Let $\triangle{ABC}$ be an equilateral triangle of side length $6.$ Let $P$ be a point inside $\triangle{ABC}$ such that $\angle{BPC}=120^\circ.$ The circle with diameter $\overline{AP}$ meets the circumcircle of $\triangle{ABC}$ again at $X \neq A.$ Given that $AX=5,$ compute $XP.$

Let $a,b, c, d,e$ be the roots of $p(x) = 2x^5 - 3x^3 + 2x -7$. Find the value of $$(a^3 - 1)(b^3 - 1)(c^3 - 1)(d^3 - 1)(e^3 - 1).$$

Find all function $ f: \mathbb{R}^{+} \to \mathbb{R}^{+}$ such that for every three real positive number $x,y,z$ : $$ x+f(y) , f(f(y)) + z , f(f(z))+f(x) $$ are length of three sides of a triangle and for every postive number $p$ , there is a triangle with these sides and perimeter $p$. [i]Proposed by Amirhossein Zolfaghari [/i]

Let $a_n$ be the number of ways to express $n$ as an ordered sum of powers of $3.$ For example $a_4=3,$ since $$4=1+1+1+1=1+3=3+1.$$ Let $b_n$ denote the remainder upon dividing $a_n$ by $3.$ Evaluate $$\sum_{n=1}^{3^{2025}} b_n.$$

Let $ \{a_n\}^{\infty}_1$ be a sequence of real numbers such that $ a_1 \equal{} 2,$ and \[ a_{n\plus{}1} \equal{} a^2_n \minus{} a_n \plus{} 1, \forall n \in \mathbb{N}.\] Prove that \[ 1 \minus{} \frac{1}{2003^{2003}} < \sum^{2003}_{i\equal{}1} \frac{1}{a_i} < 1.\]

Let $O$ be the circumcenter and $H$ the orthocenter of an acute-triangle $ABC$. The perpendicular bisector of $AO$ intersects the line $BC$ at point $S$. Let $L$ be the midpoint of $OH$. Prove that $\angle OAH = \angle LSA$.

Alice is drawing a shape on a piece of paper. She starts by placing her pencil at the origin, and then draws line segments of length one, alternating between vertical and horizontal segments. Eventually, her pencil returns to the origin, forming a closed, non-self-intersecting shape. Show that the area of this shape is even if and only if its perimeter is a multiple of eight.

Suppose that $f : \mathbb{N} \rightarrow \mathbb{N}$ is a function for which the expression $af(a)+bf(b)+2ab$ for all $a,b \in \mathbb{N}$ is always a perfect square. Prove that $f(a)=a$ for all $a \in \mathbb{N}$.

Let $ABC$ be an equilateral triangle. A point $T$ is chosen on $AC$ and on arcs $AB$ and $BC$ of the circumcircle of $ABC$, $M$ and $N$ are chosen respectively, so that $MT$ is parallel to $BC$ and $NT$ is parallel to $AB$. Segments $AN$ and $MT$ intersect at point $X$, while $CM$ and $NT$ intersect in point $Y$. Prove that the perimeters of the polygons $AXYC$ and $XMBNY$ are the same.

Let $A,B,C$ be angles of an acute triangle with \begin{align*} \cos^2 A + \cos^2 B + 2 \sin A \sin B \cos C &= \frac{15}{8} \text{ and} \\ \cos^2 B + \cos^2 C + 2 \sin B \sin C \cos A &= \frac{14}{9}. \end{align*} There are positive integers $p$, $q$, $r$, and $s$ for which \[ \cos^2 C + \cos^2 A + 2 \sin C \sin A \cos B = \frac{p-q\sqrt{r}}{s}, \] where $p+q$ and $s$ are relatively prime and $r$ is not divisible by the square of any prime. Find $p+q+r+s$. [i]Note: due to an oversight by the exam-setters, there is no acute triangle satisfying these conditions. You should instead assume $ABC$ is obtuse with $\angle B > 90^{\circ}$.[/i]

Let $ABCD$ be a parallelogram with $AB=CD=16$ and $BC=AD=24.$ Suppose the angle bisectors of $\angle A$ and $\angle D$ intersect $BC$ at $E$ and $F,$ respectively. Moreover, suppose $AE$ and $DF$ intersect at $P.$ Given that the sum of the areas of quadrilaterals $ABFP$ and $DCEP$ is $100,$ compute the area of the parallelogram.

