IMOmath

Multiple choice practice test

1. (60 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. (22 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. (5 p.)
The table \( 4\times 4 \) is filled with numbers as follows: \[ \begin{array}{|c|c|c|c|} \hline 1&2&3&4 \\ \hline 8&9&10&11\\ \hline 15&16&17&18\\ \hline 22&23&24&25\\ \hline \end{array}\] First reverse the order of numbers in the second row. Then reverse the order of numbers fourth row. Then sum the numbers on each of the diagonals. What is the positive difference between the two diagonal sums?

   A    2

   B    4

   C    6

   D    8

   E    9

   N   

4. (2 p.)
A basketball player made five successful shots during a game. Each shot was worth either 2 or 3 points. How many different numbers could represent the total points scored by the player?

   A    2

   B    3

   C    4

   D    5

   E    6

   N   

5. (8 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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