On a cartesian plane a parabola $y = x^2$ is drawn. For a given $k > 0$ we consider all trapezoids inscribed into this parabola with bases parallel to the x-axis, and the product of the lengths of their bases is exactly $k$. Prove that the diagonals of all such trapezoids share a common point.

Suppose that $a, b, c$ are real numbers such that for all positive numbers $x_1,x_2,\dots,x_n$ we have $(\frac{1}{n}\sum_{i=1}^nx_i)^a(\frac{1}{n}\sum_{i=1}^nx_i^2)^b(\frac{1}{n}\sum_{i=1}^nx_i^3)^c\ge 1$ Prove that vector $(a, b, c)$ is a nonnegative linear combination of vectors $(-2,1,0)$ and $(-1,2,-1)$.

Let $ABC$ be an acute triangle with circumcenter $O$, orthocenter $H$, and circumcircle $\Omega$. Let $M$ be the midpoint of $AH$ and $N$ the midpoint of $BH$. Assume the points $M$, $N$, $O$, $H$ are distinct and lie on a circle $\omega$. Prove that the circles $\omega$ and $\Omega$ are internally tangent to each other. [i]Dhroova Aiylam and Evan Chen[/i]

The angle of a rotation $\rho$ is $\alpha <180^\circ$ and $\rho$ maps the convex polygon $M$ in itself. Prove that there exist two circles $c_1$ and $c_2$ with radius $r$ and $2r$, so that $c_1$ is inner for $M$ and $M$ is inner for $c_2$.

Prove that there exist infinitely many triples of positive integers $(a,b,c)$ so that $a,b,c$ are pairwise coprime and $$\bigg \lfloor \frac{a^2}{2021} \bigg \rfloor + \bigg \lfloor \frac{b^2}{2021} \bigg \rfloor = \bigg \lfloor \frac{c^2}{2021} \bigg \rfloor.$$

A polynomial $P(x)$ of degree $n$ has $n$ different real roots. What is the largest number of its coefficients that can be zero?

Let $ABC$ be an equilateral triangle with side length $8.$ Let $X$ be on side $AB$ so that $AX=5$ and $Y$ be on side $AC$ so that $AY=3.$ Let $Z$ be on side $BC$ so that $AZ,BY,CX$ are concurrent. Let $ZX,ZY$ intersect the circumcircle of $AXY$ again at $P,Q$ respectively. Let $XQ$ and $YP$ intersect at $K.$ Compute $KX\cdot KQ.$

Let $a$ and $b$ be integers such that $a - b = a^2c - b^2d$ for some consecutive integers $c$ and $d$. Prove that $|a - b|$ is a perfect square.

Ahn chooses a two-digit integer, subtracts it from 200, and doubles the result. What is the largest number Ahn can get? $\textbf{(A)}\ 200 \qquad \textbf{(B)}\ 202 \qquad \textbf{(C)}\ 220 \qquad \textbf{(D)}\ 380 \qquad \textbf{(E)}\ 398$

A function $f: R \to R$ satisfies the conditions: $f (x + 19) \le f (x) + 19$ and $f (x + 94) \ge f (x) + 94$ for all $x \in R$. Prove that $f (x + 1) = f (x) + 1$ for all $x \in R$.

How many numbers of $ n $ digits formed only with $ 1,9,8 $ and $ 6 $ divide themselves by $ 3 $ ?

Let $n \geq 2$ be an integer. There are $n$ distinct circles in general position, that is, any two of them meet in two distinct points and there are no three of them meeting at one point. Those circles divide the plane in limited regions by circular edges, that meet at vertices (note that each circle have exactly $2n-2$ vertices). For each circle, temporarily color its vertices alternately black and white (note that, doing this, each vertex is colored twice, one for each circle passing through it). If the two temporarily colouring of a vertex coincide, this vertex is definitely colored with this common color; otherwise, it will be colored with gray. Show that if a circle has more than $n-2 + \sqrt{n-2}$ gray points, all vertices of some region are grey. Observation: In this problem, a region cannot contain vertices or circular edges on its interior. Also, the outer region of all circles also counts as a region.

Let $N$ be a positive integer whose digits add up to $23$. What is the greatest possible product the digits of $N$ can have?

