IKMC 2008 - Grades 11-12 - Self-study

Numbers 3, 4 and two unknown numbers fill a 2 × 2 table. The row sums are 5 and 10; one column sums to 9. What is the larger unknown?

If $x+y=0$ and x ≠ 0, find $x^{2008}/y^{2008}$.

An array has columns numbered 1 to 21 and rows numbered 1 to 33. Erase rows whose number is not a multiple of 3 and columns whose number is even. How many cells remain?

How many primes $p$ also make $p^4+1$ prime?

A river starts at A. It splits into a branch carrying 2/3 of its water and a branch carrying the rest. The first branch later splits into three carrying 1/8, 5/8 and the rest of that branch. The last of these rejoins the other original branch, as shown. What fraction of the original water flows at B?

In isosceles triangle ABC, CA = CB. Point D lies on AB with AD = AC and DB = DC. Find angle ACB.

Find the maximum of $|5\sin x-3|$ for real $x$.

A circle has diameter AB, where A = (−2, 0) and B = (8, 0). Its upper point D = (0, d) is shown. Find d.

Five distinct points A₁, A₂, A₃, A₄, A₅ lie in this order on a line, with possibly unequal gaps. Where on that line must P lie to minimize PA₁ + PA₂ + PA₃ + PA₄ + PA₅?

Shaheen fills the two blanks in 2 _ _ 8 with digits so that the four-digit number is divisible by 3. How many possibilities are there?

Six of the numbers −9, 0, −5, 5, −4, −1, −3 are arranged into three pairs with equal sums. Which number remains?

Every cube in the figure has edge length 1. Find the length AB.

Five competition problems have different positive integer point values. Nizami solved all five. The two lowest values total 10; the two highest total 18. How many points did Bill obtain? (The source changes the name here.)

Ayesha drew 36 kangaroos using yellow, brown and black. Exactly 25 contain yellow, 28 brown, 20 black, and 5 contain all three colours. How many are single-colour?

Three circles of radius r touch externally as shown. Find the area of the central region A.

The two regular hexagons shown are congruent. What fraction of the parallelogram is shaded?

A fraction has negative numerator and denominator. Its numerator is one greater than its denominator. Which statement is true?

If $x^2yz^3=7^3$ and $xy^2=7^9$, find $xyz$.

Three distinct points are selected uniformly at random from the grid shown. What is the probability that they are collinear?

Four identical dice are arranged in a row as shown. Each has faces 1 to 6, but opposite faces need not sum to 7. Find the sum on all six touching faces.