The sum of several positive numbers is equal to $10$, and the sum of their squares is greater than $20$. Prove that the sum of the cubes of these numbers is greater than $40$. [i](5 points)[/i]

Let $a$ and $b$ be positive integers. Show that if $a^3+b^3$ is the square of an integer, then $a + b$ is not a product of two different prime numbers.

In quadrilateral $ABCD$, $\angle DAB=\angle ABC=110^{\circ}$, $\angle BCD=35^{\circ}$, $\angle CDA=105^{\circ}$, and $AC$ bisects $\angle DAB$. Find $\angle ABD$.

Let be a sequence $ \left( x_n \right)_{n\ge 0} $ with $ x_0\in (0,1) $ and defined as $$ 2x_n=x_{n-1}+\sqrt{3-3x_{n-1}^2} . $$ Prove that this sequence is bounded and periodic. Moreover, find $ x_0 $ for which this sequence is convergent. [i]Ovidiu Țâțan[/i]

Let a regular $7$-gon $A_0A_1A_2A_3A_4A_5A_6$ be inscribed in a circle. Prove that for any two points $P, Q$ on the arc $A_0A_6$ the following equality holds: \[\sum_{i=0}^6 (-1)^{i} PA_i = \sum_{i=0}^6 (-1)^{i} QA_i .\]

[u]Round 1[/u] [b]p1.[/b] The fractal T-shirt for this year's Duke Math Meet is so complicated that the printer broke trying to print it. Thus, we devised a method for manually assembling each shirt - starting with the full-size 'base' shirt, we paste a smaller shirt on top of it. And then we paste an even smaller shirt on top of that one. And so on, infinitely many times. (As you can imagine, it took a while to make all the shirts.) The completed T-shirt consists of the original 'base' shirt along with all of the shirts we pasted onto it. Now suppose the base shirt requires $2011$ $cm^2$ of fabric to make, and that each pasted-on shirt requires $4/5$ as much fabric as the previous one did. How many $cm^2$ of fabric in total are required to make one complete shirt? [b]p2.[/b] A dog is allowed to roam a yard while attached to a $60$-meter leash. The leash is anchored to a $40$-meter by $20$-meter rectangular house at the midpoint of one of the long sides of the house. What is the total area of the yard that the dog can roam? [b]p3.[/b] $10$ birds are chirping on a telephone wire. Bird $1$ chirps once per second, bird $2$ chirps once every $2$ seconds, and so on through bird $10$, which chirps every $10$ seconds. At time $t = 0$, each bird chirps. Define $f(t)$ to be the number of birds that chirp during the $t^{th}$ second. What is the smallest $t > 0$ such that $f(t)$ and $f(t + 1)$ are both at least $4$? [u]Round 2[/u] [b]p4.[/b] The answer to this problem is $3$ times the answer to problem 5 minus $4$ times the answer to problem 6 plus $1$. [b]p5.[/b] The answer to this problem is the answer to problem 4 minus $4$ times the answer to problem 6 minus $1$. [b]p6.[/b] The answer to this problem is the answer to problem 4 minus $2$ times the answer to problem 5. [u]Round 3[/u] [b]p7.[/b] Vivek and Daniel are playing a game. The game ends when one person wins $5$ rounds. The probability that either wins the first round is $1/2$. In each subsequent round the players have a probability of winning equal to the fraction of games that the player has lost. What is the probability that Vivek wins in six rounds? [b]p8.[/b] What is the coefficient of $x^8y^7$ in $(1 + x^2 - 3xy + y^2)^{17}$? [b]p9.[/b] Let $U(k)$ be the set of complex numbers $z$ such that $z^k = 1$. How many distinct elements are in the union of $U(1),U(2),...,U(10)$? [u]Round 4[/u] [b]p10.[/b] Evaluate $29 {30 \choose 0}+28{30 \choose 1}+27{30 \choose 2}+...+0{30 \choose 29}-{30\choose 30}$. You may leave your answer in exponential format. [b]p11.[/b] What is the number of strings consisting of $2a$s, $3b$s and $4c$s such that $a$ is not immediately followed by $b$, $b$ is not immediately followed by $c$ and $c$ is not immediately followed by $a$? [b]p12.[/b] Compute $\left(\sqrt3 + \tan (1^o)\right)\left(\sqrt3 + \tan (2^o)\right)...\left(\sqrt3 + \tan (29^o)\right)$. [u]Round 5[/u] [b]p13.[/b] Three massless legs are randomly nailed to the perimeter of a massive circular wooden table with uniform density. What is the probability that the table will not fall over when it is set on its legs? [b]p14.[/b] Compute $$\sum^{2011}_{n=1}\frac{n + 4}{n(n + 1)(n + 2)(n + 3)}$$ [b]p15.[/b] Find a polynomial in two variables with integer coefficients whose range is the positive real numbers. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that for any three pairwise different integer numbers $x,y,z$ the expression $(x-y)^5 + (y-z)^5 + (z-x)^5$ is divisible by $5(x-y)(y-z)z-x)$.