Find all differentiable functions $f:(0,\infty) \to \mathbb{R}$ such that $$f(b)-f(a)=(b-a)f’(\sqrt{ab}) \qquad \text{for all}\qquad a,b>0.$$ [i]Proposed by Orif Ibrogimov, National University of Uzbekistan[/i]

Imagine the unit square in the plane to be a [i]carrom board[/i]. Assume the [i]striker[/i] is just a point, moving with no friction (so it goes forever), and that when it hits an edge, the angle of reflection is equal to the angle of incidence, as in real life. If the striker ever hits a corner it falls into the pocket and disappears. The trajectory of the striker is completely determined by its starting point $(x,y)$ and its initial velocity $\overrightarrow{(p,q)}$. If the striker eventually returns to its initial state (i.e. initial position and initial velocity), we define its [i]bounce number[/i] to be the number of edges it hits before returning to its initial state for the $1^{\text{st}}$ time. For example, the trajectory with initial state $[(.5,.5);\overrightarrow{(1,0)}]$ has bounce number $2$ and it returns to its initial state for the $1^{\text{st}}$ time in $2$ time units. And the trajectory with initial state $[(.25,.75);\overrightarrow{(1,1)}]$ has bounce number $4$. $\textbf{(a)}$ Suppose the striker has initial state $[(.5,.5);\overrightarrow{(p,q)}]$. If $p>q\geqslant 0$ then what is its velocity after it hits an edge for the $1^{\text{st}}$ time ? What if $q>p\geqslant 0$ ? $\textbf{(b)}$ Draw a trajectory with bounce number $5$ or justify why it is impossible. $\textbf{(c)}$ Consider the trajectory with initial state $[(x,y);\overrightarrow{(p,0)}]$ where $p$ is a positive integer. In how much time will the striker $1^{\text{st}}$ return to its initial state ? $\textbf{(d)}$ What is the bounce number for the initial state $[(x,y);\overrightarrow{(p,q)}]$ where $p,q$ are relatively prime positive integers, assuming the striker never hits a corner ?

[u]Discrete Math Round[/u] [b]p1.[/b] A class of six students has to split into two indistinguishable teams of three people. Compute the number of distinct team arrangements that can result. [b]p2.[/b] What is the probability that a randomly chosen factor of $2016$ is a perfect square? [b]p3.[/b] Compute the remainder when $$5\underbrace{666...6666}_{2016 \,\, sixes}5$$ is divided by $17$. [b]p4.[/b] At an M&M factory, two types of M&Ms are produced, red and blue. The M&Ms are transported individually on a conveyor belt. Anna is watching the conveyor belt, and has determined that four out of every five red M&Ms are followed by a blue one, while one out of every six blue M&Ms is followed by a red one. What proportion of the M&Ms are red? [b]p5.[/b] Three cards are chosen from a standard deck of $52$ without replacing them. Given that the ace of spades was chosen, what is the expected number of aces chosen? [b]p6.[/b] Moor decides that he needs a new email address, and forms the address by taking some permutation of the $12$ letters $MMMOOOOOORRR$. How many permutations of the letters will contain $MOOR$ in this exact order at least once? [b]p7.[/b] Suppose that the $8$ corners of a cube can be colored either red, green, or blue. We call a coloring of the cube rotationally symmetric if the cube can be rotated along a single axis parallel to an edge of a cube either $90^o$, $180^o$, or $270^o$, and reach the original coloring. How many rotationally symmetric colorings exist using the $3$ colors? Assume that any colorings which are identical after rotation are equivalent. [b]p8.[/b] Let $x = \frac{1}{9} + \frac{1}{99} + \frac{1}{999} + ...+ \frac{1}{999999999}$ . Compute the number of digits in the first $3000$ decimal places of the base $10$ representation of $x$ which are greater than or equal to $8$. [b]p9.[/b] Two $20$-sided dice are rolled. Their outcomes are independent and take uniformly distributed integer values from $1$ to $20$, inclusive. For each roll, let $x$ be (the sum of the dice) $\times $ (the positive difference of the dice). What is the expected value of $x$? [b]p10.[/b] Compute $$\sum^{1000}_{a=1} \sum^{1000}_{b=1} \sum^{1000}_{c=1} \left\lfloor \frac{1000}{lcm (a, b, c)} \right \rfloor \phi (a) \phi (b) \phi(c)$$ where $\phi (n) = | \{k : 1 \le k \le n, gcd (k, n) = 1\} |$ counts the integers coprime to $n$ that are less than or equal to $n$. [u]Discrete Math Tiebreaker[/u] [b]Tie 1.[/b] A certain elementary school has $48$ students in the third grade that must be organized into three classes of $16$ students each. There are three troublemakers in the grade. If the students are assigned independently and randomly to classes, what is the probability that all three trou blemakers are assigned to the same $16$ student class? [b]Tie 2.[/b] A $4$-digit number $x$ has the property that the expected value of the integer obtained from switching any two digits in $x$ is $4625$. Given that the sum of the digits of $x$ is $20$, compute $x$. [b]Tie 3.[/b] Let $S$ be the set of factors of $10^5$. The number of subsets of $S$ with a least common multiple of $10^5$ can be written as $2^n * m$, where $n$ and $m$ are positive integers and $m$ is not divisible by $2$. Compute $m + n$. PS. You should use hide for answers.

