2011 Canadian Open Math Challenge

Part of Canadian Open Math Challenge

Subcontests

(12)

1

COMC 2011 Problem 12

f(x)=x^2-ax+b

a

b

are positive integers. (a) Suppose that

a=2

b=2

. Determine the set of real roots of

f(x)-x

, and the set of real roots of

f(f(x))-x

. (b) Determine the number of positive integers

(a,b)

1\le a,b\le 2011

for which every root of

f(f(x))-x

1

COMC 2011 Problem 11

n

be a positive integer. A row of

n+ 1

squares is written from left to right, numbered

0, 1, 2, \cdots, n

Two frogs, named Alphonse and Beryl, begin a race starting at square 0. For each second that passes, Alphonse and Beryl make a jump to the right according to the following rules: if there are at least eight squares to the right of Alphonse, then Alphonse jumps eight squares to the right. Otherwise, Alphonse jumps one square to the right. If there are at least seven squares to the right of Beryl, then Beryl jumps seven squares to the right. Otherwise, Beryl jumps one square to the right. Let A(n) and B(n) respectively denote the number of seconds for Alphonse and Beryl to reach square n. For example, A(40) = 5 and B(40) = 10. (a) Determine an integer n>200 for which

B(n) <A(n)

. (b) Determine the largest integer n for which

B(n) \le A(n)

1

COMC 2011 Problem 10

Charlotte writes a test consisting of 100 questions, where the answer to each question is either TRUE or FALSE. Charlotte’s teacher announces that for every five consecutive questions on the test, the answers to exactly three of them are TRUE. Just before the test starts, the teacher whispers to Charlotte that the answers to the first and last questions are both FALSE. (a) Determine the number of questions for which the correct answer is TRUE. (b) What is the correct answer to the sixth question on the test? (c) Explain how Charlotte can correctly answer all 100 questions on the test.

1

COMC 2011 Problem 9

ABC is a triangle with coordinates A =(2, 6), B =(0, 0), and C =(14, 0). (a) Let P be the midpoint of AB. Determine the equation of the line perpendicular to AB passing through P. (b) Let Q be the point on line BC for which PQ is perpendicular to AB. Determine the length of AQ. (c) There is a (unique) circle passing through the points A, B, and C. Determine the radius of this circle.

1

COMC 2011 Problem 8

A group of n friends wrote a math contest consisting of eight short-answer problem

S_1, S_2, S_3, S_4, S_5, S_6, S_7, S_8

, and four full-solution problems

F_1, F_2, F_3, F_4

. Each person in the group correctly solved exactly 11 of the 12 problems. We create an 8 x 4 table. Inside the square located in the

i

j

th column, we write down the number of people who correctly solved both problem

S_i

F_j

. If the 32 entries in the table sum to 256, what is the value of n?

1

COMC 2011 Problem 7

In the figure, BC is a diameter of the circle, where

BC=\sqrt{901}, BD=1

DA=16

EC=x

, what is the value of x? [asy]size(2inch); pair O,A,B,C,D,E; B=(0,0); O=(2,0); C=(4,0); D=(.333,1.333); A=(.75,2.67); E=(1.8,2); draw(Arc(O,2,0,360)); draw(B--C--A--B); label("

A

",A,N); label("

B

",B,W); label("

C

",C,E); label("

D

",D,W); label("

E

",E,N); label("Figure not drawn to scale",(2,-2.5),S); [/asy]

1

COMC 2011 Problem 6

Integers a, b, c, d, and e satisfy the following three properties: (i)

2 \le a < b <c <d <e <100

\gcd (a,e) = 1

(iii) a, b, c, d, e form a geometric sequence. What is the value of c?

1

COMC 2011 Problem 5

Arthur is driving to David’s house intending to arrive at a certain time. If he drives at 60 km/h, he will arrive 5 minutes late. If he drives at 90 km/h, he will arrive 5 minutes early. If he drives at n km/h, he will arrive exactly on time. What is the value of n?

1

COMC 2011 Problem 4

In the figure, AQPB and ASRC are squares, and AQS is an equilateral triangle. If QS = 4 and BC = x, what is the value of x? [asy] unitsize(16); pair A,B,C,P,Q,R,T; A=(3.4641016151377544, 2); B=(0, 0); C=(6.928203230275509, 0); P=(-1.9999999999999991, 3.464101615137755); Q=(1.4641016151377544, 5.464101615137754); R=(8.928203230275509, 3.4641016151377544); T=(5.464101615137754, 5.464101615137754); dot(A);dot(B);dot(C);dot(P); dot(Q);dot(R);dot(T); label("

A

", (3.4641016151377544, 2),E); label("

B

", (0, 0),S); label("

C

", (6.928203230275509, 0),S); label("

P

", (-1.9999999999999991, 3.464101615137755), W); label("

Q

", (1.4641016151377544, 5.464101615137754),N); label("

R

", (8.928203230275509, 3.4641016151377544),E); label("

S

", (5.464101615137754, 5.464101615137754),N); draw(B--C--A--B); draw(B--P--Q--A--B); draw(A--C--R--T--A); draw(Q--T--A--Q); label("

x

", (3.4641016151377544, 0), S); label("

4

", (Q+T)/2, N);[/asy]

1

COMC 2011 Problem 3

The faces of a cube contain the number 1, 2, 3, 4, 5, 6 such that the sum of the numbers on each pair of opposite faces is 7. For each of the cube’s eight corners, we multiply the three numbers on the faces incident to that corner, and write down its value. (In the diagram, the value of the indicated corner is 1 x 2 x 3 = 6.) What is the sum of the eight values assigned to the cube’s corners?

1

COMC 2011 Problem 2

Carmen selects four different numbers from the set

\{1, 2, 3, 4, 5, 6, 7\}

whose sum is 11. If

l

is the largest of these four numbers, what is the value of

l

1

COMC 2011 Problem 1

r

is a number such that

r^2-6r+5=0

(r-3)^2