IKMC 2027 - Grades 9-10 - Self-study SET 8 - Thematic Batch 3

Marcus and Oscar have solid homogeneous spherical marbles. Oscar has five of diameter 2 cm, while Marcus’s marbles have diameter 1 cm. Their total marble masses are equal. How many marbles does Marcus have?

Julia cuts a rectangular card of perimeter 30 cm along one horizontal and one vertical line into four pieces as shown. One piece is a square of area 9 cm². What is the perimeter of rectangle ABCD?

Alex writes 25×2=211. He says that changing every digit by exactly one, either increasing or decreasing it, can make the equation true. How many digits must be increased by one?

Tom multiplies a two-digit number by a one-digit number, obtaining a three-digit product. Across the three numbers, each digit 1 through 6 occurs exactly once. What is the one-digit factor?

On Sunday, David visits a farm 72 km from home. On the outward journey, the car travels 12 km per litre of fuel; on the return it travels 18 km per litre. What is the average number of kilometres travelled per litre for the round trip?

Alex labels the five vertices of a quadrilateral pyramid with 1,2,3,4,5, each used once. He adds the vertex labels on each face. Four face sums are 7,8,9,10. What is the remaining sum?

How many positive integers have digit sum 2020 and digit product 4?

In the diagram, identical letters stand for the same digit. What is the result of the calculation on the right?

Mr David says about his two children: “Their ages, added together and then added to the product of their ages, give 14.” How old are his children this year?

A digital clock uses 24-hour notation and currently shows 23:32. The four digits form a palindrome, reading the same forwards and backwards. What is the shortest time until it next shows a palindrome?

How many ordered pairs of positive integers (m, n) satisfy mn=3(m+n)?

Tommy randomly chooses a nine-digit positive integer with distinct nonzero digits. What is the probability that it is divisible by 18?

Which expression equals $2^{3x}3^{2x}$?

What is the sum of the two marked angles in the diagram? The five star tips are the vertices of a regular pentagon.

Twelve real numbers are split into groups A, B, C containing three, five, four numbers, respectively. The mean of all numbers in A and B together is 40; the mean of B is −14; and A and C have equal means. What is the mean of all twelve numbers?

How many positive integers n satisfy that 3 n²−3n+20 is divisible by n−1?

A printer produces 24 pages in 90 seconds. At the same rate, how many pages can it produce in five minutes?

Sven loves the digit 7. He represents 0 by “7”, 1 by “77”, 2 by “777”, and so on. Whenever seven consecutive 7s occur, he replaces the last with S. For example, 16 is written “777777S777777S777”. How many S characters are used to write 2022?

A pair of binoculars originally costs 128 monetary units. Last month the shop applied three successive discounts of 25% each. What is the current price?

Tom uses 1×3 and 1×5 rectangular tiles to make five strips, each one square high, of lengths 10,12,16,18,20 squares. He may leave no excess pieces and may not overlap tiles. What is the minimum total number of 1×3 tiles needed?