Consider a convex polyhedron $P$ with vertices $V_1,\ldots ,V_p$. The distinct vertices $V_i$ and $V_j$ are called [i]neighbours[/i] if they belong to the same face of the polyhedron. To each vertex $V_k$ we assign a number $v_k(0)$, and construct inductively the sequence $v_k(n)\ (n\ge 0)$ as follows: $v_k(n+1)$ is the average of the $v_j(n)$ for all neighbours $V_j$ of $V_k$ . If all numbers $v_k(n)$ are integers, prove that there exists the positive integer $N$ such that all $v_k(n)$ are equal for $n\ge N$ .

Find all triples $(a,b,c)$ of positive integers such that (i) $a \leq b \leq c$; (ii) $\text{gcd}(a,b,c)=1$; and (iii) $a^3+b^3+c^3$ is divisible by each of the numbers $a^2b, b^2c, c^2a$.

The Cincinnati Reals are playing the Houston Alphas in the last game of the Swirled Series. The Alphas are leading by $1$ run in the bottom of the $9\text{th}$ (last) inning, and the Reals are at bat. Each batter has a $\dfrac{1}{3}$ chance of hitting a single and a $\dfrac{2}{3}$ chance of making an out. If the Reals hit $5$ or more singles before they make $3$ outs, they will win. If the Reals hit exactly $4$ singles before they make $3$ outs, they will tie the game and send it into extra innings, and they will have a $\dfrac{3}{5}$ chance of eventually winning the game (since they have the added momentum of coming from behind). If the Reals hit fewer than $4$ singles, they will LOSE! What is the probability that the Alphas hold off the Reals and win, sending the packed Alphadome into a frenzy? Express the answer as a fraction.

A point is picked uniformly at random inside of a square. Four segments are then drawn in connecting the point to each of the vertices of the square, cutting the square into four triangles. What is the probability that at least two of the resulting triangles are obtuse?

Let $MATH$ be a trapezoid with $MA=AT=TH=5$ and $MH=11$. Point $S$ is the orthocenter of $\triangle ATH$. Compute the area of quadrilateral $MASH$.

Assume the $n$ sets $A_1, A_2..., A_n$ are a partition of the set $A=\{1,2,...,29\}$, and the sum of any elements in $A_i$ , $(i=1,2,...,n)$ is not equal to $30$. Find the smallest possible value of $n$.

Let $ABC$ be an acute-angled triangle, and let $D, E$ and $K$ be the midpoints of its sides $AB, AC$ and $BC$ respectively. Let $O$ be the circumcentre of triangle $ABC$, and let $M$ be the foot of the perpendicular from $A$ on the line $BC$. From the midpoint $P$ of $OM$ we draw a line parallel to $AM$, which meets the lines $DE$ and $OA$ at the points $T$ and $Z$ respectively. Prove that: (a) the triangle $DZE$ is isosceles (b) the area of the triangle $DZE$ is given by the formula \[E_{DZE}=\frac{BC\cdot OK}{8}\]

Given is a $K_{2024}$ in which every edge has weight $1$ or $2$. If every cycle has even total weight, find the minimal value of the sum of all weights in the graph.

Let $ABC$ be a triangle with $AB = 16$, $AC = 10$, $BC = 18$. Let $D$ be a point on $AB$ such that $4AD = AB$ and let E be the foot of the angle bisector from $B$ onto $AC$. Let $P$ be the intersection of $CD$ and $BE$. Find the area of the quadrilateral $ADPE$.