Three Circles

Three circles of equal radius pairwise meet at tangent points A, B, and C. What's the ratio of the area of triangle ABC to the area of one of the circles?

Triangles in a Pentagon

Leo draws a convex pentagon, and then connects all its diagonal lines. How many triangles of all sizes are shown in Leo's picture?

Third Term

In a sequence such that the next term in the sequence is the sum of the previous two terms.The sum of the first ten terms in the sequence is 231. The fifth term in the sequence is 13. What is the third term?

Two Families

When Dr. White and Dr. Green met each other 5 years ago, each family had only one child. Now they each have two children under 10 years old. These are the facts about their children:
1) James is the oldest among the four;
2) Russell is 4 years older than his brother;
3) Stephen's age is half of one of Dr. White's children's age;
4) Kevin is 2 years older than Dr. Green's younger child.
If each child is a different age, how old is Stephen?

Timed Exam

Leo plans to take a timed exam. He fills the corresponding bubble in the answer sheet right after completing each problem. If he spends 2 minutes to complete each problem, he would have 6 problems on the answer sheet left empty. If he spends 1 minute on each problem, he would have 4 minutes left before the end of the exam. Let N be the number of problems in the exam. What's the least value for N?

Memorial Day

Memorial Day, observed on the last Monday of May, is a U.S. federal holiday for honoring and mourning the military personnel who had died while serving in the United States Armed Forces. One year, there were four Saturdays and four Wednesdays in the month of May. On which day of May was Memorial Day?

Greatest Possible GCF

Sum of five different positive integers is 2020. What is the greatest possible greatest common factor of the five?

Arithmetic Sequence

In the year-end ceremony, 12 top-performing CMM students form a line. Each of them holds a different number from 1 to 12. Dr. Hong calls their numbers one at a time to give them awards. At first, Dr. Hong tries to call the numbers so that no three of them form an arithmetic sequence. What's the maximum number of students Dr. Hong calls before he sees an arithmetic sequence of at least three numbers.

Circle in a Right Triangle

A circle is inscribed in a right triangle of leg lengths 8 and 15. What's the area of this circle? (Let Pi = 3.14.)

Squares and Median

The sum of 3 different positive integers is 55. The sum of any two of them is a perfect square. What is the median of these three positive integers?