Olympiad problems

2003 JHMMC 8, 19

Tags: JHMMC

Two angles are supplementary, and one angle is $9$ times as large as the other. What is the number of degrees in the measure of the larger angle?

2003 JHMMC 8, 21

Tags: JHMMC , 3D Geo

The surface area and the volume of a cube are numerically equal. Find the cube’s volume.

2003 JHMMC 8, 16

Tags: JHMMC

A lazy student used the approximation $\pi=\frac{22} 7$ to calculate the circumference of a given circle. If his answer was 6, what was the radius of the circle?

2003 JHMMC 8, 2

Tags: JHMMC

Philip has $3$ triangles and $6$ pentagons. Let $S$ be the total number of sides of the shapes he has. Let $N$ be the number of shapes he has. What is $S+N$?

The ages of Mr. and Mrs. Fibonacci are both two-digit numbers. If Mr. Fibonacci’s age can be formed by reversing the digits of Mrs. Fibonacci’s age, find the smallest possible positive difference between their ages.

2003 JHMMC 8, 25

Tags: JHMMC

Two positive whole numbers differ by $3$. The sum of their squares is $117$. Find the larger of the two numbers.

2014 JHMMC 7 Contest, 27

Tags: 3 digit friendly numbers , theorems , JHMMC , geo bashing , geo

Young Guy likes to make friends with numbers, so he calls a number “friendly” if the sum of its digits is equal to the product of its digits. How many $3 \text{digit friendly numbers}$ are there?

2014 JHMMC 7 Contest, 2

Tags: Quick NT , JHMMC

2. What’s the closest number to $169$ that’s divisible by $9$?

2014 JHMMC 7 Contest, 17

Tags: Find all such x , JHMMC

Find all $x$ such that $\frac{x^2+1}{x-1}=\frac{x^2-1}{x+1}$.

2003 JHMMC 8, 8

Tags: JHMMC , geo

What is the area of a square in square feet, if each of its diagonals is $4$ feet long?

2014 JHMMC 7 Contest, 4

Tags: French and Spanish Courses , JHMMC

$27$ students in a school take French. $32$ students in a school take Spanish. $5$ students take both courses. How many of these students in total take only $1$ language course?

2003 JHMMC 8, 27

Tags: JHMMC

A pair of positive integers $a$ and $b$ is such that their greatest common divisor is $5$ and their least common multiple is $55$. Find the smallest possible value of $a + b$.

2014 JHMMC 7 Contest, 14

Tags: consecutive integers , JHMMC

$11$ consecutive integers sum to $1331$. What is the largest of the $11$ integers?

2003 JHMMC 8, 18

Tags: JHMMC , nt

How many multiples of $17$ are there between $23$ and $227$?

2003 JHMMC 8, 32

Tags: JHMMC

Let $N$ be the product of the first nine multiples of $19$ (i.e. $N = 19\times38 \times57\times\cdots\times 152\times 171$). What is the last digit of $N$?

2014 JHMMC 7 Contest, 15

Tags: PurpleRedBLue , JHMMC

Rita the painter rolls a fair $6\text{-sided die}$that has $3$ red sides, $2$ yellow sides, and $1$ blue side. Rita rolls the die twice and mixes the colors that the die rolled. What is the probability that she has mixed the color purple?