IKMC 2009 - Grades 11-12 - Paper 02 - Self-study

There are 200 fish in an aquarium. 1% of them are blue; all the rest are yellow. How many yellow fish must be removed so that blue fish represent 2% of all the fish remaining?

Which of the following numbers is the largest?

For how many positive integers $n$ is $n^2+n$ prime?

Mari, Ville and Ossi go to a café. Each buys three glasses of juice, two ice creams and five biscuits. Which amount could be their total bill?

Pricing convention: each item price is an integer number of cents; the three identical orders are charged equally.

The solid shown has six triangular faces and five vertices. A number is assigned to each vertex. The sums of the three vertex numbers on all faces are equal. Two labels, 1 and 5, are shown. What is the sum of all five labels?

Two circles have centres $F,G$ and radii 13 and 15. They intersect at $P,Q$, where $PQ=24$. Which of the following could be $FG$?

A box contains 2 white, 3 red and 4 blue socks. One third of all socks have holes, but Liz does not know their colours. She draws socks randomly. What is the smallest number she must draw to guarantee two undamaged socks of the same colour?

The square in the figure has side length 1. What is the radius of the small circle?

Extend the sides of triangle $ABC$ in both directions to $P,Q,R,S,T,U$, as shown, so that $PA=AB=BS$, $TC=CA=AQ$, and $UC=CB=BR$. If the area of $ABC$ is 1, what is the area of hexagon $PQRSTU$?

Colour the squares of the grid with A, B, C and D so that neighbouring squares have different colours. Squares sharing only a vertex also count as neighbours. Some colours are given. Which colours are possible for the shaded square?

A small ball starts at corner A of a square billiard table of side 2 m. It follows the path shown, reflecting off three sides, and reaches corner B. At each bounce, the angle of incidence equals the angle of reflection. What is the total path length in metres?

There are 2009 kangaroos, each light or dark. One light kangaroo is taller than exactly 8 dark kangaroos, one than exactly 9, one than exactly 10, and so on, with exactly one taller than all dark kangaroos. How many are light?

Intended reading: this consecutive list accounts for all light kangaroos.

The illustrated $2\times2\times2$ cube contains four transparent unit cubes and four opaque black unit cubes, arranged so no straight line of sight passes through from front to back, left to right, or top to bottom. What is the minimum number of black unit cubes needed to make a $3\times3\times3$ cube opaque in the same three directions?

On an island, nobles always tell the truth and liars always lie. Twenty-five people stand in a queue. The first says that everyone behind him is a liar. Every other person says the person immediately in front is a liar. How many liars are in the queue?

What is the units digit of $1^2-2^2+3^2-4^2+\cdots-2008^2+2009^2$?

An equilateral triangle of side 3 and a circle of radius 1 have the same centre. What is the perimeter of their union, shown in the figure?

The graphs of real functions f and g are shown. What relation connects them?

One hundred contestants answer four problems. Respectively, 90, 85, 80 and 70 contestants solve problems 1, 2, 3 and 4. What is the smallest possible number who solve all four?

The figure gives the front view (f.v.) and top view (t.v.) of a solid with flat faces. Which of figures I–IV can be its view from the left?

Editorial correction: the Austria source prints “bird’s-eye view” below the upright trapezoid and “front elevation” below the parallelogram. These captions are exchanged relative to the intended construction. The original is retained; for this scored question use the upright trapezoid as FRONT and the parallelogram as TOP. Scoring is conditional on this correction.

The numbers in a $3\times3$ magic square have equal sums in every row, column and diagonal. Only 47 and 63 are visible in the indicated positions. What must the upper-left entry $a$ be?