IMOmath

Multiple choice practice test

1. (23 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. (27 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   

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. (21 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   

5. (26 p.)
Let \( A_0=(0,0) \). Points \( A_1 \), \( A_2 \), \( \dots \) lie on the \( x \) axis and points \( B_1 \), \( B_2 \), \( \dots \) lie on the graph of \( y=\sqrt x \). Assume that for each \( k \) the triangle \( A_{k-1}B_kA_k \) is equilateral. Find the minimal \( n \) such that \( A_0A_n\geq 100 \).

   A    13

   B    15

   C    17

   D    19

   E    21

   N   





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