Let $ABC$ be a triangle with circumcircle $\Omega$ and incentre $I$. Let the line passing through $I$ and perpendicular to $CI$ intersect the segment $BC$ and the arc $BC$ (not containing $A$) of $\Omega$ at points $U$ and $V$ , respectively. Let the line passing through $U$ and parallel to $AI$ intersect $AV$ at $X$, and let the line passing through $V$ and parallel to $AI$ intersect $AB$ at $Y$ . Let $W$ and $Z$ be the midpoints of $AX$ and $BC$, respectively. Prove that if the points $I, X,$ and $Y$ are collinear, then the points $I, W ,$ and $Z$ are also collinear. [i]Proposed by David B. Rush, USA[/i]

Determine the maximal length $L$ of a sequence $a_1,\dots,a_L$ of positive integers satisfying both the following properties: [list=disc] [*]every term in the sequence is less than or equal to $2^{2023}$, and [*]there does not exist a consecutive subsequence $a_i,a_{i+1},\dots,a_j$ (where $1\le i\le j\le L$) with a choice of signs $s_i,s_{i+1},\dots,s_j\in\{1,-1\}$ for which \[s_ia_i+s_{i+1}a_{i+1}+\dots+s_ja_j=0.\] [/list]

Given integers $a$, $ b$ and $c$, $c\ne b$. It is known that the square trinomials $ax^2 + bx + c$ and $(c-b)x^2 + (c- a)x + (a + b)$ have a common root (not necessarily integer). Prove that $a+b+2c$ is divisible by $3$.

Triangle $ABC$ has sides $AB = 25$, $BC = 30$, and $CA=20$. Let $P,Q$ be the points on segments $AB,AC$, respectively, such that $AP=5$ and $AQ=4$. Suppose lines $BQ$ and $CP$ intersect at $R$ and the circumcircles of $\triangle{BPR}$ and $\triangle{CQR}$ intersect at a second point $S\ne R$. If the length of segment $SA$ can be expressed in the form $\frac{m}{\sqrt{n}}$ for positive integers $m,n$, where $n$ is not divisible by the square of any prime, find $m+n$. [i]Victor Wang[/i]

The polynomial $ P(x) \equal{} x^3 \plus{} ax^2 \plus{} bx \plus{} c$ has the property that the mean of its zeros, the product of its zeros, and the sum of its coefficients are all equal. If the $ y$-intercept of the graph of $ y \equal{} P(x)$ is 2, what is $ b$? $ \textbf{(A)} \ \minus{} 11 \qquad \textbf{(B)} \ \minus{} 10 \qquad \textbf{(C)} \ \minus{} 9 \qquad \textbf{(D)} \ 1 \qquad \textbf{(E)} \ 5$

What is the smallest positive integer $n$ such that $2^n - 1$ is a multiple of $2015$?

Each vertex of convex pentagon $ABCDE$ is to be assigned a color. There are $6$ colors to choose from, and the ends of each diagonal must have different colors. How many different colorings are possible? $ \textbf{(A)}\ 2520 \qquad \textbf{(B)}\ 2880 \qquad \textbf{(C)}\ 3120 \qquad \textbf{(D)}\ 3250 \qquad \textbf{(E)}\ 3750 $

[hide=D stands for Dedekind, Z stands for Zermelo]they had two problem sets under those two names[/hide] [u]Set 1[/u] [b]D1 / Z1.[/b] What is $1 + 2 \cdot 3$? [b]D2.[/b] What is the average of the first $9$ positive integers? [b]D3 / Z2.[/b] A square of side length $2$ is cut into $4$ congruent squares. What is the perimeter of one of the $4$ squares? [b]D4.[/b] Find the ratio of a circle’s circumference squared to the area of the circle. [b]D5 / Z3.[/b] $6$ people split a bag of cookies such that they each get $21$ cookies. Kyle comes and demands his share of cookies. If the $7$ people then re-split the cookies equally, how many cookies does Kyle get? [u]Set 2[/u] [b]D6.[/b] How many prime numbers are perfect squares? [b]D7.[/b] Josh has an unfair $4$-sided die numbered $1$ through $4$. The probability it lands on an even number is twice the probability it lands on an odd number. What is the probability it lands on either $1$ or $3$? [b]D8.[/b] If Alice consumes $1000$ calories every day and burns $500$ every night, how many days will it take for her to first reach a net gain of $5000$ calories? [b]D9 / Z4.[/b] Blobby flips $4$ coins. What is the probability he sees at least one heads and one tails? [b]D10.[/b] Lillian has $n$ jars and $48$ marbles. If George steals one jar from Lillian, she can fill each jar with $8$ marbles. If George steals $3$ jars, Lillian can fill each jar to maximum capacity. How many marbles can each jar fill? [u]Set 3[/u] [b]D11 / Z6.[/b] How many perfect squares less than $100$ are odd? [b]D12.[/b] Jash and Nash wash cars for cash. Jash gets $\$6$ for each car, while Nash gets $\$11$ per car. If Nash has earned $\$1$ more than Jash, what is the least amount of money that Nash could have earned? [b]D13 / Z5.[/b] The product of $10$ consecutive positive integers ends in $3$ zeros. What is the minimum possible value of the smallest of the $10$ integers? [b]D14 / Z7.[/b] Guuce continually rolls a fair $6$-sided dice until he rolls a $1$ or a $6$. He wins if he rolls a $6$, and loses if he rolls a $1$. What is the probability that Guuce wins? [b]D15 / Z8.[/b] The perimeter and area of a square with integer side lengths are both three digit integers. How many possible values are there for the side length of the square? PS. You should use hide for answers. D.16-30/Z.9-14, 17, 26-30 problems have been collected [url=https://artofproblemsolving.com/community/c3h2916250p26045695]here [/url]and Z.15-25 [url=https://artofproblemsolving.com/community/c3h2916258p26045774]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In a badminton competition, $16$ players participate, of which $10$ are professionals and $6$ are amateurs. In the first phase, eight games are drawn. Among the eight winners of these games, four games are drawn. The four winners qualify for the semi-finals of the competition. Assuming that, whenever a professional player and an amateur play each other, the professional wins the game, what is the probability that an amateur player will reach the semi-finals of the competition?

