IKMC 2016 - Grades 11-12 - Self-study

Tom and Johann’s ages sum to 23, Johann and Alex’s to 24, and Alex and Tom’s to 25. How old is the oldest?

Calculate $1/10+1/100+1/1000$.

Maria wants to build a bridge across a river. From every point along one shore, the shortest possible bridge to the opposite shore has the same length. Which illustrated river definitely cannot have this property?

How many integers are strictly greater than 2015×2017 and strictly less than 2016×2016?

A scatter plot in the xy-plane depicts the kangaroo shown. Swap the x- and y-coordinates of every point. Which illustrated picture results?

What is the minimum number of planes needed to enclose a bounded three-dimensional region?

Diana writes nonnegative integers in the nine circles so that each of the eight small triangles has the same sum at its three corners. What is the greatest possible number of distinct values?

Rectangles S₁ and S₂ in the coordinate diagram have equal area. Find x:y.

If $x^2-4x+2=0$, find $x+2/x$.

A circle has centre O and a tangent at P. A, O, B and X are collinear as shown. Arc AP has length 20 and arc BP has length 16. Find angle AXP.

Positive integers a,b,c,d satisfy $a+2=b-2=2c=d/2$. Which is greatest?

In the number pyramid, every cell is the product of the two cells directly below it. The bottom three cells contain integers greater than 1. Which number cannot be at the top?

If $x_1=2$ and $x_{n+1}=x_n^{x_n}$ for $n\ge1$, find $x_4$.

In rectangle ABCD, BC is half of diagonal AC. Point X lies on side CD and satisfies AX=XC. Find angle CAX.

Diana cuts a rectangle of area 2016 into 56 identical squares. The side lengths of the rectangle and squares are integers. How many noncongruent rectangles allow this?

The pictured square has perimeter 4. What is the perimeter of the equilateral triangle?

On an island, knights always tell the truth and liars always lie. Seven people sit in a circle around a bonfire. Every person says, “I am sitting between two liars!” How many liars are there?

Three three-digit numbers use the digits 1 through 9 exactly once in total. Which number cannot be their sum?

Each of the ten points in the diagram is labelled 0, 1 or 2. The corner sum of every small white triangle is divisible by 3; the corner sum of every black triangle is not. Three labels are given. What values can label the inner point “?”?

Bettina chooses five points A,B,C,D,E on a circle and draws the tangent at A. The five marked angles x are equal. The figure is not to scale. Find angle ABD.