IKMC 2027 - Grades 7-8 - Self-study SET 9 - Thematic Batch 3

Mike has 125 small cubes. He builds the large cube shown, with nine tunnels through it. How many small cubes are not used?

Daisy has four chocolate sweets, two vanilla sweets and one ginger sweet. She wants to eat three on one weekend day and four on the other. How many ways can she distribute the sweets between the two days?

Interpretation: sweets of the same flavour are indistinguishable.

What time on Monday is 300 minutes after 22:00 on Sunday?

Exactly one of three closed rooms contains a lion. Door 1 says “This room contains a lion”. Door 2 says “This room does not contain a lion”. Door 3 says “2+3=2×3”. Exactly one statement is true. Which room contains the lion?

Let $S=\frac{\sqrt2-1}{\sqrt2}+\frac{\sqrt3-\sqrt2}{\sqrt6}+\cdots+\frac{\sqrt9-\sqrt8}{\sqrt{72}}$. Find S.

In the picture a monkey stands in a long corridor. At the monkey's position, Ank measures the pictured window height as 5 cm and the pictured monkey height as 6.45 cm. The real window is 1 m high. How tall is the monkey?

The perimeters of triangles ADF, BDE, DEF and FEC are respectively 12, 24, 19 and 24, as shown. Find the perimeter of triangle ABC.

Jant wants to place the shown L-shaped piece and plus-shaped piece on the shown board. What is the least number of dots that can be covered?

Interpretation: pieces follow the grid, may be rotated, and do not overlap.

Costi and an expedition search for treasure using a map. It shows two villages at the foot of a strangely shaped mountain, with lengths marked. The treasure is exactly halfway between the villages. Costi must find the distance between them.
After studying the map, Costi labels the points as shown. Extend rays BE and AC until they meet at F.

1. What kind of quadrilateral is CDEF? Hence find EF and CF.

2. Find AF and BF.

3. What kind of triangle is AFB? Hence find AB.

4. Find the distance from the treasure to each village.

In the cube, consider the marked angles α, β, γ and δ. Find α+β+γ+δ.

The figure shows the first four isosceles right triangles in a sequence continued by the same rule. Given AB=AC=1, find the hypotenuse of the 2014th triangle.

Electricity prices rise by 12%, then by 10% the next month, and then fall by 22%. What is the ratio of the latest price to the original price?

The large cube is made from eight identical small cubes, as shown. The total surface area of the eight separate small cubes exceeds the surface area of the large cube by 384 cm². Find the edge length of a small cube.

How many trailing zero digits does $20^{14}$ have?

Here-and-There offers a holiday trip to Corsica for EUR 850. In September the trip costs EUR 765. By what percentage is the September trip cheaper than the holiday trip?

Ank has fifteen straight sticks of lengths 1 cm, 2 cm, 3 cm,...,15 cm. What is the greatest number of congruent equilateral triangles he can make using every stick without breaking any?

Interpretation: separate triangles, no shared sticks; sides may be assembled end to end.

Joe lives on the mainland and John on an island 2 km away. Joe takes 50 minutes to row to the island; John takes 40 minutes to row to the mainland. They start simultaneously at their usual speeds. How far apart are they after 20 minutes?

Interpretation: they row towards one another along the same straight 2 km route.

How many decimal digits are in $(2^{31})^{65}(5^{65})^{31}$?

A school has 2017 lockers numbered 1 to 2017. Fox erases every digit 7 from their labels. The principal replaces every locker whose label has lost at least one digit. How many lockers are replaced?

The pair 18 and 54 has its sum (54+18=72) equal to twice its difference (54−18=36). How many pairs of numbers, both less than 100, have this property?

Original printed choices: A) 1; B) 5; C) 25; D) 33; E) 49

Editorial task: Identify the missing domain and distinguish the literal and intended counts.