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2019 Canadian Team Mathematics Contest
Individual Problems

(45 minutes)

Important Notes

Problems

  1. In the diagram, points A, B, C, and D are on a circle.

    The four distinct points are on the circumference of the circle (outside edge).

    Philippe uses a ruler to connect each pair of these points with a line segment. How many line segments does he draw?

  2. What is the smallest positive integer n for which 2019n is an integer?

  3. At 7:00 a.m. yesterday, Sherry correctly determined what time it had been 100 hours before. What was her answer? (Be sure to include “a.m.” or “p.m.” in your answer.)

  4. A standard die with six faces is tossed onto a table. Itai counts the total number of dots on the five faces that are not lying on the table. What is the probability that this total is at least 19?

  5. Suppose that a, b, c, and d are positive integers with 0<a<b<c<d<10. What is the maximum possible value of abcd ?

  6. When the line with equation y=2x+7 is reflected across the line with equation x=3, the equation of the resulting line is y=ax+b. What is the value of 2a+b?

  7. Suppose that (23)x=4096 and that y=x3. What is the ones (units) digit of the integer equal to 3y?

  8. Yasmine makes her own chocolate beverage by mixing volumes of milk and syrup in the ratio 5:2. Milk comes in 2 L bottles and syrup comes in 1.4 L bottles. Yasmine has a limitless supply of full bottles of milk and of syrup. Determine the smallest volume of chocolate beverage that Yasmine can make that uses only whole bottles of both milk and syrup.

  9. Suppose that a is an integer. A sequence x1,x2,x3,x4, is constructed with

    For example, if a=2, then x1=2x2=2x1=4x3=x21=3x4=2x3=6x5=x41=5 and so on. The integer N=578 can appear in this sequence after the 10th term (for example, x12=578 when a=10), but the integer 579 does not appear in the sequence after the 10th term for any value of a. What is the smallest integer N>1395 that could appear in the sequence after the 10th term for some value of a?

  10. In the diagram, ABCDEFGH is a rectangular prism. (Vertex H is hidden in this view.)

    ABCD is the top face of the rectangular prism, and EFGH is the bottom face of the prism. The front face (closest to you) is ABGF, and the back face of the prism is CDEH.

    IfABF=40 and ADF=20, what is the measure of BFD, to the nearest tenth of a degree?

Relay Problems

Relay #1

Seat a

If x=1 and y=630, what is the value of 2019x3y9?

Seat b

Let t be TNYWR.
At the start of 2018, the Canadian Excellent Mathematics Corporation had t employees in its Moose Jaw office, 40 employees in its Okotoks office, and no other employees. During 2018, the number of employees in the Moose Jaw office increased by 25% and the number of employees in the Okotoks office decreased by 35%. How many additional employees did the CEMC have at the end of 2018 compared to the beginning of 2018?

Seat c

Let t be TNYWR.
Kolapo lists the four-digit positive integers that can be made using the digits 2, 4, 5, and 9, each once. Kolapo lists these integers in increasing order. What is the tth number in his list?

Relay #2

Seat a

In the diagram, ABC is similar to DEF.

Triangles ABC and DEF are both right triangles, where angles ABC and DEF are both 90 degrees. Side DE is 96, EF is 24, BC is 33, and AB is x.

What is the value of x?

Seat b

Let t be TNYWR.
The sum of the even integers from 2 to 2k inclusive equals t for some positive integer k. That is, 2+4+6++(2k2)+2k=t What is the value of k?

Seat c

Let t be TNYWR.
Suppose that O is the origin. Points P(a,b) and Q(c,1) are in the first quadrant with a=2c. If the slope of OP is t and the slope of OQ is 1, what is the slope of PQ?

Relay #3

Seat a

How many perfect squares are there between 2 and 150?

Seat b

Let t be TNYWR.
The line with equation y=2x+t and the parabola with equation y=(x1)2+1 intersect at point P in the first quadrant. What is the y-coordinate of P?

Seat c

Let t be TNYWR.
The triangle in the first quadrant formed by the x-axis, the y-axis, and the line with equation (k1)x+(k+1)y=t has area 10. What is the value of k?

