IMOmath

Multiple choice practice test

1. (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   

2. (10 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   

3. (6 p.)
Points \( C \) and \( D \) are on the same side of diameter \( AB \) of circle \( k \). Assume that \( \angle AOC=30^{\circ} \) and \( \angle DOB=45^{\circ} \). Let \( \alpha_1 \) denote the area of the smaller sector \( COD \) of the circle, and let \( \alpha \) denote the area of the entire circle. Calculate the ratio \( \frac{\alpha_1}{\alpha} \).

   A    \( \frac29 \)

   B    \( \frac14 \)

   C    \( \frac5{18} \)

   D    \( \frac7{24} \)

   E    \( \frac3{10} \)

   N   

4. (34 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. (36 p.)
Two circles of radius \( 1 \) are chosen in the following way. The center of the circle \( k_0 \) is chosen uniformly at random from the line segment joining \( (0,0) \) and \( (2,0) \). Independently of this choice, the center of circle \( k_1 \) is chosen uniformly at random from the line segment joining \( (0,1) \) to \( (2,1) \). What is the probability that \( k_0 \) and \( k_1 \) intersect?

   A    \( \frac{2+\sqrt2}4 \)

   B    \( \frac{3\sqrt3+2}8 \)

   C    \( \frac{2\sqrt2-1}2 \)

   D    \( \frac{2+\sqrt3}4 \)

   E    \( \frac{4\sqrt3-3}4 \)

   N   





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