Suppose $a$, $b$, and $c$ are positive integers such that \[\dfrac ab+\dfrac bc+\dfrac ca-\dfrac{524}{abc}=\dfrac ba+\dfrac cb+\dfrac ac - \dfrac{518}{abc}=1.\] Find $a^2+b^2+c^2$. [i]Proposed by David Altizio[/i]

Let $ A_0 \equal{} (a_1,\dots,a_n)$ be a finite sequence of real numbers. For each $ k\geq 0$, from the sequence $ A_k \equal{} (x_1,\dots,x_k)$ we construct a new sequence $ A_{k \plus{} 1}$ in the following way. 1. We choose a partition $ \{1,\dots,n\} \equal{} I\cup J$, where $ I$ and $ J$ are two disjoint sets, such that the expression \[ \left|\sum_{i\in I}x_i \minus{} \sum_{j\in J}x_j\right| \] attains the smallest value. (We allow $ I$ or $ J$ to be empty; in this case the corresponding sum is 0.) If there are several such partitions, one is chosen arbitrarily. 2. We set $ A_{k \plus{} 1} \equal{} (y_1,\dots,y_n)$ where $ y_i \equal{} x_i \plus{} 1$ if $ i\in I$, and $ y_i \equal{} x_i \minus{} 1$ if $ i\in J$. Prove that for some $ k$, the sequence $ A_k$ contains an element $ x$ such that $ |x|\geq\frac n2$. [i]Author: Omid Hatami, Iran[/i]

In an acute-angled triangle $ABC$ the altitudes $AU,BV,CW$ intersect at $H$. Points $X,Y,Z$, different from $H$, are taken on segments $AU,BV$, and $CW$, respectively. (a) Prove that if $X,Y,Z$ and $H$ lie on a circle, then the sum of the areas of triangles $ABZ, AYC, XBC$ equals the area of $ABC$. (b) Prove the converse of (a).

Let $ABC$ be a triangle and let $Q$ be a point such that $\overline{AB} \perp \overline{QB}$ and $\overline{AC} \perp \overline{QC}$. A circle with center $I$ is inscribed in $\triangle ABC$, and is tangent to $\overline{BC}$, $\overline{CA}$ and $\overline{AB}$ at points $D$, $E$, and $F$, respectively. If ray $QI$ intersects $\overline{EF}$ at $P$, prove that $\overline{DP} \perp \overline{EF}$. [i]Proposed by Aaron Lin[/i]

In the product $ B2 \times 7B\equal{}6396$, $ B$ is a digit. The value of $ B$ is \[ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 8 \]

Fix a prime number $ p > 5$. Let $ a,b,c$ be integers no two of which have their difference divisible by $ p$. Let $ i,j,k$ be nonnegative integers such that $ i \plus{} j \plus{} k$ is divisible by $ p \minus{} 1$. Suppose that for all integers $ x$, the quantity \[ (x \minus{} a)(x \minus{} b)(x \minus{} c)[(x \minus{} a)^i(x \minus{} b)^j(x \minus{} c)^k \minus{} 1]\] is divisible by $ p$. Prove that each of $ i,j,k$ must be divisible by $ p \minus{} 1$. [i]Kiran Kedlaya and Peter Shor.[/i]

Let $ABC$ be an acute scalene triangle, let $D$ be a point on the $A$-altitude, and let the circle with diameter $AD$ meet $AC$, $AB$, and the circumcircle of $ABC$ at $E$, $F$, $G$, respectively. Let $O$ be the circumcenter of $ABC$, let $AO$ meet $EF$ at $T$, and suppose the circumcircles of $ABC$ and $GTO$ meet at $X \neq G$. Then, prove that $AX$, $DG$, and $EF$ concur.

On the ends of the diameter two "$1$"s are written. Each of the semicircles is divided onto two parts and the sum of the numbers of its ends (i.e. "$2$") is written at the midpoint. Then every of the four arcs is halved and in its midpoint the sum of the numbers on its ends is written. Find the total sum of the numbers on the circumference after $n$ steps.

Two lines with slopes $\dfrac{1}{2}$ and $2$ intersect at $(2,2)$. What is the area of the triangle enclosed by these two lines and the line $x+y=10 ?$ $\textbf{(A) } 4 \qquad\textbf{(B) } 4\sqrt{2} \qquad\textbf{(C) } 6 \qquad\textbf{(D) } 8 \qquad\textbf{(E) } 6\sqrt{2}$

Find values of $n\in \mathbb{N}$ for which the fraction $\frac{3^n-2}{2^n-3}$ is reducible.

Determine all triangles such that the lengths of the three sides and its area are given by four consecutive natural numbers.

Given a prime $ p = 8k+1 $ for some integer $ k $. Let $ r $ be the remainder when $ \binom{4k}{k} $ is divided by $ p $. Prove that $ \sqrt{r} $ is not an integer. [i]Proposed by Evan Chen[/i]

Let $A_1A_2A_3$ be a triangle and, for $1 \leq i \leq 3$, let $B_i$ be an interior point of edge opposite $A_i$. Prove that the perpendicular bisectors of $A_iB_i$ for $1 \leq i \leq 3$ are not concurrent.

