The product of 22 integers is 1. Show that their sum can not be 0.

How many 6-digit numbers have at least an even digit?

(a) Prove that (p^2)-1 is divisible by 24 if p is a prime number greater than 3. (b) Prove that (p^2)-(q^2) is divisible by 24 if p and q are prime numbers greater than 3.

Given 11 natural numbers under 21, show that you can choose two such that one divides the other.

Point A, which is within an acute, is reflected with respect to both sides of angle A to obtain the points B and C. the segment BC intersects the sides of angle A at points D and E respectively. Prove that BC/2>DE.

Two children take turns breaking chocolate bar that is 5*10 squares. They can only break the bar using the divisions between squares and can only do 1 break at a time.. The first player that when breaking the chocolate bar breaks off only a single square wins. Is there a winning strategy for any player?

Show that for any natural number n, n^3 + (n + 1)^3 + (n + 2)^3 is divisible by 9.

A set of ten two-digit numbers is given. Prove that one can always choose two disjoint subsets of this set such that the sum of their elements is the same. Please remember to hide your solution. (by using the hide tags of course.. I don't literally mean that you should hide it )

Find all prime numbers $p$ and $q$ such that $2^2+p^2+q^2$ is also prime. Please remember to hide your solution. (by using the hide tags of course.. I don't literally mean that you should hide it )

The number $1234$ is written on the board. A play consists of subtracting a non-zero digit of that number from that number, and replacing the number by the result. The player who writes the number (not digit) zero wins. Determine if there is a winning strategy for one of the two players who play consecutively. Please remember to hide your solution. (by using the hide tags of course.. I don't literally mean that you should hide it )

Let $P$ be a point inside the triangle $ABC$, such that the angles $\angle CBP$ and $\angle PAC$ are equal. Denote the intersection of the line $AP$ and the segment $BC$ by $D$, and the intersection of the line $BP$ with the segment $AC$ by $E$. The circumcircles of the triangles $ADC$ and $BEC$ meet at $C$ and $F$. Show that the line $CP$ bisects the angle $DFE$. Please remember to hide your solution. (by using the hide tags of course.. I don't literally mean that you should hide it )

Twelve balls are numbered by the numbers $1,2,3,\cdots,12$. Each ball is colored either red or green, so that the following two conditions are satisfied: (i) If two balls marked by different numbers $a$ and $b$ are colored red and $a+b<13$, then the ball marked by the number $a+b$ is colored red, too. (ii) If two balls marked by different numbers $a$ and $b$ are colored green and $a+b<13$, then the ball marked by the number $a+b$ is also colored green. How many ways are there of coloring the balls? Please remember to hide your solution. (by using the hide tags of course.. I don't literally mean that you should hide it )

Find all natural numbers such that each is equal to the sum of the factorials of its digits. Please remember to hide your solution. (by using the hide tags of course.. I don't literally mean that you should hide it )