Point $X$ moves along side $AB$ of triangle $ABC$, and point $Y$ moves along its circumcircle in such a way that line $XY$ passes through the midpoint of arc $AB$. Find the locus of the circumcenters of triangles $IXY$ , where I is the incenter of $ ABC$.

Let $O$ denote the circumcentre of an acute-angled triangle $ABC$. Let point $P$ on side $AB$ be such that $\angle BOP = \angle ABC$, and let point $Q$ on side $AC$ be such that $\angle COQ = \angle ACB$. Prove that the reflection of $BC$ in the line $PQ$ is tangent to the circumcircle of triangle $APQ$.

An integer $N$ which satisfies exactly three of the four following conditions is called [i]two-good[/i]. $~$ [center] (I) $N$ is divisible by $2$ (II) $N$ is divisible by $4$ (III) $N$ is divisible by $8$ (IV) $N$ is divisible by $16$ [/center]$~$ How many integers between $1$ and $100$, inclusive, are [i]two-good[/i]? $\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }10$

Let \( M \) and \( N \) be the midpoints of sides \( BC \) and \( AC \), respectively, in an acute-angled triangle \( ABC \). Suppose there exists a point \( P \) on the line segment \( AM \) such that \( \angle NPC = \angle MPC \). Let \( D \) be the intersection point of the line \( NP \) and the line parallel to \( CP \) passing through \( B \). Prove that \( AD = AB \). Proposed by Soroush Behroozifar

Suppose a sequence of reals $\{a_n\}_{n\geq 0}$ satisfies $a_0 = 0$, $\frac{100}{101} <a_{100}<1$, and $$2a_n - a_{n-1} -a_{n+1} \leq 2 (1-a_n )^3$$ for every $n\geq 1$. (1) Define a sequence $b_n = a_n - \frac{n}{n+1}$. Prove that $b_n\leq b_{n+1}$ for any $n\geq 100$. (2) Determine whether infinite series $\sum_{n=1}^\infty \frac{a_n}{n^2}$ converges or diverges.

Find all functions $f: \mathbb{Z} \to \mathbb{Z}$, such that $$f(x+y)((f(x) - f(y))^2+f(xy))=f(x^3)+f(y^3)$$ for all integers $x, y$.

[u]Round 1[/u] [b]1.1.[/b] A classroom has $29$ students. A teacher needs to split up the students into groups of at most $4$. What is the minimum number of groups needed? [b]1.2.[/b] On his history map quiz, Eric recalls that Sweden, Norway and Finland are adjacent countries, but he has forgotten which is which, so he labels them in random order. The probability that he labels all three countries correctly can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [b]1.3.[/b] In a class of $40$ sixth graders, the class average for their final test comes out to be $90$ (out of a $100$). However, a student brings up an issue with problem $5$, and $10$ students receive credit for this question, bringing the class average to a $90.75$. How many points was problem $5$ worth? [u]Round 2[/u] [b]2.1.[/b] Compute $1 - 2 + 3 - 4 + ... - 2022 + 2023$. [b]2.2.[/b] In triangle $ABC$, $\angle ABC = 75^o$. Point $D$ lies on side $AC$ such that $BD = CD$ and $\angle BDC$ is a right angle. Compute the measure of $\angle A$. [b]2.3.[/b] Joe is rolling three four-sided dice each labeled with positive integers from $1$ to $4$. The probability the sum of the numbers on the top faces of the dice is $6$ can be written as $\frac{p}{q}$ where $p$ and $q$ are relatively prime integers. Find $p + q$. [u]Round 3[/u] [b]3.1.[/b] For positive integers $a, b, c, d$ that satisfy $a + b + c + d = 23$, what is the maximum value of $abcd$? [b]3.2.[/b] A buckball league has twenty teams. Each of the twenty teams plays exactly five games with each of the other teams. If each game takes 1 hour and thirty minutes, then how many total hours are spent playing games? [b]3.3.[/b] For a triangle $\vartriangle ABC$, let $M, N, O$ be the midpoints of $AB$, $BC$, $AC$, respectively. Let $P, Q, R$ be points on $AB$, $BC$, $AC$ such that $AP =\frac13 AB$, $BQ =\frac13 BC$, and $CR =\frac13 AC$. The ratio of the areas of $\vartriangle MNO$ and $\vartriangle P QR$ can be expressed as $\frac{m}{n}$ , where $ m$ and $n$ are relatively prime positive integers. Find $m + n$. [u]Round 4[/u] [b]4.1.[/b] $2023$ has the special property that leaves a remainder of $1$ when divided by $2$, $21$ when divided by $22$, and $22$ when divided by $23$. Let $n$ equal the lowest integer greater than $2023$ with the above properties. What is $n$? [b]4.2.[/b] Ants $A, B$ are on points $(0, 0)$ and $(3, 3)$ respectively, and ant A is trying to get to $(3, 3)$ while ant $B$ is trying to get to $(0, 0)$. Every second, ant $A$ will either move up or right one with equal probability, and ant $B$ will move down or left one with equal probability. The probability that the ants will meet each other be $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Find $a + b$. [b]4.3.[/b] Find the number of trailing zeros of $100!$ in base $ 49$. PS. You should use hide for answers. Rounds 5-9 have been posted [url=https://artofproblemsolving.com/community/c3h3129723p28347714]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $\Delta ABC$ be a triangle with circumcenter $O$ and circumcircle $w$. Let $S$ be the center of the circle which is tangent with $AB$, $AC$, and $w$ (in the inside), and let the circle meet $w$ at point $K$. Let the circle with diameter $AS$ meet $w$ at $T$. If $M$ is the midpoint of $BC$, show that $K,T,M,O$ are concyclic.

