Let $ABC$ be an acute triangle with sides and area equal to $a, b, c$ and $S$ respectively. [color=#FF0000]Prove or disprove[/color] that a necessary and sufficient condition for existence of a point $P$ inside the triangle $ABC$ such that the distance between $P$ and the vertices of $ABC$ be equal to $x, y$ and $z$ respectively is that there be a triangle with sides $a, y, z$ and area $S_1$, a triangle with sides $b, z, x$ and area $S_2$ and a triangle with sides $c, x, y$ and area $S_3$ where $S_1 + S_2 + S_3 = S.$

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.

It is known that, having $6$ weighs, it is possible to balance the scales with loads, which weights are successing natural numbers from $1$ to $63$. Find all such sets of weighs.

Let $k$ be a positive even integer. Show that $$\sum_{n=0}^{k/2}(-1)^n\binom{k+2}n\binom{2(k-n)+1}{k+1}=\frac{(k+1)(k+2)}2.$$

Two cyclists leave simultaneously a point $P$ in a circular runway with constant velocities $v_1, v_2 (v_1 > v_2)$ and in the same sense. A pedestrian leaves $P$ at the same time, moving with velocity $v_3 = \frac{v_1+v_2}{12}$ . If the pedestrian and the cyclists move in opposite directions, the pedestrian meets the second cyclist $91$ seconds after he meets the first. If the pedestrian moves in the same direction as the cyclists, the first cyclist overtakes him $187$ seconds before the second does. Find the point where the first cyclist overtakes the second cyclist the first time.

Let $k$ be an odd number that is greater than or equal to $3$. Prove that there exists a $k^{th}$-degree integer-valued polynomial with non-integer-coefficients that has the following properties: (1) $f(0)=0$ and $f(1)=1$; and. (2) There exist infinitely many positive integers $n$ so that if the following equation: \[ n= f(x_1)+\cdots+f(x_s), \] has integer solutions $x_1, x_2, \dots, x_s$, then $s \geq 2^k-1$.

A set $\mathcal S$ of translates of an equilateral triangle is given in the plane, and any two have nonempty intersection. Prove that there exist three points such that every triangle in $\mathcal S$ contains one of these points.

Prove that if for natural numbers $ a, b $ the number $ \sqrt[3]{a} + \sqrt[3]{b} $ is rational, then $ a, b $ are cubes of natural numbers.

Find the number of positive integers $n$, such that $\sqrt{n} + \sqrt{n + 1} < 11$.

Find all pairs of positive integers $(x,y)$, so that the number $x^3+y^3$ is a prime.

Jon wrote the $n$ smallest perfect squares on one sheet of paper, and the $n$ smallest triangular numbers on another (note that $0$ is both square and triangular). Jon notices that there are the same number of triangular numbers on the first paper as there are squares on the second paper, but if $n$ had been one smaller, this would not have been true. If $n < 2008$, let $m$ be the greatest number Jon could have written on either paper. Find the remainder when $m$ is divided by $2008$.

Find the integer solutions of the equation \[ \left[ \sqrt{2m} \right] = \left[ n(2+\sqrt 2) \right] \]

Given integers $a> 0$, $n> 0$, suppose that $a^1 + a^2 +...+ a^n \equiv 1 \mod 10$. Prove that $a \equiv n \equiv 1 \mod 10$ .