Let $f(x)$ and $g(x)$ be strictly increasing linear functions from $\mathbb R $ to $\mathbb R $ such that $f(x)$ is an integer if and only if $g(x)$ is an integer. Prove that for any real number $x$, $f(x)-g(x)$ is an integer.

Given $n$ numbers $a_1,a_2,...,a_n$ ($n\geq 3$) where $a_i\in\{0,1\}$ for all $i=1,2.,,,.n$. Consider $n$ following $n$-tuples \[ \begin{aligned} S_1 & =(a_1,a_2,...,a_{n-1},a_n)\\ S_2 & =(a_2,a_3,...,a_n,a_1)\\ & \vdots\\ S_n & =(a_n,a_1,...,a_{n-2},a_{n-1}).\end{aligned}\] For each tuple $r=(b_1,b_2,...,b_n)$, let \[ \omega (r)=b_1\cdot 2^{n-1}+b_2\cdot 2^{n-2}+\cdots+b_n. \] Assume that the numbers $\omega (S_1),\omega (S_2),...,\omega (S_n)$ receive exactly $k$ different values. a) Prove that $k|n$ and $\frac{2^n-1}{2^k-1}|\omega (S_i)\quad\forall i=1,2,...,n.$ b) Let \[ \begin{aligned} M & =\max _{i=\overline{1,n}}\omega (S_i)\\ m & =\min _{i=\overline{1,n}}\omega (S_i). \end{aligned} \] Prove that \[ M-m\geq\frac{(2^n-1)(2^{k-1}-1)}{2^k-1}. \]

Some of the $n$ faces of a polyhedron are colored in black such that any two black-colored faces have no common vertex. The rest of the faces of the polyhedron are colored in white. Prove that the number of common sides of two white-colored faces of the polyhedron is at least $n-2$.

Compute the maximum area of a triangle having a median of length 1 and a median of length 2.

Source: 2018 Canadian Open Math Challenge Part C Problem 2 ----- Alice has two boxes $A$ and $B$. Initially box $a$ contains $n$ coins and box $B$ is empty. On each turn, she may either move a coin from box $a$ to box $B$, or remove $k$ coins from box $A$, where $k$ is the current number of coins in box $B$. She wins when box $A$ is empty. $\text{(a)}$ If initially box $A$ contains 6 coins, show that Alice can win in 4 turns. $\text{(b)}$ If initially box $A$ contains 31 coins, show that Alice cannot win in 10 turns. $\text{(c)}$ What is the minimum number of turns needed for Alice to win if box $A$ initially contains 2018 coins?

Faces of a convex polyhedron are six squares and 8 equilateral triangles and each edge is a common side for one triangle and one square. All dihedral angles obtained from the triangle and square with a common edge, are equal. Prove that it is possible to circumscribe a sphere around the polyhedron, and compute the ratio of the squares of volumes of that polyhedron and of the ball whose boundary is the circumscribed sphere.

A quadrilateral is cut from a piece of gift wrapping paper, which has equally wide white and gray stripes. The grey stripes in the quadrilateral have a combined area of $10$. Determine the area of the quadrilateral. [img]https://1.bp.blogspot.com/-ia13b4RsNs0/XzP0cepAcEI/AAAAAAAAMT8/0UuCogTRyj4yMJPhfSK3OQihRqfUT7uSgCLcBGAsYHQ/s0/2020%2Bmohr%2Bp2.png[/img]