- 1.
Determine the values of the positive integer n for which the number 9n3 − 9n2 + n − 1 is a prime number.
- 題型：問答題
- 難易度：尚未記錄
- 2.
The nine squares of a 3 × 3 checkerboard must be painted so that each row, each column, and each of the two diagonals have no two squares of the same colour. What is the least number of colours needed?
- 題型：問答題
- 難易度：尚未記錄
- 3.
What is the largest positive integer n such that the arithmetic mean of the numbers from {12, 22, …, (n − 1)2, n2} is strictly less than 2010?
- 題型：問答題
- 難易度：尚未記錄
- 4.
Tom has drunk too much, and the starts to walk home in an odd fashion:
− he takes 1 step forward;
− then he turns by 90° to the right and takes 2 steps forward;
− then he turns by 90° to the right and takes 1 step forward;
− then he turns by 90° to the left and takes 1 step forwards;
− then he repeats.
Each step is 1 meter long. After 172 steps he collapses to the ground. How many meters is Tom from the starting place when his walk ends?
- 題型：問答題
- 難易度：尚未記錄
- 5.
Let f : {x, y, z} → {1, 2, …, 50} is a strictly increasing function such that 2f(y) ≠ f(x) − f(z). How many different f's are there?
- 題型：問答題
- 難易度：尚未記錄
Let H and K be subgroups of a group G, where e is the identity of G.
- 6.
Prove that if |H| = 10 and |K| = 21, then H ∩ K = {e}.
- 題型：問答題
- 難易度：尚未記錄
Let H and K be subgroups of a group G, where e is the identity of G.
- 7.
If |H| = m and |K| = n, with gcd(m, n) = 1, prove that H ∩ K = {e}.
- 題型：問答題
- 難易度：尚未記錄