A decorative window is made up of a rectangle with semicircles at either end. The ratio of $AD$ to $AB$ is $3:2$. And $AB$ is 30 inches. What is the ratio of the area of the rectangle to the combined area of the semicircle. [asy] import graph; size(5cm); real lsf=0; pen dps=linewidth(0.7)+fontsize(8); defaultpen(dps); pen ds=black; real xmin=-4.27,xmax=14.73,ymin=-3.22,ymax=6.8; draw((0,4)--(0,0)); draw((0,0)--(2.5,0)); draw((2.5,0)--(2.5,4)); draw((2.5,4)--(0,4)); draw(shift((1.25,4))*xscale(1.25)*yscale(1.25)*arc((0,0),1,0,180)); draw(shift((1.25,0))*xscale(1.25)*yscale(1.25)*arc((0,0),1,-180,0)); dot((0,0),ds); label("$A$",(-0.26,-0.23),NE*lsf); dot((2.5,0),ds); label("$B$",(2.61,-0.26),NE*lsf); dot((0,4),ds); label("$D$",(-0.26,4.02),NE*lsf); dot((2.5,4),ds); label("$C$",(2.64,3.98),NE*lsf); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);[/asy] $ \textbf{(A)}\ 2:3 \qquad\textbf{(B)}\ 3:2\qquad\textbf{(C)}\ 6:\pi \qquad\textbf{(D)}\ 9: \pi \qquad\textbf{(E)}\ 30 : \pi$

Let $a,b \in \mathbb{R}$ with $a < b,$ 2 real numbers. We say that $f: [a,b] \rightarrow \mathbb{R}$ has property $(P)$ if there is an integrable function on $[a,b]$ with property that \[ f(x) - f \left( \frac{x + a}{2} \right) = f \left( \frac{x + b}{2} \right) - f(x) , \forall x \in [a,b]. \] Show that for all real number $t$ there exist a unique function $f:[a,b] \rightarrow \mathbb{R}$ with property $(P),$ such that $\int_{a}^{b} f(x) \text{dx} = t.$

Let $a_1,a_2, \cdots$ be a sequence of integers that satisfies: $a_1=1$ and $a_{n+1}=a_n+a_{\lfloor \sqrt{n} \rfloor} , \forall n\geq 1 $. Prove that for all positive $k$, there is $m \geq 1$ such that $k \mid a_m$.

$P$ is an arbitrary point inside acute triangle $ABC$. Let $A_1,B_1,C_1$ be the reflections of point $P$ with respect to sides $BC,CA,AB$. Prove that the centroid of triangle $A_1B_1C_1$ lies inside triangle $ABC$.

Prove that if the polynomial $ f(x) = ax^3 + bx^2 + cx + d $ with integer coefficients takes odd values for $ x = 0 $ and $ x = 1 $, then the equation $ f(x) = 0 $ has no integer roots.

Let $ABCD$ be a parallelogram with $AC=BC.$ A point $P$ is chosen on the extension of ray $AB$ past $B.$ The circumcircle of $ACD$ meets the segment $PD$ again at $Q.$ The circumcircle of triangle $APQ$ meets the segment $PC$ at $R.$ Prove that lines $CD,AQ,BR$ are concurrent.

Let $x$ be a complex number such that $x^{2011}=1$ and $x\neq 1$. Compute the sum \[\dfrac{x^2}{x-1}+\dfrac{x^4}{x^2-1}+\dfrac{x^6}{x^3-1}+\cdots+\dfrac{x^{4020}}{x^{2010}-1}.\]

Prove that an octagon, whose all angles are equal and all sides have rational length, has a center of symmetry.

Let $n \geq 2, n \in \mathbb{N}$, $a,b,c,d \in \mathbb{N}$, $\frac{a}{b} + \frac{c}{d} < 1$ and $a + c \leq n,$ find the maximum value of $\frac{a}{b} + \frac{c}{d}$ for fixed $n.$

