Let integer $a$, $b$, and $c$ satisfy $a+b+c=0$, prove $|a^{1999}+b^{1999}+c^{1999}|$ is a composite number.
The number $2^{29}$ is a nine-digit number whose digits are all distinct. Which digit of $0$ to $9$ does not appear?
If $3x + 2 = 17$, what is the value of $x$?
Let $a_1$, $a_2$, $a_3$, $\cdots$, $a_n$ be a random permutation of $1$, $2$, $3$, .., $n$, where $n$ is an odd number. Prove $$(a_1-1)(a_2-2)\cdots(a_n-n)$$ is an even number.
David, Hikmet, Jack, Marta, Rand, and Todd were in a $12$-person race with $6$ other people. Rand finished $6$ places ahead of Hikmet. Marta finished $1$ place behind Jack. David finished $2$ places behind Hikmet. Jack finished $2$ places behind Todd. Todd finished $1$ place behind Rand. Marta finished in $6^{th}$ place. Who finished in $8^{th}$ place?
The Tigers beat the Sharks $2$ out of $3$ times they played. They then played $N$ more times, and the Sharks ended up winning at least $95\%$ of all the games played. What is the minimum possible value for $N$?
Larry and Julius are playing a game, taking turns throwing a ball at a bottle sitting on a ledge. Larry throws first. The winner is the first person to knock the bottle off the ledge. At each turn the probability that a player knocks the bottle off the ledge is $\tfrac{1}{2}$, independently of what has happened before. What is the probability that Larry wins the game?
In equilateral triangle $ABC$, shown here, each downward pointing black triangle has its vertices at the midpoints of the sides of a larger upward pointing white triangle. What fraction of the area of $DABC$ is white? Express your answer as a common fraction.
What is the units digit of the sum of the squares of the integers from $1$ to $2015$, inclusive?
What is the radius of a circle inscribed in a triangle with sides of length $5$, $12$ and $13$ units?
For how many positive integers $n$ is $\frac{n}{30-n}$ also a positive integer?
Julia's age is a two-digit multiple of $5$, and when Julia's age is divided by $2$, $3$, $4$, $6$ or $8$, the remainder is always $1$. If Julia is five times as old as Bart, how old is Bart?
Two integers have a sum of $26$. when two more integers are added to the first two, the sum is $41$. Finally, when two more integers are added to the sum of the previous $4$ integers, the sum is $57$. What is the minimum number of even integers among the $6$ integers?
There are $5$ coins placed flat on a table according to the figure. What is the order of the coins from top to bottom?
A palindrome, such as $83438$, is a number that remains the same when its digits are reversed. The numbers $x$ and $(x+32)$ are three-digit and four-digit palindromes, respectively. What is the sum of the digits of $x$?
Triangle $ABC$ has vertices $A = (3,0)$, $B = (0,3)$, and $C$, where $C$ is on the line $x + y = 7$. What is the area of $\triangle ABC$?
A class collects $50$ dollars to buy flowers for a classmate who is in the hospital. Roses cost $3$ dollars each, and carnations cost $2$ dollars each. No other flowers are to be used. How many different bouquets could be purchased for exactly $50$ dollars?
Triangle $ABC$ has side lengths $AB = 5$, $BC = 6$, and $AC = 7$. Two bugs start simultaneously from $A$ and crawl along the sides of the triangle in opposite directions at the same speed. They meet at point $D$. What is $BD$?
A line passes through $A\ (1,1)$ and $B\ (100,1000)$. How many other points with integer coordinates are on the line and strictly between $A$ and $B$?
What is the minimum number of small squares that must be colored black so that a line of symmetry lies on the diagonal $\overline{BD}$ of square $ABCD$?
Soda is sold in packs of $6$, $12$ and $24$ cans. What is the minimum number of packs needed to buy exactly $90$ cans of soda?
Suppose $d$ is a digit. For how many values of $d$ is $2.00d5 > 2.005$?
Bill walks $\tfrac12$ mile south, then $\tfrac34$ mile east, and finally $\tfrac12$ mile south. How many miles is he, in a direct line, from his starting point?
The Little Twelve Basketball Conference has two divisions, with six teams in each division. Each team plays each of the other teams in its own division twice and every team in the other division once. How many conference games are scheduled?
Alice and Bob play a game involving a circle whose circumference is divided by $12$ equally-spaced points. The points are numbered clockwise, from $1$ to $12$. Both start on point $12$. Alice moves clockwise and Bob, counterclockwise. In a turn of the game, Alice moves $5$ points clockwise and Bob moves $9$ points counterclockwise. The game ends when they stop on the same point. How many turns will this take?