IMSO 2009 Mathematics - International Short Answer

A big square consists of 25 small squares, some black. What is the least number of additional squares to colour black so that the pattern is symmetric about both diagonals?

If $A\star B=\frac{A+2B}{3}$, find $(2\star3)\star(4\star5)$.

From November 8 through November 14 inclusive, an athlete eats 154 eggs. Each day he eats 6 more than the previous day. How many eggs does he eat on November 13?

Calculate $\frac{3+6+9+12+\cdots+75}{4+8+12+16+\cdots+100}$.

Find the shaded area in the 5 by 7 unit grid.

Danny cuts 4 cm × 6 cm cards from a 21 cm × 29 cm sheet. What is the greatest number of cards?

Mrs Anna has four children: Alex, Brad, Christine and Dennis. Alex is not the youngest, but is younger than Dennis. Brad’s age is the mean of Alex’s and Dennis’s ages. Who is oldest?

A correct answer earns 5 points and a wrong answer earns −2 points. Tom answers all 35 questions and scores 140. How many answers are correct?

The boundary uses only horizontal and vertical segments. Find its perimeter from the labelled lengths.

Use the digits 2, 3, 4, 5, 7, 8 exactly once to make three-digit numbers P and Q. Find the smallest positive P−Q.

The natural numbers greater than 1 are arranged in five columns as shown. In which column and row does 2009 appear?

The number 8 has two properties: 8+1=9 is a square, and 8/2=4 is a square. Find the next natural number after 8 with both properties.

ABCD is a trapezoid with AB parallel to CD and AB:CD=3:1. P lies on CD. Find the ratio of the area of triangle APB to the area of trapezoid ABCD.

If I put 9 marbles in each filled box, two boxes remain empty. If I put 6 in every box, three marbles remain. How many boxes are there?

Arrange 1,1,2,2,3,3,4,4,5,5 as shown. Every interior number is the sum or the absolute difference of its two neighbours. Find X.

A football team earns 3 points for a win, 1 for a draw and 0 for a loss. After 20 matches it has 53 points. What is its least possible number of losses?

Of eight points in a plane, five lie on one line. What is the greatest number of triangles whose vertices are chosen from the eight points?

Place the digits 0 through 9 once each in the ten squares. Each arrowed line has sum 20; 8 and 4 are fixed. Find the question-mark digit.

ABCD is a square with AT=TD, DS=SC, CR=BR and AP=PQ=QB. Using the nine marked points as vertices, how many different right triangles have at least one of P,Q,R,S,T as a vertex?

ABCDE is a five-digit positive integer. The number ABCDE1 is three times 1ABCDE. Find ABCDE.