Each point in the plane is labeled with a real number. Show that there exist two distinct points $P$ and $Q$ whose labels differ by less than the distance from $P$ to $Q$. [i]Holden Mui[/i]

Let $x, y, z, t$ be real numbers such that $(x^2 + y^2 -1)(z^2 + t^2 - 1) > (xz + yt -1)^2$. Prove that $x^2 + y^2 > 1$.

How many pairs $ (m,n)$ of positive integers with $ m < n$ fulfill the equation $ \frac {3}{2008} \equal{} \frac 1m \plus{} \frac 1n$?

Let $ABC$ be an acute-angled triangle with circumcircle $\omega$ and circumcentre $O$. Points $D\neq B$ and $E\neq C$ lie on $\omega$ such that $BD\perp AC$ and $CE\perp AB$. Let $CO$ meet $AB$ at $X$, and $BO$ meet $AC$ at $Y$. Prove that the circumcircles of triangles $BXD$ and $CYE$ have an intersection lie on line $AO$. [i]Ivan Chan Kai Chin, Malaysia[/i]

Let $ABCD$ be a cyclic quadrilateral inscirbed in a circle $O$ ($AB> AD$), and let $E$ be a point on segment $AB$ such that $AE = AD$. Let $AC \cap DE = F$, and $DE \cap O = K(\ne D)$. The tangent to the circle passing through $C,F,E$ at $E$ hits $AK$ at $L$. Prove that $AL = AD$ if and only if $\angle KCE = \angle ALE$.

Ria has $4$ green marbles and 8 red marbles. She arranges them in a circle randomly, if the probability that no two green marbles are adjacent is $\frac{p}{q}$ where the positive integers $p,q$ have no common factors other than $1$, what is $p+q$?

Let $f$ and $g$ be two nonzero polynomials with integer coefficients and $\deg f>\deg g$. Suppose that for infinitely many primes $p$ the polynomial $pf+g$ has a rational root. Prove that $f$ has a rational root.

Fifteen boys are standing on a field, and each of them has a ball. No two distances between two of the boys are equal. Each boy throws his ball to the boy standing closest to him. (a) Show that one of the boys does not get any ball. (b) Prove that none of the boys gets more than five balls.

Stekel and Prick play a game on an $ m \times n$ board, where $m$ and $n$ are positive are integers. They alternate turns, with Stekel starting. Spine bets on his turn, he always takes a pawn on a square where there is no pawn yet. Prick does his turn the same, but his pawn must always come into a square adjacent to the square that Spike just placed a pawn in on his previous turn. Prick wins like the whole board is full of pawns. Spike wins if Prik can no longer move a pawn on his turn, while there is still at least one empty square on the board. Determine for all pairs $(m, n)$ who has a winning strategy.

Prove that if \( a \), \( b \), and \( c \) are positive real numbers, then the following inequality holds: \[ \frac{a + 3c}{a + b} + \frac{c + 3a}{b + c} + \frac{4b}{c + a} \geq 6. \]

Suppose that $$f(x) = \sum_{i=0}^\infty c_ix^i$$ is a power series for which each coefficient $c_i$ is $0$ or $1$. Show that if $f(2/3) = 3/2$, then $f(1/2)$ must be irrational.

For a positive integer $k$, let $d(k)$ denote the number of divisors of $k$ and let $s(k)$ denote the digit sum of $k$. A positive integer $n$ is said to be [i]amusing[/i] if there exists a positive integer $k$ such that $d(k)=s(k)=n$. What is the smallest amusing odd integer greater than $1$?

Let $M$be an interior point of triangle $ABC$. Let the line $AM$ intersect the circumcircle of the triangle $MBC$ for the second time at point $D$, the line $BM$ intersect the circumcircle of the triangle $MCA$ for the second time at point $E$, and the line $CM$ intersect the circumcircle of the triangle $MAB$ for the second time at point $F$. Prove that $\frac{AD}{MD} + \frac{BE}{ME} + \frac{CF}{MF} \geq \frac{9}{2}$. [i]Proposed by Nairy Sedrakyan[/i]

What number is exactly halfway between $\frac 1 6$ and $\frac 1 4$?

