In triangle $ABC$, $AC = 13$, $BC = 14$, and $AB=15$. Points $M$ and $D$ lie on $AC$ with $AM=MC$ and $\angle ABD = \angle DBC$. Points $N$ and $E$ lie on $A$B with $AN=NB$ and $\angle ACE = \angle ECB$. Let $P$ be the point, other than $A$, of intersection of the circumcircles of $\triangle AMN$ and $\triangle ADE$. Ray $AP$ meets $BC$ at $Q$. The ratio $\frac{BQ}{CQ}$ can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m-n$.
Prove: randomly select 51 numbers from $1, 2, 3, \cdots, 100$, at least two of them must be relatively prime to each other.
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?
True or false: Let $\tau({n})$ denotes the number of positive divisors that $n$ has. Then $$\tau({1}) + \tau({2}) +\tau({3}) +\cdots + \tau({2015})$$ is an odd number.
Is it possible to equally divide the set {$1, 2, 3, \cdots, 972$} into 12 non-intersect subsets so that each subset has exactly 81 elements, and the sums of those subsets are all equal?
There are $n$ points, $A_1$, $A_2$, $\cdots$, $A_n$ on a line segment, $\overline{A_0A_{n+1}}$. The point $A_0$ is black, $A_{n+1}$ is white, and the rest points are colored randomly either black or white. Prove: among these $n+1$ line segments $A_kA_{k+1}$, where $k=0, 1, \cdots, n$, the number of those with different colored ending points is odd.
If $3x + 2 = 17$, what is the value of $x$?
A mountain bike, originally priced at \$650, is on sale for 20% off. After the fixed shipping cost of \$25 is added to the new price, what is the final cost of the bike?
Margie's car can go 32 miles on a gallon of gas, and gas currently costs \$4 per gallon. How many miles can Margie drive on \$20 worth of gas?
Joe wrote $3$ positive integers on the whiteboard, then erased one number and replaced with the sum of the other two numbers minus $1$. He continued doing this and stopped when there're $17$, $1967$ and $1983$ on the whiteboard. Is it possible that the initial three numbers Joe wrote are $2$, $2$ and $2$?
Eleven members of the Middle School Math Club each paid the same amount for a guest speaker to talk about problem solving at their math club meeting. They paid their guest speaker \$ $\underline{1}\underline{A}\underline{2}$. What is the missing digit A of this 3-digit number?
In $\bigtriangleup ABC$, $D$ is a point on side $\overline{AC}$ such that $BD=DC$ and $\angle BCD$ measures $70^\circ$. What is the degree measure of $\angle ADB$?
Rectangle $ABCD$ and right triangle $DCE$ have the same area. They are joined to form a trapezoid, as shown. What is $DE$?
The circumference of the circle with center $O$ is divided into 12 equal arcs, marked the letters $A$ through $L$ as seen below. What is the number of degrees in the sum of the angles $x$ and $y$?
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.
Rectangle $ABCD$ has sides $CD=3$ and $DA=5$. A circle of radius $1$ is centered at $A$, a circle of radius $2$ is centered at $B$, and a circle of radius $3$ is centered at $C$. Which of the following is closest to the area of the region inside the rectangle but outside all three circles?
A 2-digit number is such that the product of the digits plus the sum of the digits is equal to the number. What is the unit digit of the number?
A straight one-mile stretch of highway, $40$ feet wide, is closed. Robert rides his bike on a path composed of semicircles as shown. If he rides at $5$ miles per hour, how many hours will it take to cover the one-mile stretch? Note: $1$ mile= $5280$ feet
In the table shown, $y =\frac{3x-1}{2}$. What is the value of $t$?
Two of the three sides of a triangle are 20 and 15. Which of the following numbers is not a possible perimeter of the triangle?
Mr. Patrick teaches math to 15 students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was 80. after he graded Payton's test, the class average became 81. What was Payton's score on the test?
A collection of circles in the upper half-plane, all tangent to the $x$-axis, is constructed in layers as follows. Layer $L_0$ consists of two circles of radii $70^2$ and $73^2$ that are externally tangent. For $k\ge1$, the circles in $\bigcup_{j=0}^{k-1}L_j$ are ordered according to their points of tangency with the $x$-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer $L_k$ consists of the $2^{k-1}$ circles constructed in this way. Let $S=\bigcup_{j=0}^{6}L_j$, and for every circle $C$ denote by $r(C)$ its radius. What is \[\sum_{C\in S} \frac{1}{\sqrt{r(C)}}?\]
Amelia needs to estimate the quantity $\frac{a}{b} - c$, where $a, b,$ and $c$ are large positive integers. She rounds each of the integers so that the calculation will be easier to do mentally. In which of these situations will her answer necessarily be greater than the exact value of $\frac{a}{b} - c$?
A box contains 2 red marbles, 2 green marbles, and 2 yellow marbles. Carol takes 2 marbles from the box at random; then Claudia takes 2 of the remaining marbles at random; and then Cheryl takes the last 2 marbles. What is the probability that Cheryl gets 2 marbles of the same color?