First Name _____________________________________
Last Name _____________________________________
Grade __________________________________________
School __________________________________________
City ___________________________________________
RSM Branch____________________________________
1
2
There are 7 cards. Each card has one digit on it. Overall, there are only three different digits,
and each digit appears on a different number of cards. The cards are arranged to make a
7-digit number. What is the smallest possible value of this number?
If the sum of the 3-digit numbers RSM, SMR, and MRS is equal to 1998, compute
R + S + M.
4
A pentagon has side lengths 4, 14, 14, 2014, and X, where X is an integer. Find the least
possible value of X.
Amy thought of a 2-digit natural number. One of the following statements is false, the other
three are true:
a) her number is divisible by 4;
b) her number is divisible by 5;
c) her number is divisible by 6;
d) her number is divisible by 7.
Find the sum of all possible distinct values of Amy’s number.
5
6
The length of one of the legs of a right triangle was increased by 40%, and the length of
its other leg was decreased by 40%. The hypotenuse of the new right triangle has the same
length as the hypotenuse of the original triangle. Compute 900r 2, where r is the ratio of the
longer leg of the original triangle to its shorter leg.
There are many chameleons in the RSM Zoo. Before breakfast some of them were red and all
the others were blue. After breakfast, half of the red chameleons became blue. After dinner,
half of all chameleons which were blue at that time became red, bringing the total number
of red chameleons to the before-breakfast level. What was the ratio of the number of red
chameleons to the number of blue chameleons before breakfast?
Please fold over on line. Write answers on back.
3
Grades
8-9
First Name _____________________________________
Last Name _____________________________________
Grade __________________________________________
School __________________________________________
City ___________________________________________
____________________________________________
RSM Branch____________________________________
7
Peter walked half of the distance from home to school at a constant speed. Half of the
remaining distance he ran at a constant speed, twice as fast as he walked. The rest of his
way to school Peter rode a skateboard at a constant speed, twice as fast as he ran. He took
22 minutes to get from home to school. On the way home (using the same route), Peter rode
a skateboard half of the distance, at the same speed as he rode on the way to school. Half of
the remaining distance he ran at the same speed as he ran on the way to school. The rest of
his way home Peter walked at the same speed as he walked on the way to school. How many
minutes did it take him to get home from school?
8
A rectangle with one side 10 and adjacent side X can be cut into exactly 4 squares. Compute
the sum of all possible distinct values of X.
10
11
12
All 5-digit positive integers containing each of the digits 1, 2, 3, 4, 5 exactly once are written
down on the board. Compute the sum of all these numbers.
Alex, Felix, and Phoenix have 2014 apples altogether. One of them has R apples, another one
has S apples, and the remaining one has M apples, where R, S, and M are natural numbers.
If Alex gives 10 of his apples to Felix, Felix gives 20 of his apples to Phoenix, and Phoenix
gives 40 of his apples to Alex, they would again have R, S, and M apples (in some other
order). Find the median of the numbers R, S, and M.
In triangle RSM with area 800, points R1, S1, and M1 are the midpoints of SM, MR, and RS
respectively, points R2, S2, and M2 are the midpoints of MR1, RS1, and SM1 respectively.
Compute the area of triangle R2S2M2.
Recall that symbol x˩ means the largest integer not greater than the number x,
and symbol x˥ means the smallest integer not less than the number x. For instance,
2˩ = 2˥ = 2, 2.5˩ = 2, 2.5˥ = 3. For how many positive integers x not greater than
2014 is the number
x/2˥ /3˩ /2˥ even?
Please fold over on line. Write answers on back.
9
Grades
8-9