Team Problems

(45 minutes)

Important Notes

Problems

  1. For what value of x is 4x8+3x=12+5x ?

  2. What is the value of 3.5×2.5+6.5×2.5 ?

  3. Ada is younger than Darwyn. Max is younger than Greta. James is older than Darwyn. Max and James are the same age. Which of the five people is the oldest?

  4. Determine the average (mean) of 12, 14 and 18 as a fraction in lowest terms.

  5. Suppose that M=15+24×3342÷51N=1524×33+42÷51 What is the value of M+N?

  6. How many four-digit palindromes abba have the property that the two-digit integer ab and the two-digit integer ba are both prime numbers? (For example, 2332 does not have this property, since 23 is prime but 32 is not.)

  7. Adia writes a list in increasing order of the integers between 1 and 100, inclusive, that cannot be written as the product of two consecutive positive integers. What is the 40th integer in her list?

  8. For how many ordered pairs of positive integers (a,b) is 1<a+b<22?

  9. Shelly-Ann normally runs along the Laurel Trail at a constant speed of 8 m/s. One day, one-third of the trail is covered in mud, through which Shelly-Ann can only run one-quarter of her normal speed, and it takes her 12 s to run the entire length of the trail. How long is the trail, in metres?

  10. Determine the value of a for which 5a+5a+1=4500.

  11. Ezekiel has a rectangular piece of paper with an area of 40. The width of the paper is more than twice the height. He folds the bottom left and top right corners at 45 and creates a parallelogram with an area of 24.

    The first fold line originates from the top left corner of the page and is 45 degrees down from the top edge. The second fold line originates from the bottom right corner and is 45 degrees above the bottom edge of the paper.

    What is the perimeter of the original rectangle?

  12. What is the value of 1234562123455×123457 ?

  13. Determine the value of (log24)(log46)(log68).

  14. The integers x,y and z satisfy x5=6y=z2. What is the largest possible value of x+y+z?

  15. Suppose that G=10100. (G is known as a googol.) How many times does the digit 9 occur in the integer equal to G10092 ?

  16. Suppose that f(x)=x4x31 and g(x)=x8x62x4+1. If g(x)=f(x)h(x), determine the polynomial function h(x).

  17. In the diagram, pentagon ABCDE is symmetrical about altitude CF.

    A complete description of the pentagon follows.

    Also, AE=200, CF=803, ABC=150, and BCD=120. Determine the vertical distance between AE and BD.

  18. The height of Cylinder A is equal to its diameter. The height and diameter of Cylinder B are each twice those of Cylinder A. The height of Cylinder C is equal to its diameter. The volume of Cylinder C is the sum of the volumes of Cylinders A and B. What is the ratio of the diameter of Cylinder C to the diameter of Cylinder A?

  19. Suppose that f(x)=a(xb)(xc) is a quadratic function where a, b and c are distinct positive integers less than 10. For each choice of a, b and c, the function f(x) has a minimum value. What is the minimum of these possible minimum values?

  20. The integers from 1 to k are concatenated to form the integer N=123456789101112. Determine the smallest integer value of k>2019 such that N is divisible by 9.

  21. The real numbers x1,x2,x3,,xn are the consecutive terms of an arithmetic sequence. If x2x1+x3+x3x2+x4+x4x3+x5++xn2xn3+xn1+xn1xn2+xn=1957 what is the value of n?

  22. Suppose that f1(x)=12x. For each positive integer n2, define fn(x)=f1(fn1(x)) for all real numbers x in the domain of f1(fn1(x)). The value of f2019(4) can be written as ab where a and b are positive integers with no common divisor larger than 1. What is (a,b)?

  23. The numbers p, q, r, and t satisfy p<q<r<t. When these numbers are paired, each pair has a different sum and the four largest sums are 19, 22, 25, and 28. What is the the sum of the possible values for p?

  24. Suppose that α and β are the two positive roots of the equation x213xlog13x=0 Determine the value of αβ.

  25. In the diagram, ABCD is a square. Point F is on AB with BF=2AF. Point E is on AD with FEC=BCE.

    If 0<ECD<45, what is the value of tan(ECD)?