IKMC 2027 - Grades 5-6 - Self-study SET 3 - Thematic Batch 3

Forty children take part in two adventure rides at Discovery Park. Twenty-five ride the Tornado and eighteen go parasailing. How many children take part in both rides?



Figure labels: Lốc xoáy = Tornado; Dù bay = parasailing.

Two tanks initially contain 100 litres and 120 litres of water. Their bottom outlets drain at 1 litre per hour and 3 litres per hour, respectively, as shown. If both outlets are opened together, after how many hours will the remaining volumes be equal?



Editorial clarification: Find the first time the remaining volumes are equal.

Five people are in a room. Each always tells the truth or always lies. Four leave, one after another, and each says after leaving, “There are more liars than truth-tellers left in the room.” How many truth-tellers were there initially?

The picture shows the distant skyline of a castle. Which of the pieces below does not occur in that skyline?

Start with a 3 × 3 grid filled with 1s. At each step, select a 2 × 2 region and add 1 to all four entries in it. Which of the following grids can result after several steps?



Source correction for this exercise: Replace the initial value 1 by 0 in every cell. As originally printed with 1, none of A–E is possible. Answer the corrected version; the original wording is retained above for provenance.

An open rectangular box measures 5 × 6 × 7. Its base is 5 × 6, two side walls are 5 × 7, and the other two are 6 × 7. It is filled by 210 unit cubes of size 1 × 1 × 1. How many cubes have a face touching one of the enclosing faces of the box?



Editorial clarification: The listed box has a base and four side walls; no lid is included.

Tom has four cards bearing 1, 3, 6, 7. How many four-digit numbers can he arrange using the cards?



Editorial clarification for the intended source answer: The 6 card may be turned upside down to show 9. Without this convention the answer is 24.

Equilateral triangle ABC has area 27. Parallel lines divide each side into three equal parts, as shown. What is the area of the white region?

Sam uses at most 50 identical solid square tiles to form a rectangle with no holes. The number of tiles on the outer border equals the number entirely inside. How many tiles does she use?

In the diagram, AC=10 m, BD=15 m and AD=22 m. How long is BC in metres?

In a laboratory, a beneficial bacterium described in the source as countering Covid-19 is cultured in favourable conditions. In one hour it produces two new bacteria and then stops reproducing. In the next hour, each newly born bacterium produces two more and then stops reproducing. This process repeats, and all bacteria are assumed to live forever. How many bacteria are there after four hours?



Editorial note: This is a fictional growth model. The source’s Covid-19 claim is not validated medical information.

Tom has two analogue clocks. One gains 30 minutes each day and the other loses 18 minutes each day. He sets both correctly. If he never resets them, after at least how many days will both again show the correct time?



Editorial clarification: Read ordinary 12-hour clock faces, without date or a.m./p.m. indicators.

Peter lists the positive integers from 1 to 2006. He underlines each multiple of 2, then each multiple of 3, and finally each multiple of 4. How many numbers are underlined exactly twice?

The outline of a mound is shown, with a frog initially at the topmost M. It repeatedly jumps clockwise over one vertex to the next, until its visited-letter sequence contains “MATH” 2020 times. How many complete circuits around the mound has it made by then?



Editorial clarification: Count complete circuits; include the starting M in the visited-letter sequence.

Tom can take the school bus each day. Walking to school and returning by bus takes 50 minutes altogether. Taking the bus both ways takes only 20 minutes. How long would it take to walk both ways? Assume both the bus and Tom move at constant speeds.

At 6:30, a cyclist travelling at 12 km/h and a three-wheeled motorcyclist travelling at 28 km/h set off from A towards B. Half an hour later, a motorcyclist travelling at 24 km/h sets off on the same route. At what time is the motorcycle exactly halfway between the bicycle and the three-wheeled motorcycle?

During a break, Quantik invites some pupils to take anniversary photographs at three locations. Eighteen are photographed at the school gate, twenty in the tradition room, and fourteen on the sports field. Tom says ten pupils appear at all three locations, eight other pupils at exactly two locations, and all the remaining pupils at exactly one. How many pupils take part?

A magician’s hat contains 12 grey, 16 white and 20 yellow rabbits. What is the smallest number the magician must take out to be certain of having at least two rabbits of every colour?

Bin says that three squares drawn on paper can intersect to create other shapes. When three identical squares are drawn, what is the greatest total number of outlined squares, of whatever size, that can occur?



Source error: The printed key C (7) is contradicted by an arrangement with 8 outlined squares. The original question and choices are retained for review; do not treat the printed key as verified. Count all outlined squares, allowing equal side lengths.

Original choices: A. 3; B. 5; C. 7; D. 9; E. 11.

Editorial self-study task: Give a configuration of three congruent squares that disproves the printed claim that the maximum is 7. List the squares in your configuration and explain why this does not, by itself, determine the true maximum.

Four positive integers have the property that the difference of any two is divisible by 5. What remainder can their product leave on division by 5?