Let $K$ and $L$ be the centers of the excircles of a non-isosceles triangle $ABC$ opposite $B$ and $C$ respectively. Let $B_1$ and $C_1$ be the midpoints of the sides $AC$ and $AB$ respectively Let $M$ and $N$ be symmetric to $B$ and $C$ about $B_1$ and $C_1$ respectively. Prove that the lines $KM$ and $LN$ meet on $BC$.

A jar contains 4 blue marbles, 3 green marbles, and 5 red marbles. If Helen reaches in the jar and selects a marble at random, then the probability that she selects a red marble can be expressed as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

The following operation is performed over points $O_1, O_2, O_3$ and $A$ in space. The point $A$ is reflected with respect to $O_1$, the resultant point $A_1$ is reflected through $O_2$, and the resultant point $A_2$ through $O_3$. We get some point $A_3$ that we will also consecutively reflect through $O_1, O_2, O_3$. Prove that the point obtained last coincides with $A$..

[b]Problem 3[/b] : Let $x_1,x_2,\cdots,x_{2022}$ be non-negative real numbers such that $$x_k + x_{k+1}+x_{k+2} \leq 2$$ for all $k = 1,2,\cdots,2020$. Prove that $$\sum_{k=1}^{2020}x_kx_{k+2}\leq 1010$$

[b]p1.[/b] Darth Vader urgently needed a new Death Star battle station. He sent requests to four planets asking how much time they would need to build it. The Mandalorians answered that they can build it in one year, the Sorganians in one and a half year, the Nevarroins in two years, and the Klatoonians in three years. To expedite the work Darth Vader decided to hire all of them to work together. The Rebels need to know when the Death Star is operational. Can you help the Rebels and find the number of days needed if all four planets work together? We assume that one year $= 365$ days. [b]p2.[/b] Solve the inequality: $\left( \sin \frac{\pi}{12} \right)^{\sqrt{1-x}} > \left( \sin \frac{\pi}{12} \right)^x$ [b]p3.[/b] Solve the equation: $\sqrt{x^2 + 4x + 4} = x^2 + 3x - 6$ [b]p4.[/b] Solve the system of inequalities on $[0, 2\pi]$: $$\sin (2x) \ge \sin (x)$$ $$\cos (2x) \le \cos (x)$$ [b]p5.[/b] The planet Naboo is under attack by the imperial forces. Three rebellian camps are located at the vertices of a triangle. The roads connecting the camps are along the sides of the triangle. The length of the first road is less than or equal to $20$ miles, the length of the second road is less than or equal to $30$ miles, and the length of the third road is less than or equal to $45$ miles. The Rebels have to cover the area of this triangle by a defensive field. What is the maximal area that they may need to cover? [b]p6.[/b] The Lake Country on the planet Naboo has the shape of a square. There are nine roads in the country. Each of the roads is a straight line that divides the country into two trapezoidal parts such that the ratio of the areas of these parts is $2:5$. Prove that at least three of these roads intersect at one point. PS. You should use hide for answers.

Let $D$ be the foot of the internal angle bisector of the angle $A$ of a triangle $ABC$. Let $E$ be a point on side $[AC]$ such that $|CE|= |CD|$ and $|AE|=6\sqrt 5$; let $F$ be a point on the ray $[AB$ such that $|DB|=|BF|$ and $|AB|<|AF| = 8\sqrt 5$. What is $|AD|$? $ \textbf{(A)}\ 10\sqrt 5 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 4\sqrt{15} \qquad\textbf{(D)}\ 7\sqrt 5 \qquad\textbf{(E)}\ \text{None of above} $

For positive integers $a, b, c$ (not necessarily distinct), suppose that $a+bc, b+ac, c+ab$ are all perfect squares. Show that $$a^2(b+c)+b^2(a+c)+c^2(a+b)+2abc$$ can be written as sum of two squares.

