IMOmath

Multiple choice practice test

1. (40 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   

2. (30 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   

3. (11 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   

4. (12 p.)
Two points \( B \) and \( C \) are located on the segment \( AD \). The length of \( AB \) is 4 times the length of \( BD \), and the length of \( AC \) is 9 times the length of \( CD \). Determine \( \frac{BC}{AD} \).

   A    \( \frac1{36} \)

   B    \( \frac1{13} \)

   C    \( \frac1{10} \)

   D    \( \frac5{36} \)

   E    \( \frac15 \)

   N   

5. (4 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   





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