IMOmath

Multiple choice practice test

1. (20 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. (5 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. (55 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   

4. (17 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   

5. (2 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   





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