The fraction $$\left(\frac{\frac13+1}{3} +\frac{1+ \frac13}{3} \right) / \left(\frac{3}{\frac{1}{3+1}+\frac{ 1}{1+3}}\right)$$ can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Let the incircle of $\triangle ABC$ meet $BC,CA,AB$ at $J,K,L$. Let $D(\ne B, J),E(\ne C,K), F(\ne A,L)$ be points on $BJ,CK,AL$. If the incenter of $\triangle ABC$ is the circumcenter of $\triangle DEF$ and $\angle BAC = \angle DEF$, prove that $\triangle ABC$ and $\triangle DEF$ are isosceles triangles.

You want to color a flag like the one shown in the following image, for which four different colors are available. Two regions of the flag that share a side (or a segment of a side) must have different colors. The flag cannot be flipped, rotated, or reflected. How many different flags can be colored with these conditions? [img]https://cdn.artofproblemsolving.com/attachments/4/9/879d1e144acdbc63ee2ffe34cf13a920d5d836.png[/img]

Wendy takes Honors Biology at school, a smallish class with only fourteen students (including Wendy) who sit around a circular table. Wendy's friends Lucy, Starling, and Erin are also in that class. Last Monday none of the fourteen students were absent from class. Before the teacher arrived, Lucy and Starling stretched out a blue piece of yarn between them. Then Wendy and Erin stretched out a red piece of yarn between them at about the same height so that the yarn would intersect if possible. If all possible positions of the students around the table are equally likely, let $m/n$ be the probability that the yarns intersect, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.

Let $n{}$ be a non-negative integer and consider the standard power expansion of the following polynomial \[\sum_{k=0}^n\binom{n}{k}^2(X+1)^{2k}(X-1)^{2(n-k)}=\sum_{k=0}^{2n}a_kX^k.\]The coefficients $a_{2k+1}$ all vanish since the polynomial is invariant under the change $X\mapsto -X.$ Prove that the coefficients $a_{2k}$ are all positive.

A rhombus with sidelength $1$ has an inscribed circle with radius $\frac{1}{3}.$ If the area of the rhombus can be expressed as $\frac{a}{b}$ for relatively prime, positive $a,b,$ evaluate $a+b.$ [i]Proposed by Alex Li[/i]

there are $n$ points on the plane,any two vertex are connected by an edge of red,yellow or green,and any triangle with vertex in the graph contains exactly $2$ colours.prove that $n<13$

Find all positive integers $ k$ such that for any positive numbers $ a, b$ and $ c$ satisfying the inequality \[ k(ab \plus{} bc \plus{} ca) > 5(a^2 \plus{} b^2 \plus{} c^2),\] there must exist a triangle with $ a, b$ and $ c$ as the length of its three sides respectively.

In the $xyz$ space, consider a right circular cylinder with radius of base 2, altitude 4 such that \[\left\{ \begin{array}{ll} x^2+y^2\leq 4 &\quad \\ 0\leq z\leq 4 &\quad \end{array} \right.\] Let $V$ be the solid formed by the points $(x,\ y,\ z)$ in the circular cylinder satisfying \[\left\{ \begin{array}{ll} z\leq (x-2)^2 &\quad \\ z\leq y^2 &\quad \end{array} \right.\] Find the volume of the solid $V$.

Call a convex polyhedron a [i]footballoid [/i] if it has the following properties. (1) Any face is either a regular pentagon or a regular hexagon. (2) All neighbours of a pentagonal face are hexagonal (a [i]neighbour [/i] of a face is a face that has a common edge with it). Find all possibilities for the number of pentagonal and hexagonal faces of a footballoid.

Let \( f: \mathbb{R} \to \mathbb{R} \) be a differentiable function such that \( f(0) = 0 \) and \( 0 < f'(t) \leq 1 \) for all \( t \in [0, 1] \). Show that: \[ \left( \int_0^1 f(t) \, dt \right)^2 \geq \int_0^1 f(t)^3 \, dt. \]

Let $ABCD$ be the trapezoid such that $AB\parallel CD$. Let $E$ be an arbitrary point on $AC$. point $F$ lies on $BD$ such that $BE\parallel CF$. Prove that circumcircles of $\triangle ABF,\triangle BED$ and the line $AC$ are concurrent.

