IMOmath

Multiple choice practice test

1. (38 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. (5 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   

3. (1 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   

4. (12 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. (40 p.)
A parking lot has 16 spaces in a row. Each of the twelve cars took one parking space, and their drivers chose spaces at random from among the available spaces. After that a big van arrived and it requires 2 adjacent spaces to park. What is the probability that the van will be able to park?

   A    \( \frac{11}{20} \)

   B    \( \frac{4}{7} \)

   C    \( \frac{81}{140} \)

   D    \( \frac35 \)

   E    \( \frac{17}{28} \)

   N   





2005-2020 IMOmath.com | imomath"at"gmail.com | Math rendered by MathJax
Home | Olympiads | Book | Training | IMO Results | Forum | Links | About | Contact us