A natural number $n$ is given. For all integer triplets $(a,b,c)$ such that $0 < |a| , |b|, |c| < 2023$ and satisfying below, show that the product of all possible integer $a$ is a perfect square. (The value of $a$ allows duplication) $$(a+nb)(a-nc) + abc = 0$$

Choose positive integers $b_1, b_2, \dotsc$ satisfying \[1=\frac{b_1}{1^2} > \frac{b_2}{2^2} > \frac{b_3}{3^2} > \frac{b_4}{4^2} > \dotsb\] and let $r$ denote the largest real number satisfying $\tfrac{b_n}{n^2} \geq r$ for all positive integers $n$. What are the possible values of $r$ across all possible choices of the sequence $(b_n)$? [i]Carl Schildkraut and Milan Haiman[/i]

Let $ A_1,B_1,C_1 $ be the middlepoints of the sides of a triangle $ ABC $ and let $ A_2,B_2,C_2 $ be on the middle of the paths $ CAB,ABC,BCA, $ respectively. Prove that $ A_1A_2,B_1B_2,C_1C_2 $ are concurrent.

There are $10$ cities in each of the three countries. Each road connects two cities from two different countries (there is at most one road between any two cities.) There are more than $200$ roads between these three countries. Prove that three cities, one city from each country, can be chosen such that there is a road between any two of these cities.

Compute the least possible value of $ABCD - AB \times CD$, where $ABCD$ is a 4-digit positive integer, and $AB$ and $CD$ are 2-digit positive integers. (Here $A$, $B$, $C$, and $D$ are digits, possibly equal. Neither $A$ nor $C$ can be zero.)

A game consists of a grid of $4\times 4$ and tiles of two colors (Yellow and White). A player chooses a type of token and gives it to the second player who places it where he wants, then the second player chooses a type of token and gives it to the first who places it where he wants, They continue in this way and the one who manages to form a line with three tiles of the same color wins (horizontal, vertical or diagonal and regardless of whether it is the tile you started with or not). Before starting the game, two yellow and two white pieces are already placed as shows the figure below. [img]https://cdn.artofproblemsolving.com/attachments/b/5/ba11377252c278c4154a8c3257faf363430ef7.png[/img] Yolanda and Xinia play a game. If Yolanda starts (choosing the token and giving it to Xinia for this to place) indicate if there is a winning strategy for either of the two players and, if any, describe the strategy.

There are 60 empty boxes $B_1,\ldots,B_{60}$ in a row on a table and an unlimited supply of pebbles. Given a positive integer $n$, Alice and Bob play the following game. In the first round, Alice takes $n$ pebbles and distributes them into the 60 boxes as she wishes. Each subsequent round consists of two steps: (a) Bob chooses an integer $k$ with $1\leq k\leq 59$ and splits the boxes into the two groups $B_1,\ldots,B_k$ and $B_{k+1},\ldots,B_{60}$. (b) Alice picks one of these two groups, adds one pebble to each box in that group, and removes one pebble from each box in the other group. Bob wins if, at the end of any round, some box contains no pebbles. Find the smallest $n$ such that Alice can prevent Bob from winning. [i]Czech Republic[/i]

Let $I$ be the incenter of a triangle $ABC$. The lines passing through $A$ and parallel to $BI, CI$ meet the perpendicular bisector to $AI$ at points $S, T$ respectively. Let $Y$ be the common point of $BT$ and $CS$, and $A^*$ be a point such that $BICA^*$ is a parallelogram. Prove that the midpoint of segment $YA^*$ lies on the excircle of the triangle touching the side $BC$.

Evaluate $(2-\sec^2{1^\circ})(2-\sec^2{2^\circ})(2-\sec^2{3^\circ})\cdots(2-\sec^2{89^\circ}).$

The points $A, B, C, D$ are marked on the straight line in this order. Circle $\omega_1$ passes through points $A$ and $C$, and the circle $\omega_2$ passes through points $B$ and $D$. On the circle $\omega_2$, the point $E$ is marked so that $AB = BE$, and on the circle $\omega_1$, the point $F$ is marked so that $CD = CF$. The line $AE$ intersects the circle $\omega_2$ a second time at point $X$, and the line $DF$ intersects the circle $\omega_1$ at point $Y$. Prove that the $XY$ lines and $AD$ is perpendicular. [i]A.D.Tereshin[/i]

There are four lines in the plane, no three concurrent, no two parallel, and no three forming an equilateral triangle. If one of them is parallel to the Euler line of the triangle formed by the other three lines, prove that a similar statement holds for each of the other lines.

