IMOmath

Multiple choice practice test

1. (10 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   

2. (38 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   

3. (9 p.)
A postman has a pedometer to count his steps. The pedometer records up to 99999 steps, then flips over to 000000 on the next step. The postman plans to determine his mileage for a year. On January 1 the postman sets the pedometer to 00000. During the year, the pedometer flips from 99999 to 00000 forty-four times. On December 31 the pedometer reads 50000. The postman takes 1800 steps per mile. Which of the following is closest to the number of miles the postman has walked over the year?

   A    2500

   B    3000

   C    3500

   D    4000

   E    4500

   N   

4. (33 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   

5. (7 p.)
There are two kinds of flowers in a shop. Roses cost 3 dollars each while carnations cost 2 dollars each. How many different kinds of bouquets can be bought with exactly 50 dollars?

   A    1

   B    7

   C    9

   D    16

   E    17

   N   





2005-2021 IMOmath.com | imomath"at"gmail.com | Math rendered by MathJax
Home | Olympiads | Book | Training | IMO Results | Forum | Links | About | Contact us