Let $a$ be a real number and $A = \{(x, y) \in R \times R | \, x + y = a\}$, $B = \{(x,y) \in R \times R | \, x^3 + y^3 < a\}$ . Find all values of $a$ such that $A \cap B = \emptyset$ .

$a_1=b_1=1$ $a_{n+1}=b_n+\frac{1}{n}$ $b_{n+1}=a_n-\frac{1}{n}$ Prove that $a_n$, $b_n$ is not convergent, but $a_nb_n$ is convergent Laurentin Panaitopol

For all positive integers $n > 1$, let $f(n)$ denote the largest odd proper divisor of $n$ (a proper divisor of $n$ is a positive divisor of $n$ except for $n$ itself). Given that $N=20^{23}\cdot23^{20}$, compute \[\frac{f(N)}{f(f(f(N)))}.\]

A deck of $52$ cards is given. There are four suites each having cards numbered $1,2,\dots, 13$. The audience chooses some five cards with distinct numbers written on them. The assistant of the magician comes by, looks at the five cards and turns exactly one of them face down and arranges all five cards in some order. Then the magician enters and with an agreement made beforehand with the assistant, he has to determine the face down card (both suite and number). Explain how the trick can be completed.

Choose nine of the digits from $0$ to $9$ and place them in the boxes in the figure so that there are no repeated digits and the indicated sum is correct. [img]https://cdn.artofproblemsolving.com/attachments/6/2/7f06575ec70eb9ddd58c6cf9dd3cb60d306e7c.png[/img] Which digit was not used? You can fill in the boxes so that the unused digit is other?

In an exam every question is solved by exactly four students, every pair of questions is solved by exactly one student, and none of the students solved all of the questions. Find the maximum possible number of questions in this exam.

Find all ordered triples $(a,b,c)$ of positive integers such that the value of the expression \[\left (b-\frac{1}{a}\right )\left (c-\frac{1}{b}\right )\left (a-\frac{1}{c}\right )\] is an integer.

Given a checkered rectangle of size n × m. Is it always possible to mark $3$ or $4$ nodes of a rectangle so that at least one of the marked nodes lay on each straight line containing the side of the rectangle, and the non-self-intersecting polygon with vertices at these nodes has an area equal to $$\dfrac{1}{2}\min \left ( \text{gcd}(n, m), \dfrac{n+m}{\text{gcd}(n, m)} \right)$$?