IMOmath

Multiple choice practice test

1. (35 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. (42 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. (7 p.)
Points \( C \) and \( D \) are on the same side of diameter \( AB \) of circle \( k \). Assume that \( \angle AOC=30^{\circ} \) and \( \angle DOB=45^{\circ} \). Let \( \alpha_1 \) denote the area of the smaller sector \( COD \) of the circle, and let \( \alpha \) denote the area of the entire circle. Calculate the ratio \( \frac{\alpha_1}{\alpha} \).

   A    \( \frac29 \)

   B    \( \frac14 \)

   C    \( \frac5{18} \)

   D    \( \frac7{24} \)

   E    \( \frac3{10} \)

   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. (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   





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