Let $\Phi$ denote the Euler totient function. Prove that for infinitely many $k$ we have $\Phi (2^k+1) < 2^{k-1}$ and that for infinitely many $m$ one has $\Phi (2^m+1) > 2^{m-1}$

Can the number \( \overline{abc} + \overline{bca} + \overline{cab} \) be a perfect square?

In a trapezoid $ABCD$, the legs $AB$ and $CD$ meet in $M$ and the diagonals $AC$ and $BD$ meet in $N.$ Let $AC=a$ and $BC=b.$ Find the area of triangles $AMD$ and $AND$ in terms of $a$ and $b.$

Let $a$ be a positive integer and $(a_n)_{n\geqslant 1}$ be a sequence of positive integers satisfying $a_n<a_{n+1}\leqslant a_n+a$ for all $n\geqslant 1$. Prove that there are infinitely many primes which divide at least one term of the sequence. [i]Moldavia Olympiad, 1994[/i]

Let $A,B$ be two distinct points on a given circle $O$ and let $P$ be the midpoint of the line segment AB. Let $O_1$ be the circle tangent to the line $AB$ at $P$ and tangent to the circle $O$. Let $l$ be the tangent line, different from the line $AB$, to $O_1$ passing through $A$. Let $C$ be the intersection point, different from $A$, of $l$ and $O$. Let $Q$ be the midpoint of the line segment $BC$ and $O_2$ be the circle tangent to the line $BC$ at $Q$ and tangent to the line segment $AC$. Prove that the circle $O_2$ is tangent to the circle $O$.

Let $x_1\leq x_2\leq \cdots\leq x_{100}$ be real numbers such that $|x_1| + |x_2| + \cdots + |x_{100}| = 1$ and $x_1 + x_2 + \cdots + x_{100} = 0$. Among all such $100$-tuples of numbers, the greatest value that $x_{76} - x_{16}$ can achieve is $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Let $n$ be a positive integer. Define a chameleon to be any sequence of $3n$ letters, with exactly $n$ occurrences of each of the letters $a, b,$ and $c$. Define a swap to be the transposition of two adjacent letters in a chameleon. Prove that for any chameleon $X$ , there exists a chameleon $Y$ such that $X$ cannot be changed to $Y$ using fewer than $3n^2/2$ swaps.

Let $n$ be an even positive integer. Write the numbers $1, 2, \cdots, n^2$ in the squares of an $n \times n$ grid so that the $k$th row, from left to right, is \[ (k-1)n + 1, \ (k-1)n + 2, \ \cdots, \ (k-1)n + n. \] Color the squares of the grid so that half of the squares in each row and in each column are red and the other half are black (a checkerboard coloring is one possibility). Prove that for each coloring, the sum of the numbers on the red squares is equal to the sum of the numbers on the black squares.

Define the sequence $a_1, a_2, a_3, \ldots$ by $a_1 = 1$ and, for $n > 1$, \[a_n = a_{\lfloor n/2 \rfloor} + a_{\lfloor n/3 \rfloor} + \ldots + a_{\lfloor n/n \rfloor} + 1.\] Prove that there are infinitely many $n$ such that $a_n \equiv n \pmod{2^{2010}}$.

Let $n$ be a positive integer and let $k$ be an integer between $1$ and $n$ inclusive. There is a white board of $n \times n$. We do the following process. We draw $k$ rectangles with integer sides lenghts and sides parallel to the ones of the $n \times n$ board, and such that each rectangle covers the top-right corner of the $n \times n$ board. Then, the $k$ rectangles are painted of black. This process leaves a white figure in the board. How many different white figures are possible to do with $k$ rectangles that can't be done with less than $k$ rectangles? Proposed by David Torres Flores

Suppose that $a,b$ are real numbers, function $f(x) = ax+b$ satisfies $\mid f(x) \mid \leq 1$ for any $x \in [0,1]$. Find the range of values of $S= (a+1)(b+1).$

We call the [i]tribi [/i] of a positive integer $k$ (denoted $T(k)$) the number of all pairs $11$ in the binary representation of $k$. e.g $$T(1)=T(2)=0,\, T(3)=1, \,T(4)=T(5)=0,\,T(6)=1,\,T(7)=2.$$ Calculate $S_n=\sum_{k=1}^{2^n}T(K)$.

Determine the value of $(101 \times 99)$ - $(102 \times 98)$ + $(103 \times 97)$ − $(104 \times 96)$ + ... ... + $(149 \times 51)$ − $(150 \times 50)$.