IKMC 2011 - Grades 11-12 - Paper 02 - Self-study

Write a number at every dot of the lattice so that the sum at the two ends of each line segment is always the same. Two numbers, 4 and 1, are given. What number belongs at x?

Michael, Fernando and Sebastian begin a race in that order. Michael and Fernando change places 9 times, Fernando and Sebastian 10 times, Michael and Sebastian 11 times. In what order do they finish?

Given $2^x=15$ and $15^y=32$, find xy.

Jan sketched his village inaccurately, but the relative positions of the houses and street crossings are correct. Three roads are straight; only Quirkwik Street is curved. Who lives on Quirkwik Street?

All four-digit numbers with digit sum 4 are listed in descending order. What is the position of 2011?

The shape consists of a regular hexagon of side 1, six squares and six equilateral triangles, as shown. What is its perimeter?

A rectangular sheet is wrapped around a cylinder. A plane cuts the cylinder and paper through X and Y as shown. The bottom part of the paper is unwrapped. Which picture could result?

Quadrilateral PQRS satisfies PS=SR, angles PSR=PQR=90°, ST perpendicular to PQ, and ST=5. Find its area.

Editorial note: the source prints an infinity-like symbol between ST and PQ. The right-angle mark in the figure establishes ST perpendicular to PQ; that is the condition used here.

Andrew writes all odd numbers from 1 through 2011 on a board. Bob erases all multiples of 3. How many numbers remain?

Max and Hugo roll some fair independent dice. With no sixes Max jumps first into a cold lake; with exactly one six Hugo jumps first; with at least two sixes neither swims. How many dice give them equal chances of jumping first?

Three rectangles are joined without gaps or overlap into one rectangle. Two have dimensions 7×11 and 4×8. The third is chosen to have the greatest possible area. What are its dimensions?

Fill the five empty cells of the 3×3 grid with nonnegative integers so every 2×2 square has sum 10. Four entries are given. Which listed value could be the sum of the five missing entries?

Forty-eight children go skiing. Six have exactly one sibling on the trip, nine have exactly two siblings, and four have exactly three siblings. The others have no siblings there. How many families are represented?

How many of the graphs $y=x^2$, $y=-x^2$, $y=\sqrt x$, $y=-\sqrt x$, $y=\sqrt{-x}$, $y=-\sqrt{-x}$, $y=\sqrt{|x|}$, $y=-\sqrt{|x|}$ are included in the figure?

A rear-window wiper has blade RW and rod OR of equal length, joined at fixed angle α. It rotates about O and clears the shaded area. Find angle β between the right-hand edge of the cleared area and the tangent to its upper curved edge.

Three horizontal lines and three parallel slanting lines are drawn. Each circle is tangent to four lines. X,Y,Z are the shaded areas; D is the area of parallelogram PQRS. What is the smallest number of areas among X,Y,Z,D that must be known to calculate the indicated area T?

On a standard coordinate plane, A=(1,−10) was marked on $y=ax^2+bx+c$. The axes and most of the parabola were erased, leaving the shown part. Which statement can be false?

All six sides AB,BC,CD,DE,EF,FA of a hexagon are tangent to one circle. The first five side lengths are 4,5,6,7,8 respectively. Find FA.

Different letters stand for distinct nonzero digits, and repeated letters have the same digit. What is the smallest positive integer value of $\dfrac{K\cdot A\cdot N\cdot G\cdot A\cdot R\cdot O\cdot O}{G\cdot A\cdot M\cdot E}$?

The brothers Gerhard and Günther truthfully describe their chess club. Gerhard says all members except five girls are boys. Günther says every group of six members includes at least four girls. How many members are there?