In the figure, $ABC$ and $CDE$ are right-angled and isosceles triangles. Segments $AD$ and $BC$ intersect at $P$, and segments $CD$ and $BE$ intersect at $Q$. (a) Show that segment$ PQ$ is parallel to segment $AE$. (b) If $BP = 4$ and $DQ = 9$, find the measure of segment $BD$. [img]https://cdn.artofproblemsolving.com/attachments/d/3/4c2c7514d71bbac68d58fc6de9ec2649e58957.png[/img]

Points $M$, $N$, $P$ are selected on sides $\overline{AB}$, $\overline{AC}$, $\overline{BC}$, respectively, of triangle $ABC$. Find the area of triangle $MNP$ given that $AM=MB=BP=15$ and $AN=NC=CP=25$. [i]Proposed by Evan Chen[/i]

Let $ABC$ be a triangle, and let $P$ be a point inside it such that $\angle PAC = \angle PBC$. The perpendiculars from $P$ to $BC$ and $CA$ meet these lines at $L$ and $M$, respectively, and $D$ is the midpoint of $AB$. Prove that $DL = DM.$

Given positive integer $n$, find the biggest real number $C$ which satisfy the condition that if the sum of the reciprocals of a set of integers (They can be the same.) that are greater than $1$ is less than $C$, then we can divide the set of numbers into no more than $n$ groups so that the sum of reciprocals of every group is less than $1$.

Let $\Omega$ be the circumcircle of a triangle $ABC$ with $\angle BAC > 90^{\circ}$ and $AB > AC$. The tangents of $\Omega$ at $B$ and $C$ cross at $D$ and the tangent of $\Omega$ at $A$ crosses the line $BC$ at $E$. The line through $D$, parallel to $AE$, crosses the line $BC$ at $F$. The circle with diameter $EF$ meets the line $AB$ at $P$ and $Q$ and the line $AC$ at $X$ and $Y$. Prove that one of the angles $\angle AEB$, $\angle PEQ$, $\angle XEY$ is equal to the sum of the other two.

Let $n$ points $A_1, A_2, \ldots, A_n$, ($n>2$), be considered in the space, where no four points are coplanar. Each pair of points $A_i, A_j$ are connected by an edge. Find the maximal value of $n$ for which we can paint all edges by two colors – blue and red such that the following three conditions hold: [b]I.[/b] Each edge is painted by exactly one color. [b]II.[/b] For $i = 1, 2, \ldots, n$, the number of blue edges with one end $A_i$ does not exceed 4. [b]III.[/b] For every red edge $A_iA_j$, we can find at least one point $A_k$ ($k \neq i, j$) such that the edges $A_iA_k$ and $A_jA_k$ are blue.

Let $[0, 1]$ be the set $\{x \in \mathbb{R} : 0 \leq x \leq 1\}$. Does there exist a continuous function $g : [0, 1] \to [0, 1]$ such that no line intersects the graph of $g$ infinitely many times, but for any positive integer $n$ there is a line intersecting $g$ more than $n$ times? [i]Proposed by Ethan Tan[/i]

