IMOmath

Multiple choice practice test

1. (28 p.)
A function \( f \) is defined by \( f(z)=(4+i)z^2+\alpha z+ \gamma \) for all complex numbers \( z \), where \( \alpha \) and \( \gamma \) are complex numbers. Given that \( f(1) \) and \( f(i) \) are both real, find the smallest possible value for \( |\alpha|+|\gamma| \).

   A    \( 1 \)

   B    \( \sqrt2 \)

   C    \( 2 \)

   D    \( 2\sqrt2 \)

   E    \( 4 \)

   N   

2. (34 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   

3. (31 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   

4. (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   

5. (3 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   





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