IMOmath

Multiple choice practice test

1. (17 p.)
The operation \( \circ \) is defined on the set of real number as \( a\circ b=(a-b)^2 \). What is \( (x-y)^2\circ(y-x)^2 \)?

   A    0

   B    \( x^2+y^2 \)

   C    \( 2x^2 \)

   D    \( 2y^2 \)

   E    \( 4xy \)

   N   

2. (51 p.)
A tourist walks at a rate 5 feet per second along a straight path. Trash bins are located every 200 feet along the path. A garbage truck travels 10 feet per second in the same direction as the tourist and stops for 30 seconds at each of the garbage bins. When the tourist started the walk, she noticed the truck ahead of her just leaving the next bin. How many times will the truck and the tourist meet?

   A    4

   B    5

   C    6

   D    7

   E    8

   N   

3. (10 p.)
Points \( C \) and \( D \) are on the same side of diameter \( AB \) of circle \( k \). Assume that \( \angle AOC=30^{\circ} \) and \( \angle DOB=45^{\circ} \). Let \( \alpha_1 \) denote the area of the smaller sector \( COD \) of the circle, and let \( \alpha \) denote the area of the entire circle. Calculate the ratio \( \frac{\alpha_1}{\alpha} \).

   A    \( \frac29 \)

   B    \( \frac14 \)

   C    \( \frac5{18} \)

   D    \( \frac7{24} \)

   E    \( \frac3{10} \)

   N   

4. (7 p.)
In a sport competition, each of participating teams has 21 players. Each player has to be paid at least 15000 dollars. However, in each of the teams, the total amount of all players’ salaries cannot exceed 700000 dollars. What is the maximal possible salary that a single player can have?

   A    270000

   B    385000

   C    400000

   D    430000

   E    700000

   N   

5. (12 p.)
There are two kinds of flowers in a shop. Roses cost 3 dollars each while carnations cost 2 dollars each. How many different kinds of bouquets can be bought with exactly 50 dollars?

   A    1

   B    7

   C    9

   D    16

   E    17

   N   





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