IKMC 2027 - Grades 5-6 - Self-study SET 9 - Thematic Batch 3

Mr Beaver has a field of area 6700 m², shaped as shown. He grows wheat in the black overlap. Which listed length could be the width of that wheat-growing region?



Diagram conventions: width means the shorter overlap side; both overlap dimensions are strictly less than 60 m, as the partially shifted squares show.

In 2018 Alice notices that the sum of the year’s digits, 2+0+1+8=11, is prime. What is the smallest number of years after 2018 for which this happens again?

After school, Nikita and Sasha decide to run around the school football field. They start together and run in the same direction. Nikita completes a lap in 3 minutes and Sasha in 4 minutes. When will they first meet again after starting?



Starting convention: both begin at the same point at the same instant.

The six tips of the star shown are the midpoints of the sides of the surrounding regular hexagon. If the star’s area is 6 cm², what is the hexagon’s area?

Alice, Ben, Cherry and David each have a different favourite pet: a cat, a dog, a goldfish or a bird. Ben’s pet is furry; David’s has four legs; Cherry has the bird; and neither Alice nor Ben likes cats. Which statement below is incorrect?

Five standard dice are stacked vertically, each with its 4-dot face upwards. Opposite faces sum to 7. How many dots are visible around the outside of the stack?



Visibility convention: count all four lateral sides and the exposed top face; the bottom rests on a surface and is hidden.

This morning, sisters Susan and Maggie decide to compare two different walking routes to school. The diagram shows their routes. Which sister takes the longer route?

Ant Anne starts at vertex A of the wire pyramid shown. She follows edges and returns to A, using each edge at most once. For example, A→O→I→A and A→U→E→I→A qualify. Reverse routes such as A→O→I→A and A→I→O→A count as different. How many routes are possible?



Original source choices (retained):

A. 8

B. 12

C. 16

D. 24

E. 32

Source consistency task: count routes that stop at their first return to A, then examine what changes if later returns are allowed while every edge is still used at most once. Explain which interpretation gives the printed key D.

Three identical dice are rolled. How many possibilities give a total of 9 dots?



Counting convention: permutations of the same three face values are treated as one possibility.

Kiên has filled five of the ten circles shown. He wants the three numbers on every side of the pentagon to have the same sum. What number must go in the circle marked x?

Every cell of a 5×5 board contains one ladybug. At a signal, each moves diagonally to a cell sharing a corner with its starting cell, as illustrated. Some cells may receive several ladybugs and some become empty. What is the greatest possible number of empty cells?



Original source choices (retained):

A. 1

B. 3

C. 5

D. 6

E. 7

Source consistency task: solve the literal maximum problem, give a bound and construction, and explain why the printed source choice C=5 does not answer it. The original stem and choices are retained.

A garden has 15 plots. Coffee is grown on 7 plots, pepper on 10 and avocado on 4; plots may contain several crops or other crops. What is the greatest number of plots that can contain all three named crops?

Yesterday, Susan spent one hour completing all her homework. One third of that time is for mathematics, then two fifths of the remaining time is for geography. How many minutes does she spend on other subjects?

At a sea festival, octopuses make up 1/5 of the guards, crabs 1/4 and turtles 1/6; the rest are rays. The total is less than 100. How many rays are there?

In Kangcity, IKMC is held on the third Sunday of September. In 2020 it was on 20 September. On what September date will the 2021 contest take place?

Peter loves Rock music and the Kangaroo mathematics competition. He enjoys making rules that assign digits to letters. Under his rule, ROCKGANG corresponds to 76243153. What digit sequence corresponds to KANGAROO?

A baseball breaks Mr Kaminari’s window. Exactly one of Nobita, Gian and Suneo caused it. Nobita says, “I did not hit that ball.” Gian says, “I did not break the window.” Suneo says, “Nobita did it.” Exactly two children are lying.

1. If Nobita did it, how many statements are false?
2. If Gian did it, how many are false?
3. If Suneo did it, how many are false?
4. Which assumption gives exactly two liars, and who broke the window?

A bee moves through a grid of regular hexagonal tiles according to the path shown. Which labelled cell will it enter next?



Pattern convention: repeat the displayed block of two left turns followed by three right turns, each 60 degrees.

A tree holds 13 magpie-robins and 21 tailorbirds. Half of all the birds fly away to feed. What is the greatest number of tailorbirds that can remain?

Hippos Tata, Titi and Toto walk in that order: Tata first, Titi second, Toto last. Tata is 500 kg heavier than Titi, and Toto is 1000 kg heavier than Titi. Which picture correctly shows their walking order?