[b]p1. [/b] At Duke, $1/2$ of the students like lacrosse, $3/4$ like football, and $7/8$ like basketball. Let $p$ be the proportion of students who like at least all three of these sports and let $q$ be the difference between the maximum and minimum possible values of $p$. If $q$ is written as $m/n$ in lowest terms, find the value of $m + n$. [b]p2.[/b] A [i]dukie [/i]word is a $10$-letter word, each letter is one of the four $D, U, K, E$ such that there are four consecutive letters in that word forming the letter $DUKE$ in this order. For example, $DUDKDUKEEK$ is a dukie word, but $DUEDKUKEDE$ is not. How many different dukie words can we construct in total? [b]p3.[/b] Rectangle $ABCD$ has sides $AB = 8$, $BC = 6$. $\vartriangle AEC$ is an isosceles right triangle with hypotenuse $AC$ and $E$ above $AC$. $\vartriangle BFD$ is an isosceles right triangle with hypotenuse $BD$ and $F$ below $BD$. Find the area of $BCFE$. [b]p4.[/b] Chris is playing with $6$ pumpkins. He decides to cut each pumpkin in half horizontally into a top half and a bottom half. He then pairs each top-half pumpkin with a bottom-half pumpkin, so that he ends up having six “recombinant pumpkins”. In how many ways can he pair them so that only one of the six top-half pumpkins is paired with its original bottom-half pumpkin? [b]p5.[/b] Matt comes to a pumpkin farm to pick $3$ pumpkins. He picks the pumpkins randomly from a total of $30$ pumpkins. Every pumpkin weighs an integer value between $7$ to $16$ (including $7$ and $16$) pounds, and there’re $3$ pumpkins for each integer weight between $7$ to $16$. Matt hopes the weight of the $3$ pumpkins he picks to form the length of the sides of a triangle. Let $m/n$ be the probability, in lowest terms, that Matt will get what he hopes for. Find the value of $m + n$ [b]p6.[/b] Let $a, b, c, d$ be distinct complex numbers such that $|a| = |b| = |c| = |d| = 3$ and $|a + b + c + d| = 8$. Find $|abc + abd + acd + bcd|$. [b]p7.[/b] A board contains the integers $1, 2, ..., 10$. Anna repeatedly erases two numbers $a$ and $b$ and replaces it with $a + b$, gaining $ab(a + b)$ lollipops in the process. She stops when there is only one number left in the board. Assuming Anna uses the best strategy to get the maximum number of lollipops, how many lollipops will she have? [b]p8.[/b] Ajay and Joey are playing a card game. Ajay has cards labelled $2, 4, 6, 8$, and $10$, and Joey has cards labelled $1, 3, 5, 7, 9$. Each of them takes a hand of $4$ random cards and picks one to play. If one of the cards is at least twice as big as the other, whoever played the smaller card wins. Otherwise, the larger card wins. Ajay and Joey have big brains, so they play perfectly. If $m/n$ is the probability, in lowest terms, that Joey wins, find $m + n$. [b]p9.[/b] Let $ABCDEFGHI$ be a regular nonagon with circumcircle $\omega$ and center $O$. Let $M$ be the midpoint of the shorter arc $AB$ of $\omega$, $P$ be the midpoint of $MO$, and $N$ be the midpoint of $BC$. Let lines $OC$ and $PN$ intersect at $Q$. Find the measure of $\angle NQC$ in degrees. [b]p10.[/b] In a $30 \times 30$ square table, every square contains either a kit-kat or an oreo. Let $T$ be the number of triples ($s_1, s_2, s_3$) of squares such that $s_1$ and $s_2$ are in the same row, and $s_2$ and $s_3$ are in the same column, with $s_1$ and $s_3$ containing kit-kats and $s_2$ containing an oreo. Find the maximum value of $T$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider the isosceles right triangle $ABC$ ($AB = AC$) and the points $M, P \in [AB]$ so that $AM = BP$. Let $D$ be the midpoint of the side $BC$ and $R, Q$ the intersections of the perpendicular from $A$ on$ CM$ with $CM$ and $BC$ respectively. Prove that a) $\angle AQC = \angle PQB$ b) $\angle DRQ = 45^o$

Given are two distinct monic cubics $F(x)$ and $G(x)$. All roots of the equations $F(x)=0$, $G(x)=0$ and $F(x)=G(x)$ are written down. There are eight numbers written. Prove that the greatest of them and the least of them cannot be both roots of the polynomial $F(x)$.