Ten distinct points are placed on a circle. All ten of the points are paired so that the line segments connecting the pairs do not intersect. In how many different ways can this pairing be done? [asy] import graph; size(12cm); real labelscalefactor = 0.5; pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); pen dotstyle = black; draw((2.46,0.12)--(3.05,-0.69)); draw((2.46,1.12)--(4,-1)); draw((5.54,0.12)--(4.95,-0.69)); draw((3.05,1.93)--(5.54,1.12)); draw((4.95,1.93)--(4,2.24)); draw((8.05,1.93)--(7.46,1.12)); draw((7.46,0.12)--(8.05,-0.69)); draw((9,2.24)--(9,-1)); draw((9.95,-0.69)--(9.95,1.93)); draw((10.54,1.12)--(10.54,0.12)); draw((15.54,1.12)--(15.54,0.12)); draw((14.95,-0.69)--(12.46,0.12)); draw((13.05,-0.69)--(14,-1)); draw((12.46,1.12)--(14.95,1.93)); draw((14,2.24)--(13.05,1.93)); label("1",(-1.08,2.03),SE*labelscalefactor); label("2",(-0.3,1.7),SE*labelscalefactor); label("3",(0.05,1.15),SE*labelscalefactor); label("4",(0.00,0.38),SE*labelscalefactor); label("5",(-0.33,-0.12),SE*labelscalefactor); label("6",(-1.08,-0.4),SE*labelscalefactor); label("7",(-1.83,-0.19),SE*labelscalefactor); label("8",(-2.32,0.48),SE*labelscalefactor); label("9",(-2.3,1.21),SE*labelscalefactor); label("10",(-1.86,1.75),SE*labelscalefactor); dot((-1,-1),dotstyle); dot((-0.05,-0.69),dotstyle); dot((0.54,0.12),dotstyle); dot((0.54,1.12),dotstyle); dot((-0.05,1.93),dotstyle); dot((-1,2.24),dotstyle); dot((-1.95,1.93),dotstyle); dot((-2.54,1.12),dotstyle); dot((-2.54,0.12),dotstyle); dot((-1.95,-0.69),dotstyle); dot((4,-1),dotstyle); dot((4.95,-0.69),dotstyle); dot((5.54,0.12),dotstyle); dot((5.54,1.12),dotstyle); dot((4.95,1.93),dotstyle); dot((4,2.24),dotstyle); dot((3.05,1.93),dotstyle); dot((2.46,1.12),dotstyle); dot((2.46,0.12),dotstyle); dot((3.05,-0.69),dotstyle); dot((9,-1),dotstyle); dot((9.95,-0.69),dotstyle); dot((10.54,0.12),dotstyle); dot((10.54,1.12),dotstyle); dot((9.95,1.93),dotstyle); dot((9,2.24),dotstyle); dot((8.05,1.93),dotstyle); dot((7.46,1.12),dotstyle); dot((7.46,0.12),dotstyle); dot((8.05,-0.69),dotstyle); dot((14,-1),dotstyle); dot((14.95,-0.69),dotstyle); dot((15.54,0.12),dotstyle); dot((15.54,1.12),dotstyle); dot((14.95,1.93),dotstyle); dot((14,2.24),dotstyle); dot((13.05,1.93),dotstyle); dot((12.46,1.12),dotstyle); dot((12.46,0.12),dotstyle); dot((13.05,-0.69),dotstyle);[/asy]

Let $x$, $y$ and $z$ be positive real numbers such that $xyz = 1$. Prove that $$\left(1+x\right)\left(1+y\right)\left(1+z\right)\geq 2\left(1+\sqrt[3]{\frac{x}{z}}+\sqrt[3]{\frac{y}{x}}+\sqrt[3]{\frac{z}{y}}\right).$$

Suppose $f: \mathbb{R}^{+} \mapsto \mathbb{R}^{+}$ is a function such that $\frac{f(x)}{x}$ is increasing on $\mathbb{R}^{+}$. For $a,b,c>0$, prove that $$2\left (\frac{f(a)+f(b)}{a+b} + \frac{f(b)+f(c)}{b+c}+ \frac{f(c)+f(a)}{c+a} \right) \geq 3\left(\frac{f(a)+f(b)+f(c)}{a+b+c}\right) + \frac{f(a)}{a}+ \frac{f(b)}{b}+ \frac{f(c)}{c}$$

Let $O, I$ be the circumcenter and the incenter of a right-angled triangle, $R, r$ be the radii of respective circles, $J$ be the reflection of the vertex of the right angle in $I$. Find $OJ$.

Given triangle $ABC$, with $AC> BC$, and the it's circumcircle centered at $O$. Let $M$ be the point on the circumcircle of triangle $ABC$ so that $CM$ is the bisector of $\angle ACB$. Let $\Gamma$ be a circle with diameter $CM$. The bisector of $BOC$ and bisector of $AOC$ intersect $\Gamma$ at $P$ and $Q$, respectively. If $K$ is the midpoint of $CM$, prove that $P, Q, O, K$ lie at one point of the circle.

