IMOmath

Multiple choice practice test

1. (24 p.)
A postman has a pedometer to count his steps. The pedometer records up to 99999 steps, then flips over to 000000 on the next step. The postman plans to determine his mileage for a year. On January 1 the postman sets the pedometer to 00000. During the year, the pedometer flips from 99999 to 00000 forty-four times. On December 31 the pedometer reads 50000. The postman takes 1800 steps per mile. Which of the following is closest to the number of miles the postman has walked over the year?

   A    2500

   B    3000

   C    3500

   D    4000

   E    4500

   N   

2. (20 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   

3. (16 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. (28 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   

5. (12 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   





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