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2023 Canadian Team Mathematics Contest
Individual Problems

(45 minutes)

Important Notes

Problems

  1. Ingrid starts with n chocolates, while Jin and Brian each start with 0 chocolates. Ingrid gives one third of her chocolates to Jin. Jin gives 8 chocolates to Brian and then Jin eats half of her remaining chocolates. Jin now has 5 chocolates. What is the value of n?

  2. For what value of k is k% of 25 equal to 20% of 30?

  3. It is now 1:00 a.m. What time will it be 2023 minutes from now?

  4. A group of eight students have lockers that are arranged as shown, in two rows of four lockers with one row directly on top of the other. The students are allowed to paint their lockers either blue or red according to two rules. The first rule is that there must be two blue lockers and two red lockers in each row. The second rule is that lockers in the same column must have different colours. How many ways are there for the students to paint their lockers according to the rules?

  5. In trapezoid ABCD, AB=4, CD=6, DAB=90°, BCD=45°, and AB is parallel to CD. What is the length of BD?

  6. A train is traveling from City A to City B. If the train travels at a speed of 80 km/h, it will arrive 24 minutes late. If it travels at a speed of 90 km/h, it will arrive 32 minutes early. At what speed in km/h should the train travel in order to arrive on time?

  7. In ABC, tanBCA=1 and tanBAC=17. The perimeter of ABC is 24+182. The altitude from B to AC has length h and intersects AC at D. What is the value of h?

  8. A Tim number is a five-digit positive integer with the property that it is a multiple of 15, its hundreds digit is 3, and its tens digit is equal to the sum of its first (leftmost) three digits. How many Tim numbers are there?

  9. The real numbers x, y and z satisfy both of the equations below: 4x+7y+z=113x+y+5z=15 Given that x+y+z=pq where p and q are positive integers and the fraction pq is in lowest terms, what is the value of pq?

  10. For every positive integer n, let Sn={1,2,3,,n}; that is, Sn is the set of integers from 1 to n inclusive. There are 2n subsets of Sn. If each subset has the same likelihood of being chosen, let p(n) be the probability that a chosen subset does not contain two integers with a sum of n+1.

    For example, the subsets of S2 are (the empty set), {1}, {2}, and {1,2}. Of these four subsets, only {1,2} contains a pair of integers with a sum of 2+1=3. The other three subsets do not contain such a pair, so p(2)=34.

    What is the smallest even positive integer n for which p(n)<14?

Relay Problems

Relay #1

Seat a

A square garden has an area of 9 square metres. The perimeter of the garden is N metres. What is the value of N?

Seat b

Let t be TNYWR.
In the diagram, square ABCD has side-length t. Two vertical lines and two horizontal lines divide square ABCD into nine equal smaller squares, and each of these smaller squares is cut in half by a diagonal, as shown. Some of the triangular regions defined by these lines are shaded.

Eight of the smaller squares have one half shaded. The ninth square is unshaded.

What is the total area of the shaded parts of the square?

Seat c

Let t be TNYWR.
If t=n(n1)(n+1)+n, what is the value of n?

Relay #2

Seat a

How many integers n with n>0 satisfy 1n+1>429?

Seat b

Let t be TNYWR.
A water tank initially contains x litres of water. Pablo adds t2 litres of water to the tank, making the tank 20% full. After Pablo adds water to the tank, Chloe then adds t24 litres of water to the tank, making the tank 50% full. What is the value of x?

Seat c

Let t be TNYWR.
Point O is at the origin and points P(a,b) and Q(c,d) are in the first quadrant, as shown. The slope of OP is 125 and the length of OP is 13t. The slope of OQ is 34 and the length of OQ is 10t. What is a+c?

Relay #3

Seat a

Three of the positive divisors of 24 are 1, 8, and 24. What is the sum of all of the positive divisors of 24?

Seat b

Let t be TNYWR.
The numbers a and b satisfy both of the following equations. at6b=20at5b=10 What is the value of b?

Seat c

Let t be TNYWR.
The parabola with equation y=ax2+bx+c passes through (4,0), (t3,0), and (0,60). What is the value of a?

Team Problems

(45 minutes)

Important Notes

Problems

  1. A fish and chips truck sells three kinds of fish: cod, haddock, and halibut. During a music festival, the truck sold 220 pieces of fish, of which 40% were haddock and 40% were halibut. How many pieces of cod did the truck sell?

  2. If x25=9, what is the value of 7x?

  3. Point A has coordinates (13,23) and point B has coordinates (33,43). What is the slope of a line that is perpendicular to the line segment AB?

