IMOmath

Multiple choice practice test

1. (20 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   

2. (18 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   

3. (21 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. (20 p.)
Suppose that the sum of base-10 logarithms of the divisors of \( 10^n \) is 792. Determine \( n \).

   A    11

   B    12

   C    13

   D    14

   E    15

   N   

5. (19 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   





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