IMOmath

Multiple choice practice test

1. (33 p.)
Two circles of radius \( 1 \) are chosen in the following way. The center of the circle \( k_0 \) is chosen uniformly at random from the line segment joining \( (0,0) \) and \( (2,0) \). Independently of this choice, the center of circle \( k_1 \) is chosen uniformly at random from the line segment joining \( (0,1) \) to \( (2,1) \). What is the probability that \( k_0 \) and \( k_1 \) intersect?

   A    \( \frac{2+\sqrt2}4 \)

   B    \( \frac{3\sqrt3+2}8 \)

   C    \( \frac{2\sqrt2-1}2 \)

   D    \( \frac{2+\sqrt3}4 \)

   E    \( \frac{4\sqrt3-3}4 \)

   N   

2. (12 p.)
Two points \( B \) and \( C \) are located on the segment \( AD \). The length of \( AB \) is 4 times the length of \( BD \), and the length of \( AC \) is 9 times the length of \( CD \). Determine \( \frac{BC}{AD} \).

   A    \( \frac1{36} \)

   B    \( \frac1{13} \)

   C    \( \frac1{10} \)

   D    \( \frac5{36} \)

   E    \( \frac15 \)

   N   

3. (39 p.)
Let \( ABCD \) be a trapezoid with \( AB\|CD \), \( AB=11 \), \( BC=5 \), \( CD=19 \), and \( DA=7 \). The bisectors of \( \angle A \) and \( \angle D \) meet at \( P \), and bisectors of \( \angle B \) and \( \angle C \) meet at \( Q \). Find the area of the hexagon \( ABQCDP \).

   A    \( 28\sqrt 3 \)

   B    \( 30\sqrt3 \)

   C    \( 32\sqrt 3 \)

   D    \( 35\sqrt 3 \)

   E    \( 36\sqrt 3 \)

   N   

4. (4 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. (9 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   





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