IMOmath

Multiple choice practice test

1. (38 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   

2. (5 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   

3. (40 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   

4. (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   

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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