IKMC 2027 - Grades 7-8 - Self-study SET 7 - Thematic Batch 3

Two squares of different sizes are drawn inside an equilateral triangle as shown. One side of one square lies on a side of the triangle. Find the shaded angle.

A heat wave in Dallol, Ethiopia lasts three months and rapidly evaporates a reservoir. In its first peak week, half the original water is lost; the second week loses another $1/6$ of the original amount, the third $1/12$, the fourth $1/20$, the fifth $1/30$, and so on. What fraction of the original water remains after the heat wave? Take one month as four weeks.

In chess a bishop can move any number of squares diagonally. Two bishops attack one another if they lie on the same diagonal, as in the example. What is the greatest number of bishops that can be placed on an $8\times8$ board with no two attacking?

Seven table-tennis balls numbered 1 through 7 are in a box. Hak first takes any three, then Ank takes two of the remaining balls. Neither knows the other's numbers. Looking only at his own three balls, Hak is certain that Ank's two numbers have an even sum. What is the sum on Hak's three balls?

In trapezium ABCD, M and N are the midpoints of BC and CD respectively. Given $BC=3AD$ and the trapezium's area is 32, find the area of triangle AMN.

A ship sails along a straight course past a lighthouse. Its light is visible within 17 miles. The ship's closest distance to the lighthouse is 8 miles. For how many miles of its journey can the captain see the lighthouse?

Three girls, Eva, Irene and Olga, and two boys, Adam and Yrban, play with a ball. A girl may throw it to a boy or another girl, but never back to the person who just threw it to her. A boy throws it to the other boy. Eva makes the fifth throw. Who received the ball from the first throw?

How many ways can the numbers 1, 2 and 3 fill the displayed $2\times2$ grid? The top-left cell is already 1. Cells sharing a horizontal or vertical side may not contain equal numbers.

John writes a number on each edge of a cube, then labels each vertex with the sum on its three incident edges. The labels at A, B, C, D, E, F and G are respectively 16, 17, 20, 26, 24, 22 and 14. What is the label at H?

Romeo writes the twenty-digit number $20202020202020202020$ on the board and divides it by 808. What is the remainder?

A rectangular block is cut from a large cube, leaving the shape shown. The original cube has volume $512\,\mathrm{dm}^3$. Find the surface area of the remaining solid.

The Ek family has more children than the Lind family, but the Lind family has more boys than the Ek family. Which statement about their numbers of girls must be true?

Rotate equilateral triangle RZT about Z to obtain equilateral triangle ZAC as drawn, with $\angle CZR=70^\circ$. Find $\angle CAR$.

A company is upgrading its warehouse. Cabinet A costs 1 million, holds 800 files and occupies $6\,\mathrm{m}^2$. Cabinet B costs 2 million, holds 1200 files and occupies $8\,\mathrm{m}^2$. The budget is 14 million and the free floor area is $72\,\mathrm{m}^2$. What is the greatest additional file capacity?

Two people each fire ten shots. Altogether they have more hits than misses. Their numbers of hits are in the ratio $3:5$. How many hits do they have in total?

Numbers in a $3\times3$ grid have been hidden by an ink spill. The row sums are 24, 26 and 40, and the first two column sums are 27 and 20, as shown. What is the third column sum?

The figure shows some dimensions of an unfolded rectangular prism. Find the volume of the original prism.

In the diagram $\angle BAC=\angle DBC$, $BD=CB=4$ and $AB=8$. Find AC.

The figure consists of 36 equal small triangles. What is the least number of such triangles that must be added to make a hexagon?

Editorial interpretation: Use the convex, straight-sided grid hexagon shown by the source solution.

The living-room clock is faulty. For every real hour it gains 36 minutes. Exactly one hour before it stops, it shows 8:24 a.m. It showed the correct time at 2:00 a.m. What time is it now?

Editorial interpretation: “Now” is the moment the clock stops.