There are $n$ line segments on the plane, no three intersecting at a point, and each pair intersecting once in their respective interiors. Tony and his $2n - 1$ friends each stand at a distinct endpoint of a line segment. Tony wishes to send Christmas presents to each of his friends as follows: First, he chooses an endpoint of each segment as a “sink”. Then he places the present at the endpoint of the segment he is at. The present moves as follows : $\bullet$ If it is on a line segment, it moves towards the sink. $\bullet$ When it reaches an intersection of two segments, it changes the line segment it travels on and starts moving towards the new sink. If the present reaches an endpoint, the friend on that endpoint can receive their present. Prove that Tony can send presents to exactly $n$ of his $2n - 1$ friends.

In a particular game, each of $4$ players rolls a standard $6{ }$-sided die. The winner is the player who rolls the highest number. If there is a tie for the highest roll, those involved in the tie will roll again and this process will continue until one player wins. Hugo is one of the players in this game. What is the probability that Hugo's first roll was a $5,$ given that he won the game? $(\textbf{A})\: \frac{61}{216}\qquad(\textbf{B}) \: \frac{367}{1296}\qquad(\textbf{C}) \: \frac{41}{144}\qquad(\textbf{D}) \: \frac{185}{648}\qquad(\textbf{E}) \: \frac{11}{36}$

Call two circles in three-dimensional space pairwise tangent at a point $ P$ if they both pass through $ P$ and lines tangent to each circle at $ P$ coincide. Three circles not all lying in a plane are pairwise tangent at three distinct points. Prove that there exists a sphere which passes through the three circles.

Let $ABC$ be an equilateral triangle of unit side length and suppose $D$ is a point on segment $\overline{BC}$ such that $DB<DC.$ Let $M$ and $N$ denote the midpoints of $\overline{AB}$ and $\overline{AC},$ respectively. Suppose $X$ and $Y$ are the intersections of lines $AB$ and $ND,$ and lines $AC$ and $MD,$ respectively. Given that $XY=4,$ what is the value of $\frac{DB}{DC}?$ [i]Proposed by Kyle Lee[/i]

Let $ABCD$ be a square. For a point $P$ inside $ABCD$, a $\emph{windmill}$ centred at $P$ consists of two perpendicular lines $l_1$ and $l_2$ passing through $P$, such that $\quad\bullet$ $l_1$ intersects the sides $AB$ and $CD$ at $W$ and $Y$, respectively, and $\quad\bullet$ $l_2$ intersects the sides $BC$ and $DA$ at $X$ and $Z$, respectively. A windmill is called $\emph{round}$ if the quadrilateral $WXYZ$ is cyclic. Determine all points $P$ inside $ABCD$ such that every windmill centred at $P$ is round.

Find all the polynomials $P(x)$ of a degree $\leq n$ with real non-negative coefficients such that $P(x) \cdot P(\frac{1}{x}) \leq [P(1)]^2$ , $ \forall x>0$.

For every $ n\in\mathbb{N}$ let $ d(n)$ denote the number of (positive) divisors of $ n$. Find all functions $ f: \mathbb{N}\to\mathbb{N}$ with the following properties: [list][*] $ d\left(f(x)\right) \equal{} x$ for all $ x\in\mathbb{N}$. [*] $ f(xy)$ divides $ (x \minus{} 1)y^{xy \minus{} 1}f(x)$ for all $ x$, $ y\in\mathbb{N}$.[/list] [i]Proposed by Bruno Le Floch, France[/i]

Let $n$ be a positive integer and let $P(x)$ be a monic polynomial of degree $n$ with real coefficients. Also let $Q(x)=(x+1)^2(x+2)^2\dots (x+n+1)^2$. Consider the minimum possible value $m_n$ of $\displaystyle\sum_{i=1}^{n+1} \dfrac{i^2P(i^2)^2}{Q(i)}$. Then there exist positive constants $a,b,c$ such that, as $n$ approaches infinity, the ratio between $m_n$ and $a^{2n} n^{2n+b} c$ approaches $1$. Compute $\lfloor 2019 abc^2\rfloor$. [i]Proposed by Vincent Huang[/i]

Let $ABCD$ be a square with side $20$ and $T_1, T_2, ..., T_{2000}$ are points in $ABCD$ such that no $3$ points in the set $S = \{A, B, C, D, T_1, T_2, ..., T_{2000}\}$ are collinear. Prove that there exists a triangle with vertices in $S$, such that the area is less than $1/10$.

Let $P$ be the parabola in the plane determined by the equation $y = x^2$ . Suppose a circle $C$ in the plane intersects $P$ at four distinct points. If three of these points are $(-28, 784)$,$(-2, 4)$, and $(13, 169)$, find the sum of the distances from the focus of $P$ to all four of the intersection points

How many five-digit multiples of 3 end with the digit 6 ?

For some positive integer $n,$ Elmo writes down the equation \[x_1+x_2+\dots+x_n=x_1+x_2+\dots+x_n.\] Elmo inserts at least one $f$ to the left side of the equation and adds parentheses to create a valid functional equation. For example, if $n=3,$ Elmo could have created the equation \[f(x_1+f(f(x_2)+x_3))=x_1+x_2+x_3.\] Cookie Monster comes up with a function $f: \mathbb{Q}\to\mathbb{Q}$ which is a solution to Elmo's functional equation. (In other words, Elmo's equation is satisfied for all choices of $x_1,\dots,x_n\in\mathbb{Q})$. Is it possible that there is no integer $k$ (possibly depending on $f$) such that $f^k(x)=x$ for all $x$? [i]Srinivas Arun[/i]