IMOmath

Multiple choice practice test

1. (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   

2. (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   

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. (32 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. (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   





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