  4. What is the integer equal to 119217211917102 ?

  5. If p+q+r=70, p=2q, and q=3r, what is the value of p?

  6. How many ordered triples (a,b,c) of integers satisfy 1a<b<c10 and ba=cb?

  7. The distinct prime factors of 18 are 2 and 3. What is the sum of all of the distinct prime factors of 4446?

  8. Using the diagram below, a seven digit integer can be created as follows: trace a path that uses each line segment exactly once and use the labels on the line segments as digits.

    A description of the diagram follows.

    For example, the path that goes from C to A to B to C to E to B to D and finally to E gives the positive integer 3264715. What is the largest possible positive integer that can be created in this way?

  9. Alheri has three hoses. Water flows out of each hose at the same constant rate. Using all three hoses, she starts to fill a swimming pool at 6:00 a.m. and calculates that the pool will be full at exactly 6:00 p.m. on the same day. At 11:00 a.m., one of the hoses unexpectedly stops working. Assuming water still flows out of the other two hoses at the same rate as before, at what time will the pool be full?

  10. The lines with the two equations below intersect at the point (2,3). (a2+1)x2by=4(1a)x+by=9 What are the possible ordered pairs (a,b)?

  11. What is the integer equal to 20234(2022)(2024)(1+20232)?

  12. For how many integers x is the expression 75xx25 equal to an integer?

  13. A positive integer is called mystical if it has at least two digits and every pair of two consecutive digits, read from left to right, forms a perfect square. For example, 364 is a mystical integer because 36 and 64 are both perfect squares, but 325 is not mystical because 32 is not a perfect square. What is the largest mystical integer?

  14. In a 3×3 grid, there are four 2×2 subgrids, each of which is bordered by a thick line in one of the four grids below.

    In one grid, the four squares in the top left corner of the grid are bordered by a thick line. In the other three grids, the squares bordered by a thick line are the four squares in the top right, then the four squares in the bottom left, then the four
squares in the bottom right.

    Kasun wants to place an integer from 1 to 4 inclusive in each cell of a 3×3 grid so that every 2×2 subgrid contains each integer exactly once. For example, the grid below and on the left satisfies the condition, but the grid below and on the right does not.

    The grid on the left. The entries in the top row, from left to right, are 1,2,3. The middle row has 3,4,1 and the bottom row has 1,2,3.     The grid on the right. The entries in the top row, from left to right, are 1,3,2. The middle row has 2,4,1 and the bottom row has 1,3,3. The four squares in the bottom right corner are bordered with a thick line. These four squares have entries 4,1,3,3.

    In how many ways can Kasun place integers in the grid so that they satisfy the condition?

  15. There are exactly three real numbers x for which (x5x) is the reciprocal of (x4). What is the sum of these three real numbers?

  16. In ABC, AB=8, BC=11, and AC=6. The points P and Q are on BC such that PBA and QAC are each similar to ABC. What is the length of PQ?

  17. Square ABCD has A and B on the x-axis and C and D below the x-axis on the parabola with equation y=x24. What is the area of ABCD?

  18. Let a=log49 and b=108log38. What is the integer equal to ab?

  19. Jolene and Tia are playing a two-player game at a carnival. In one bin, there are five red balls numbered 5, 10, 15, 20 and 25. In another bin, there are 25 green balls numbered 1 through 25. In the first stage of the game, Jolene chooses one of the red balls at random. Next, the carnival worker removes the green ball with the same number as the ball Jolene chose. Tia then chooses one of the 24 remaining green balls at random.

    Jolene and Tia win if the number on the ball chosen by Tia is a multiple of 3. What is the probability that they will win?

  20. The positive integer d has the property that each of 468, 636, and 867 has the same remainder, r, when divided by d. What is the largest possible value of d+r?

  21. Square ABCD has centre O. Points P and Q are on AB, R and S are on BC, T and U are on CD, and V and W are on AD, as shown, so that APW, BRQ, CTS, and DVU are isosceles and POW, ROQ, TOS, and VOU are equilateral.

    What is the ratio of the area of PQO to that of BRQ?

  22. The sequence a1,a2,a3, is an arithmetic sequence with common difference 3 and a1=1. The sequence b1,b2,b3, is an arithmetic sequence with common difference 10 and b1=2. What is the smallest integer larger than 2023 that appears in both sequences?

  23. In ABC, AB=10 and sin3A+3sinC=4. What is the length of the altitude from C to AB?

  24. The real numbers x, y, and z satisfy both of the equations below. x+y+z=2xy+yz+xz=0 Let a be the minimum possible value of z and b be the maximum possible value of z. What is the value of ba?

  25. Let f(x) be a function with the property that f(x)+f(x13x2)=x for all real numbers x other than 23. What is the sum f(0)+f(1)+f(2)?