IKMC 2027 - Grades 7-8 - Self-study SET 14 - Thematic Batch 3

Peter wants to invite his seven friends A,B,C,D,E,F,G to dinner, but his table seats only four people. With Peter present, he can invite three friends per dinner. He wants every pair of friends to dine with him together at least once. What is the fewest dinners he must arrange?

The solid shown consists of four small cubes. Each small cube has total surface area 24 cm². What is the total surface area of the solid?

In triangle ABC, AB = AC. Choose E on AC and D on BC so that AD = AE. Given $\angle BAD=30^\circ$, find $\angle CDE$.

Evaluate $2^5+0^1+1^0+5^2$.

Ann replaces the letters in KAN − GA + ROO with digits, equal letters receiving equal digits and different letters different digits. If GA is a two-digit number with both digits nonzero, what is the largest possible value?

Twelve cubes are arranged in a row: three blue, two yellow, three red and four green, in an unspecified order. One end cube is yellow and the other is red. All red cubes are consecutive, and all green cubes are consecutive. The tenth cube is blue. What colour is the sixth cube?

What is the units digit of $2^{36}-1$?

F and G are the midpoints of BC and BE, respectively. Which listed path from A to D is shortest?

A two-minute sand timer consists of two identical cubes. When all its sand is in one cube, that cube is half full. After sand has flowed for 20 seconds, the layer in the lower cube is 1 cm deep. What is the total sand volume?

Editorial model: Sand flows at constant volume per second, starts entirely in the upper cube, and forms a level layer. A dot between numbers in the original choices means multiplication.

The grey area is 18 square units. What is the area of parallelogram ABCD?

A rectangle’s length is increased by 60% and its width by 40%. By what percentage does its area increase?

The beavers want to connect seven islands to the coast by bridges, as shown. Dashed lines are possible bridges, and the adjacent numbers give the number of trees required. They must make every island reachable from the coast. What is the least number of trees needed?

Evaluate $\frac{1\cdot2\cdot3+2\cdot4\cdot6+4\cdot8\cdot12+7\cdot14\cdot21}{1\cdot3\cdot5+2\cdot6\cdot10+4\cdot12\cdot20+7\cdot21\cdot25}$.

Original printed choices: A) $\frac15$; B) $\frac25$; C) $\frac35$; D) $\frac45$; E) $\frac65$

Editorial task: Evaluate the expression as printed, then explain the discrepancy with the source’s intended factorization.

Andrew, Boris, Victor and David are in Grades 1,2,3,4, one in each grade, in an unspecified order. Boris is in neither Grade 1 nor Grade 4. Victor is not in Grade 4, and Andrew is in Grade 2. Who is in Grade 3?

How many four-digit numbers have digits that are consecutive integers decreasing from left to right?

Editorial reading: The source phrase “digits with four digits” means four-digit numbers.

A train has five distinct coaches A,B,C,D,E. In how many orders can they be arranged so that coach A is always closer to the engine than coach B?

Writing 2 February 2020 as 02.02.2020 uses four zeros and four twos. How many dates can be written using exactly these eight digits?

Editorial convention: Use DD.MM.YYYY, with a genuine four-digit positive year (1000–9999), as in the source’s list; leading zeros in the year are excluded.

One litre of lemonade contains 10% fruit extract, 15% sugar and 75% water. Add a quarter litre consisting of 50% water and 50% fruit extract. What percentage of the resulting mixture is sugar?

A strong southerly wind is blowing. An aircraft flies from Amsterdam to Paris at 700 km/h in 36 minutes. Another aircraft flies from Paris to Amsterdam at 900 km/h. How long does that flight take?

Editorial interpretation: The stated 700 and 900 km/h are speeds along the route relative to the ground; the wind does not require an extra unknown correction.

The figure contains an overlapping equilateral triangle and regular pentagon. What is x, in degrees?