IKMC 2007 - Grades 9-10 - Self-study

Asif, Usman and Sami have 30 balls altogether. Usman gives Sami 5, Sami gives Asif 4, and Asif gives Usman 2. They then have equal numbers. How many did Asif have initially?

Five faces of the two dice are visible. What is the sum of spots on their seven invisible faces?

Winning tickets have at least five digits and at most three digits larger than 2. The drawn tickets are 1022,22222,102334,213343,3042531. How many win?

In triangle ABC, D is the midpoint of AB, E is the midpoint of DB, and F is the midpoint of BC. The area of ABC is 96. What is the area of AEF?

Meera Jamal puts 2007 balls equally into bags A, B, C. She moves two thirds of the balls from A to C. What is the resulting ratio A: C?

A committee has 32 members and grows by 50% each year relative to the previous year. How many members will it have in three years? Source typo: “numbers” means members.

How many shortest routes can a king take from the top-left square to the bottom-right square of the grid? It may move to any adjacent square, including diagonally.

Points A (2006,2007), B (2007,2006), C (−2006, −2007), D (2006 – 2007), E (2007, −2006) are marked. Which segment is horizontal? Source note: D lacks a coordinate separator; the intended pair is D (2006, −2007).

Different letters denote different digits 0–9. Find the least value of 2007−KAN−GA−ROO.

Two lines from each of A and B to the opposite sides divide triangle ABC into nine regions as shown. If four lines are drawn from each of A and B, how many regions result?

Twelve islanders include both liars (always lying) and nobles (always truthful). Two say “Exactly two of us are liars”, four say “Exactly four”, and six say “Exactly six”. How many liars are there?

To obtain $8^8$, to what power must $4^4$ be raised?

Seven consecutive natural numbers sum to 105. How many are divisible by 5?

The number of boys who solved a problem equals the number of girls who did not. Which group is larger: all solvers or all girls?

A 10 m rope is fastened to a corner of a 6 m by 4 m house as shown. A dog is at the other end. Find the perimeter of the accessible area.

At 9:00 AM I drive at 100 km/h. At this speed my fuel suffices for 80 km, but the nearest station is 100 km away. Fuel used per kilometre is inversely proportional to speed. I want to arrive as soon as possible. When can I arrive?

KANGAROO is repeated 20 times. Delete all letters in odd positions, then repeat on the remaining sequence until one letter remains. Which letter?

A mother is 21 years older than her five-year-old son. After how many years will he be one third of her age?

Two schools have five table-tennis players each. Only doubles are played; every pair from one school plays every pair from the other exactly once. How many matches does each student play? Source ends with the incomplete phrase “Each student should play.”

Every row and column must contain two red (R) and two green (G) squares. What are X and Y, in that order?