 9 numbers. Each number is halved to obtain the number on the left. Each number is multiplied by 1/3 to obtain the number above. If the sum of the numbers is 182, what is the number in the top right corner?
Show answer
Answer: 8
Let the bottom-right number be x. Moving right doubles a number, moving down triples it. Every cell is x × 2^(columns to the right) × 3^(rows below-ish); adding all nine cells gives a total of 91x. So 91x = 182, x = 2, and the top-right corner is x × 4 = 8.
28
geometry
hard
A square is divided into three congruent rectangles as shown in figure 1. If the middle rectangle is rearranged to form figure 2, what is the ratio of the perimeter of figure 1 to the perimeter of figure 2?
Show answer
Answer: 3:5
Let each rectangle be 3 units by 1 unit, so the square has side 3 and perimeter 12. Moving the middle strip out to form the I-shape adds extra edges; tracing figure 2 gives a perimeter of 20 units. The ratio is 12 : 20 = 3 : 5.
29
algebra
easy
In the zoo, there are 45 heads of monkeys and lions. There are twice as many monkeys as lions. How many lions are there?
Show answer
Answer: 15
If there are L lions there are 2L monkeys, so L + 2L = 45, giving 3L = 45 and L = 15 lions.
30
algebra
easy
In a group of 35 students, there are 9 girls more than boys. What is the number of girls?
Show answer
Answer: 22
If there are b boys there are b + 9 girls, so b + (b + 9) = 35, giving 2b = 26, b = 13. Girls = 13 + 9 = 22.
31
sequences
medium
Jason has eaten 45 chocolates in 5 days. Every day, he ate 2 chocolates more than the previous day. How many chocolates did he eat on the first day?
Show answer
Answer: 5
If he ate a on day 1, the five days are a, a+2, a+4, a+6, a+8, summing to 5a + 20 = 45. So 5a = 25 and a = 5.
32
patterns
medium
Find the missing numbers and complete the number pattern: 14, ?, 53, ?, 144.
Show answer
Answer: 40 and 92
Each term is the sum of the two before it minus 1. Then 53 = ? + 14 − 1 gives the 2nd term = 40; the 4th term = 53 + 40 − 1 = 92; check: 92 + 53 − 1 = 144. So the missing numbers are 40 and 92.
33
logic
easy
The sum of Tom's age and Jerry's age is 25. What will the sum of their ages be after nine years?
Show answer
Answer: 43
Each of the two people ages 9 years, so the sum increases by 2 × 9 = 18. New sum = 25 + 18 = 43.
34
logic
medium
The star (☆) has a number value of? Given: ○ + △ = 26, △ + △ = 34, △ + □ = 35, ○ + □ = ☆.
Show answer
Answer: 27
△ + △ = 34 gives △ = 17. Then ○ = 26 − 17 = 9 and □ = 35 − 17 = 18. So ☆ = ○ + □ = 9 + 18 = 27.
35
logic
medium
There were 17 pieces of paper. Some of them were cut into four parts. Altogether, there are now 32 pieces of paper. How many pieces were cut into four parts?
Show answer
Answer: 5
Cutting one piece into four adds 3 pieces. The total grew by 32 − 17 = 15 pieces, so 15 ÷ 3 = 5 pieces were cut.
36
logic
medium
What is the smallest possible number of children in a family if each child has at least two brothers and two sisters?
Show answer
Answer: 6
A child needs 2 other boys (so at least 3 boys) and 2 other girls (so at least 3 girls). The smallest family is 3 boys + 3 girls = 6 children.
37
counting
easy
In a box, there are six smaller boxes, each one of them contains seven even smaller boxes. How many boxes are there in total?
Show answer
Answer: 49
The outer box (1) + the six smaller boxes (6) + the seven-in-each inner boxes (6 × 7 = 42) = 1 + 6 + 42 = 49 boxes. (If the outermost box is not counted, the answer is 48.)
38
arithmetic
easy
What is the value of the following? 75 + 75 + 75 + … (twenty-four 75's added together).
Show answer
Answer: 1800
There are 24 copies of 75 (four rows of five plus a final row of four). 24 × 75 = 1800.
39
patterns
hard
Observe the following number pattern and fill in the box with the correct answer. First box: top 11 and 35, bottom 1645. Second box: top 32 and 46, bottom 3966. Third box: top 34 and 25, bottom is unknown.
Show answer
Answer: (pattern unresolved — see solution)
The intended rule linking the two top numbers to the bottom number could not be reliably determined from the scanned figure: no consistent formula (product, sum of squares, digit combination, or linear rule) fits both given boxes 11,35→1645 and 32,46→3966. Please check against an official SASMO 2013 key. If you can confirm the rule, the third box (34, 25) can then be filled in.
40
algebra
medium
There are 47 boys and 29 girls in a group. Every week, 6 more boys and 9 more girls join the group. After how many weeks will there be the same number of boys and girls in the group?
Show answer
Answer: 6
After w weeks: boys = 47 + 6w, girls = 29 + 9w. Setting them equal: 47 + 6w = 29 + 9w, so 18 = 3w and w = 6 weeks.
41
logic
medium
A box has five pens in five colours: red, green, blue, white and yellow. The green pen is the first pen on the left. The yellow pen is next to the green pen and the red pen. The white pen is next to the red pen and the blue pen. What is the order of the pens in that box?
Show answer
Answer: Green, Yellow, Red, White, Blue
Green is leftmost. Yellow is between green and red, so: Green, Yellow, Red. White is next to red and blue, so White follows Red and Blue follows White: Green, Yellow, Red, White, Blue.
42
counting
medium
How many rectangles are in this picture?
Show answer
Answer: 30
The figure is a 2-row by 4-column grid. The number of rectangles is (ways to pick 2 of the 3 horizontal lines) × (ways to pick 2 of the 5 vertical lines) = 3 × 10 = 30.
43
patterns
hard
What is the next number in the following pattern: 1, 3, 11, 123, ?
Show answer
Answer: 15131
Each term is the square of the previous term plus 2: 1²+2 = 3, 3²+2 = 11, 11²+2 = 123, so 123²+2 = 15129 + 2 = 15131.
44
algebra
hard
There are parrots and rabbits in a pet store. The number of parrots is 50 more than that of rabbits and the total number of legs of the parrots is 40 more than that of the rabbits. How many parrots are there in the pet store?
Show answer
Answer: 80
Let rabbits = r, parrots = r + 50. Parrot legs = 2(r + 50); rabbit legs = 4r. Then 2(r + 50) = 4r + 40, so 2r + 100 = 4r + 40, giving 2r = 60, r = 30. Parrots = 30 + 50 = 80.
45
counting
hard
How many three-digit numbers are there in which all digits on the left are larger than digits on the right? (Hint: Look for a pattern.)
Show answer
Answer: 120
The digits must be strictly decreasing (hundreds > tens > units). Any choice of 3 different digits from 0–9 can be arranged in decreasing order in exactly one way, and the largest (leading) digit is automatically non-zero. So the count is C(10,3) = 120.