Prove that for all positive integers $n\geq 1$ the number $\prod^n_{k=1} k^{2k-n-1}$ is also an integer number. [i]Laurentiu Panaitopol[/i].

Let $p=4k+1$ be a prime. Show that \[\sum^{k}_{i=1}\left \lfloor \sqrt{ ip }\right \rfloor = \frac{p^{2}-1}{12}.\]

An infinite arithmetic progression whose terms are positive integers contains the square of an integer and the cube of an integer. Show that it contains the sixth power of an integer.

Let $S$ be a unit circle and $K$ a subset of $S$ consisting of several closed arcs. Let $K$ satisfy the following properties: $(\text{i})$ $K$ contains three points $A,B,C$, that are the vertices of an acute-angled triangle $(\text{ii})$ For every point $A$ that belongs to $K$ its diametrically opposite point $A'$ and all points $B$ on an arc of length $\frac{1}{9}$ with center $A'$ do not belong to $K$. Prove that there are three points $E,F,G$ on $S$ that are vertices of an equilateral triangle and that do not belong to $K$.

The positive integers $a,b,c$ are pairwise relatively prime, $a$ and $c$ are odd and the numbers satisfy the equation $a^2+b^2=c^2$. Prove that $b+c$ is the square of an integer.

Consider the square $ABCD$ and the point $E$ inside the angle $CAB$, such that $\angle BAE =15^o$, and the lines $BE$ and $BD$ are perpendicular. Prove that $AE = BD$.

Prove that $1^{1987} + 2^{1987} + ... + n^{1987}$ is divisible by $n+2$.

Let $x$, $y$, and $z$ be real numbers such that $x<y<z<6$. Solve the system of inequalities: \[\left\{\begin{array}{cc} \dfrac{1}{y-x}+\dfrac{1}{z-y}\le 2 \\ \dfrac{1}{6-z}+2\le x \\ \end{array}\right.\]

Determine all pairs $(a, b)$ of real numbers for which there exists a unique symmetric $2\times 2$ matrix $M$ with real entries satisfying $\mathrm{trace}(M)=a$ and $\mathrm{det}(M)=b$. (Proposed by Stephan Wagner, Stellenbosch University)

A square $ ABCD$ is inscribed in a circle. Let $ \alpha \equal{} \angle DAB, \beta \equal{} \angle BDA,$ and $ \gamma \equal{} \angle CDB$. Then $ \angle DBC$ equals A. $ \alpha \minus{} \beta$ B. $ \alpha \minus{} \gamma$ C. $ 90^\circ \minus{} \alpha \plus{} \beta$ D. $ 90^\circ \minus{} \alpha \plus{} \gamma$ E. $ 180^\circ \minus{} \alpha \minus{} \gamma$

In a summer camp about Applied Maths, there are $8m+1$ boys (with $m > 5$) and some girls. Every girl is friend with exactly $3$ boys and for any $2$ boys, there is exactly $1$ girl who is their common friend. Let $n$ be the greatest number of girls that can be chosen from the camp to form a group such that every boy is friend with at most $1$ girl in the group. Prove that $n \geq 2m+1$.

A square $ABCD$ is divided into $(n - 1)^2$ congruent squares, with sides parallel to the sides of the given square. Consider the grid of all $n^2$ corners obtained in this manner. Determine all integers $n$ for which it is possible to construct a non-degenerate parabola with its axis parallel to one side of the square and that passes through exactly $n$ points of the grid.

let $x_1$, $x_2$ .... $x_6$ be non-negative reals such that $x_1+x_2+x_3+x_4+x_5+x_6=1$ and $x_1x_3x_5$ + $x_2x_4x_6$ $\geq$ $\frac{1}{540}$. Let $p$ and $q$ be relatively prime integers such that $\frac{p}{q}$ is the maximum value of $x_1x_2x_3+x_2x_3x_4+x_3x_4x_5+x_4x_5x_6+x_5x_6x_1+x_6x_1x_2$. Find $p+q$

In the convex quadrilateral $ ABCD $, the corresponding points $ M $ and $ N $ are chosen on the adjacent sides $ \overline{AB} $ and $ \overline{BC} $ and the intersection point of the segments $ AN $ and $ GM $ is marked by 0. Prove that if circles can be inscribed in the quadrilaterals $ AOCD $ and $ BMON $, then a circle can also be inscribed in the quadrilateral $ ABCD $.