Determine each real root of \[x^4-(2\cdot10^{10}+1)x^2-x+10^{20}+10^{10}-1=0\] correct to four decimal places.

Find all real $x,y,z$ such that $\frac{x-2y}{y}+\frac{2y-4}{x}+\frac{4}{xy}=0$ and $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=2$.

Let $k$ and $n$ be positive integers with $n>2$. Show that the equation: \[x^n-y^n=2^k\] has no positive integer solutions.

The positive integers from $ 1$ to $n$ inclusive are written on a whiteboard. After one number is erased, the average (arithmetic mean) of the remaining $n - 1$ numbers is $22$. Knowing that $n$ is odd, determine $n$ and the number that was erased. Explain your reasoning.

Let $a_1 , a_2 ,... , a_n$ be an arrangement of the integers $1,2,..., n$. Let $$S=\frac{a_1}{1}+\frac{a_2}{2}+\frac{a_3}{3}+...+\frac{a_n}{1}.$$ Find a natural number $n$ such that among the values of $S$ for all arrangements $a_1 , a_2 ,... , a_n$ , all the integers from $n$ to $n + 100$ appear .

Let $(a_n)_{n\ge 1}$ be a sequence of real numbers such that \[a_1\binom{n}{1}+a_2\binom{n}{2}+\ldots+a_n\binom{n}{n}=2^{n-1}a_n,\ (\forall)n\in \mathbb{N}^*\] Prove that $(a_n)_{n\ge 1}$ is an arithmetical progression. [i]Lucian Dragomir[/i]

Given a $m \times n$ rectangle where $m,n\geq 2023$. The square in the $i$-th row and $j$-th column is filled with the number $i+j$ for $1\leq i \leq m, 1\leq j \leq n$. In each move, Alice can pick a $2023 \times 2023$ subrectangle and add $1$ to each number in it. Alice wins if all the numbers are multiples of $2023$ after a finite number of moves. For which pairs $(m,n)$ can Alice win? [i]Proposed by Boon Qing Hong[/i]

Prove that there exists a constant $C > 0$ such that, for any integers $m, n$ with $n \geq m > 1$ and any real number $x > 1$, $$\sum_{k=m}^{n}\sqrt[k]{x} \leq C\bigg(\frac{m^2 \cdot \sqrt[m-1]{x}}{\log{x}} + n\bigg)$$

Let \( \triangle ABC \) be an acute triangle with orthocenter \( H \), and let \( M \) be the midpoint of \( BC \). Let \( P \) be the foot of the perpendicular from \( H \) to \( AM \). Prove that \( AM \cdot MP = BM^2 \).

A circle whose center is on the side $ED$ of the cyclic quadrilateral $BCDE$ touches the other three sides. Prove that $EB+CD = ED.$

The five sides and five diagonals of a regular pentagon are drawn on a piece of paper. Two people play a game, in which they take turns to colour one of these ten line segments. The first player colours line segments blue, while the second player colours line segments red. A player cannot colour a line segment that has already been coloured. A player wins if they are the first to create a triangle in their own colour, whose three vertices are also vertices of the regular pentagon. The game is declared a draw if all ten line segments have been coloured without a player winning. Determine whether the first player, the second player, or neither player can force a win.

A new meme is circling around social media known as the [i]DaDerek Convertible[/i]. The license plate number of the [i]DaDerek Convertible[/i] is such that the product of its nonzero digits times $5$ is equal to itself. Given that its license plate number has less than or equal to $3$ digits and that it has at least one nonzero digit, find the [i]DaDerek Convertible[/i]'s license plate number.

Find all triplets of real numbers $(x,y,z)$ that are solutions to the system of equations $x^2+y^2+25z^2=6xz+8yz$ $ 3x^2+2y^2+z^2=240$