IKMC 2006 - Grades 9-10 - Self-study

What is halfway between 2006 and 6002?

How many four-digit numbers with four distinct digits are divisible by 2006?

What is the least ten-digit number formed by concatenating the six numbers 309,41,5,7,68,2 in some order?

How many times from 00:00 to 23:59 does a digital watch display exactly the digits 2,0,0,6 in any order?

A flag has three stripes of equal width divided into two, three and four equal parts respectively. What fraction of the flag is grey?

Grandma’s watch gains one minute per hour; Grandpa’s loses half a minute per hour. I synchronize them on leaving and return when their readings differ by exactly one hour. How long until I return?

Ali says 25% of his books are novels and 1/9 are poetry. He has between 50 and 100 books. How many does he have?

A packet of Chocofruit costs 10 Rs and contains a coupon. Three coupons can be exchanged for another packet. How many packets can you obtain for 150 Rs?

Positive numbers a,b,c,d,e satisfy ab=2, bc=3, cd=4, de=5. What is $e/a$?

The star shown is formed from four equal circles of radius 5 cm, a square and four equilateral triangles. What is its perimeter?

A three-digit number ends in 2. Moving this last digit to the front reduces the number by 36. What is the sum of its digits?

Sara says: “If I live to one hundred, my present age is four thirds of half my remaining time.” How old is Sara?

The rectangle is divided into six squares. The smallest square has side 1 cm. What is the side of the largest square?

In the displayed addition each letter represents a different digit and each digit a different letter. What digit is G?

Raza correctly concludes: (1) If A is true then B is true. (2) If C is false then B is false. (3) If B is false then D and E are false. Exactly one answer is true. Which is it?

What is the maximum number of digits a number can have if every pair of consecutive digits is a perfect square?

A square of area $125\text{ cm}^2$ is divided into four squares and one L-shaped region of equal areas, as shown. Find the shortest side of the L-shaped region.

Three positive numbers sum to 20. The product of the two largest is always:

Knights always tell the truth and liars always lie. A wise man asks X about X and Y: “Are you both knights?” The answer is insufficient to identify them. He then asks X: “Are you of the same type?” and can identify both. What are they?

Two identical equilateral triangles, each with perimeter 18 cm, overlap with corresponding sides parallel as shown. What is the perimeter of the intersection hexagon?