Each box in the equation $$\square \times \square \times \square - \square \times \square \times \square = 9$$ is filled in with a different number in the list 2, $3, 4, 5, 6, 7, 8$ so that the equation is true. Which number in the list is not used to fill in a box?

Suppose $p(x)$ is a polynomial with integer coefficients. It is known that $p(a) - p(b) = 1$ (where $a$ and $b$ are integers). Prove that $a$ and $b$ differ by $1$ . (Folklore)

Find all pairs of integers ($a, b$) such that $a^2 + b = b^{1999}$ .

Let $ABCD$ be a quadrilateral inscribed in a circle. Show that the centroids of triangles $ABC,$ $CDA,$ $BCD,$ $DAB$ lie on one circle.

A triangular piece of sheet metal weighs $900$ g. Prove that by cutting this sheet metal along a straight line passing through the center of gravity of the triangle, it is impossible to cut off a piece weighing less than $400$ g.

Betty used a calculator to find the product $ 0.075\times 2.56 $. She forgot to enter the decimal points. The calculator showed $ 19200 $. If Betty had entered the decimal points correctly, the answer would have been $ \text{(A)}\ .0192\qquad\text{(B)}\ .192\qquad\text{(C)}\ 1.92\qquad\text{(D)}\ 19.2\qquad\text{(E)}\ 192 $

Let $\Gamma$ be a circle with center $O$ and $AE$ be a diameter. Point $D$ lies on segment $OE$ and point $B$ is the midpoint of one of the arcs $\widehat{AE}$ of $\Gamma$. Construct point $C$ such that $ABCD$ is a parallelogram. Lines $EB$ and $CD$ meet at $F$. Line $OF$ meets the minor arc $\widehat{EB}$ at $I$. Prove that $EI$ bisects $\angle BEC$.

Let $ABC$ be an acute, scalene triangle with orthocentre $H$. Let $\ell_a$ be the line through the reflection of $B$ with respect to $CH$ and the reflection of $C$ with respect to $BH$. Lines $\ell_b$ and $\ell_c$ are defined similarly. Suppose lines $\ell_a$, $\ell_b$, and $\ell_c$ determine a triangle $\mathcal T$. Prove that the orthocentre of $\mathcal T$, the circumcentre of $\mathcal T$, and $H$ are collinear. [i]Fedir Yudin, Ukraine[/i]

Consider the points $O = (0,0), A = (- 2,0)$ and $B = (0,2)$ in the coordinate plane. Let $E$ and $F$ be the midpoints of $OA$ and $OB$ respectively. We rotate the triangle $OEF$ with a center in $O$ clockwise until we obtain the triangle $OE'F'$ and, for each rotated position, let $P = (x, y)$ be the intersection of the lines $AE'$ and $BF'$. Find the maximum possible value of the $y$-coordinate of $P$.

The point $O$ is the center of the circle circumscribed of the acute triangle $ABC$, and $H$ is the point of intersection of the heights of this triangle. Let $A_1, B_1, C_1$ be the points diametrically opposed to the vertices $A, B , C$ respectively of the triangle, and $A_2, B_2, C_2$ be the midpoints of the segments $[AH], [BH] ¸[CH]$ respectively . Prove that the lines $A_1A_2, B_1B_2, C_1C_2$ are concurrent .

Consider all $26^{26}$ words of length 26 in the Latin alphabet. Define the $\emph{weight}$ of a word as $1/(k+1)$, where $k$ is the number of letters not used in this word. Prove that the sum of the weights of all words is $3^{75}$. Proposed by Fedor Petrov, St. Petersburg State University

In a contest there are $ n$ yes-no problems. We know that no two contestants have the same set of answers. To each question we give a random uniform grade of set $ \{1,2,3,\dots,2n\}$. Prove that the probability that exactly one person gets first is at least $ \frac12$.

Study the derivatives of the function \[y=\sqrt{x^3-4x}\] and sketch its graph on the real line.

An acute-angled unequal triangle $ABC$ is drawn with its circumcircle and circumcenter $O$. The incenter $I$ is also marked. Using only a ruler (without divisions), construct the symedian (a line symmetrical to the median relative to the corresponding bisector) of the triangle, drawing no more than four lines.

Let $k,n\ge 1$ be relatively prime integers. All positive integers not greater than $k+n$ are written in some order on the blackboard. We can swap two numbers that differ by $k$ or $n$ as many times as we want. Prove that it is possible to obtain the order $1,2,\dots,k+n-1, k+n$.

For a positive number $n$, we write $d (n)$ for the number of positive divisors of $n$. Determine all positive integers $k$ for which exist positive integers $a$ and $b$ with the property $k = d (a) = d (b) = d (2a + 3b)$.