IKMC 2015 - Grades 11-12 - Self-study

Andrea was born sometime in 1997 and her sister Charlotte sometime in 2001. What is certainly true of their age difference?

Calculate $(a-b)^3+(b-a)^3$.

How many real solutions does $2^{2x}=4^{x+1}$ have?

Diana’s bar chart shows counts of four types of trees on a biology trip. Heinz suggests a pie chart to show the same proportions. Which of the illustrated pie charts is correct?

Add all integers from 2001 to 2031 inclusive, then divide the sum by 31. What is the result?

How many of the four illustrated shapes can be drawn with one continuous line, without lifting the pencil or tracing any line twice?

A quadrilateral is convex when every interior angle is less than 180°. If n is its number of right angles, which list contains all possible values of n?

A drinking glass has the shape of a truncated cone. Its outside, excluding the upper and lower circles, is to be covered completely with coloured paper without overlap. Which cut-out shape is needed?

The diameters of three semicircles are the sides of a right triangle. Their areas are X, Y and Z square centimetres as shown. Which relation is necessarily true?

A square sheet is folded along the dashed lines, in some order and direction, to form the small square shown. One corner of that small square is cut off. After unfolding, how many holes are in the interior of the original sheet?

Calculate $\sqrt{\begin{gathered}(2015+2015)+(2015-2015)\\{}+(2015\times2015)+(2015\div2015)\end{gathered}}$.

The x-axis and graphs $f(x)=2-x^2$, $g(x)=x^2-1$ divide the coordinate plane into how many regions?

Ella wants to put a number in every circle so each equals the sum of its two direct neighbours. The numbers 3 and 5 are given. What must go in the circle marked “?”?

Five distinct positive integers a,b,c,d,e satisfy $b=c\div e$, $d=a+b$, and $a=e-d$. Which is largest?

The geometric mean of n positive numbers is $\sqrt[n]{x_1x_2\cdots x_n}$. Six numbers are split into two groups of three with geometric means 3 and 12. What is the geometric mean of all six?

The figure shows three concentric circles and two perpendicular common diameters. The three grey sections have equal areas. The smallest circle has radius 1. What is the product of all three radii?

Over 20 years, Arnberg’s population increased by 40% and Berghausen’s by 60%. Their combined population increased by 54%. What was the ratio of their populations 20 years ago?

Bibi rolls a fair die labelled 1,2,3,4,5,6. Tina independently rolls a fair die labelled 2,2,2,5,5,5. Tina wins only if her number is greater. What is her winning probability?

Marbles are numbered 1 to 2015. Two marbles have the same colour exactly when their digit sums are equal. How many colours occur?

Opposite faces of a standard die sum to 7. Two identical standard dice are shown. How many dots can be on the hidden right-hand face marked “?”?