[hide=E stands for Euclid , L stands for Lobachevsky]they had two problem sets under those two names[/hide] [b]E1.[/b] How many positive divisors does $72$ have? [b]E2 / L2.[/b] Raymond wants to travel in a car with $3$ other (distinguishable) people. The car has $5$ seats: a driver’s seat, a passenger seat, and a row of $3$ seats behind them. If Raymond’s cello must be in a seat next to him, and he can’t drive, but every other person can, how many ways can everyone sit in the car? [b]E3 / L3.[/b] Peter wants to make fruit punch. He has orange juice ($100\%$ orange juice), tropical mix ($25\%$ orange juice, $75\%$ pineapple juice), and cherry juice ($100\%$ cherry juice). If he wants his final mix to have $50\%$ orange juice, $10\%$ cherry juice, and $40\%$ pineapple juice, in what ratios should he mix the $3$ juices? Please write your answer in the form (orange):(tropical):(cherry), where the three integers are relatively prime. [b]E4 / L4.[/b] Points $A, B, C$, and $D$ are chosen on a circle such that $m \angle ACD = 85^o$, $m\angle ADC = 40^o$,and $m\angle BCD = 60^o$. What is $m\angle CBD$? [b]E5.[/b] $a, b$, and $c$ are positive real numbers. If $abc = 6$ and $a + b = 2$, what is the minimum possible value of $a + b + c$? [b]E6 / L5.[/b] Circles $A$ and $B$ are drawn on a plane such that they intersect at two points. The centers of the two circles and the two intersection points lie on another circle, circle $C$. If the distance between the centers of circles $A$ and $B$ is $20$ and the radius of circle $A$ is $16$, what is the radius of circle $B$? [b]E7.[/b] Point $P$ is inside rectangle $ABCD$. If $AP = 5$, $BP = 6$, and $CP = 7$, what is the length of $DP$? [b]E8 / L6.[/b] For how many integers $n$ is $n^2 + 4$ divisible by $n + 2$? [b]E9. [/b] How many of the perfect squares between $1$ and $10000$, inclusive, can be written as the sum of two triangular numbers? We define the $n$th triangular number to be $1 + 2 + 3 + ... + n$, where $n$ is a positive integer. [b]E10 / L7.[/b] A small sphere of radius $1$ is sitting on the ground externally tangent to a larger sphere, also sitting on the ground. If the line connecting the spheres’ centers makes a $60^o$ angle with the ground, what is the radius of the larger sphere? [b]E11 / L8.[/b] A classroom has $12$ chairs in a row and $5$ distinguishable students. The teacher wants to position the students in the seats in such a way that there is at least one empty chair between any two students. In how many ways can the teacher do this? [b]E12 / L9.[/b] Let there be real numbers $a$ and $b$ such that $a/b^2 + b/a^2 = 72$ and $ab = 3$. Find the value of $a^2 + b^2$. [b]E13 / L10.[/b] Find the number of ordered pairs of positive integers $(x, y)$ such that $gcd \, (x, y)+lcm \, (x, y) =x + y + 8$. [b]E14 / L11.[/b] Evaluate $\sum_{i=1}^{\infty}\frac{i}{4^i}=\frac{1}{4} +\frac{2}{16} +\frac{3}{64} +...$ [b]E15 / L12.[/b] Xavier and Olivia are playing tic-tac-toe. Xavier goes first. How many ways can the game play out such that Olivia wins on her third move? The order of the moves matters. [b]L1.[/b] What is the sum of the positive divisors of $100$? [b]L13.[/b] Let $ABCD$ be a convex quadrilateral with $AC = 20$. Furthermore, let $M, N, P$, and $Q$ be the midpoints of $DA, AB, BC$, and $CD$, respectively. Let $X$ be the intersection of the diagonals of quadrilateral $MNPQ$. Given that $NX = 12$ and $XP = 10$, compute the area of $ABCD$. [b]L14.[/b] Evaluate $(\sqrt3 + \sqrt5)^6$ to the nearest integer. [b]L15.[/b] In Hatland, each citizen wears either a green hat or a blue hat. Furthermore, each citizen belongs to exactly one neighborhood. On average, a green-hatted citizen has $65\%$ of his neighbors wearing green hats, and a blue-hatted citizen has $80\%$ of his neighbors wearing blue hats. Each neighborhood has a different number of total citizens. What is the ratio of green-hatted to blue-hatted citizens in Hatland? (A citizen is his own neighbor.) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that all positive real $x, y, z$ satisfy the inequality $x^y + y^z + z^x > 1$. (D. Bazylev)

Given a quadrilateral \(ABCD\), point \(M\) is the midpoint of the side \(CD\). It turns out that \(\angle BMA = 90^{\circ}\) and \(\angle MAB = \angle CBD\). Prove that \(AC = AB\). [i]Proposed by Anton Trygub[/i]

Show that \[ \min \{ |PA|, |PB|, |PC| \} + |PA| + |PB| + |PC| < |AB|+|BC|+|CA| \] if $P$ is a point inside $\triangle ABC$.