Given a set $M$ of $1985$ positive integers, none of which has a prime divisor larger than $26$, prove that the set has four distinct elements whose geometric mean is an integer.

Let $a, b, c, d$ be reals such that $a \ge b \ge c \ge d$ and \[ (a - b)^3 + (b - c)^3 + (c - d)^3 - 2(d - a)^3 - 12(a - b)^2 - 12(b - c)^2 - 12(c - d)^2 + 12(d - a)^2 - 2020(a - b)(b - c)(c - d)(d - a) = 1536. \] Find the minimum possible value of $d - a$.

Triangle $ABC,$ with $BC = 48,$ is inscribed in a circle $\Omega$ of radius $49\sqrt{3}.$ There is a unique circle $\omega$ that is tangent to $\overline{AB}$ and $\overline{AC}$ and internally tangent to $\Omega.$ Let $D,$ $E,$ and $F$ be the points at which $\omega$ is tangent to $\Omega,$ $\overline{AB},$ and $\overline{AC},$ respectively. The rays $\overrightarrow{DE}$ and $\overrightarrow{DF}$ intersect $\Omega$ at points $X$ and $Y,$ respectively, such that $X \neq D$ and $Y \neq D.$ Compute $XY.$

Let $a,b$ be positive real numbers. Define $m(a,b)$ as the minimum of $\[ a,\frac{1}{b} \text{ and } \frac{1}{a}+b.\]$ Find the maximum of $m(a,b).$

Let $\omega$ be a circle with positive integer radius $r$. Suppose that it is possible to draw isosceles triangle with integer side lengths inscribed in $\omega$. Compute the number of possible values of $r$ where $1 \le r \le 2023^2$. Submit your answer as a positive integer $E$. If the correct answer is $A$, your score for this question will be $\max \left( 0, 25\left(3 - 2 \max \left( \frac{A}{E} , \frac{E}{A}\right)\right)\right)$, rounded to the nearest integer.

Find the greatest natural number $n$ such there exist natural numbers $x_{1}, x_{2}, \ldots, x_{n}$ and natural $a_{1}< a_{2}< \ldots < a_{n-1}$ satisfying the following equations for $i =1,2,\ldots,n-1$: \[x_{1}x_{2}\ldots x_{n}= 1980 \quad \text{and}\quad x_{i}+\frac{1980}{x_{i}}= a_{i}.\]

Fiona has a deck of cards labelled $1$ to $n$, laid out in a row on the table in order from $1$ to $n$ from left to right. Her goal is to arrange them in a single pile, through a series of steps of the following form: [list] [*]If at some stage the cards are in $m$ piles, she chooses $1\leq k<m$ and arranges the cards into $k$ piles by picking up pile $k+1$ and putting it on pile $1$; picking up pile $k+2$ and putting it on pile $2$; and so on, working from left to right and cycling back through as necessary. [/list] She repeats the process until the cards are in a single pile, and then stops. So for example, if $n=7$ and she chooses $k=3$ at the first step she would have the following three piles: $ \begin{matrix} 7 & \ &\ \\ 4 & 5 & 6 \\ 1 &2 & 3 \\ \hline \end{matrix} $ If she then chooses $k=1$ at the second stop, she finishes with the cards in a single pile with cards ordered $6352741$ from top to bottom. How many different final piles can Fiona end up with?

A regular octagon $ ABCDEFGH$ has an area of one square unit. What is the area of the rectangle $ ABEF$? [asy]unitsize(8mm); defaultpen(linewidth(.8pt)+fontsize(6pt)); pair C=dir(22.5), B=dir(67.5), A=dir(112.5), H=dir(157.5), G=dir(202.5), F=dir(247.5), E=dir(292.5), D=dir(337.5); draw(A--B--C--D--E--F--G--H--cycle); label("$A$",A,NNW); label("$B$",B,NNE); label("$C$",C,ENE); label("$D$",D,ESE); label("$E$",E,SSE); label("$F$",F,SSW); label("$G$",G,WSW); label("$H$",H,WNW);[/asy]$ \textbf{(A)}\ 1\minus{}\frac{\sqrt2}{2} \qquad \textbf{(B)}\ \frac{\sqrt2}{4} \qquad \textbf{(C)}\ \sqrt2\minus{}1 \qquad \textbf{(D)}\ \frac12 \qquad \textbf{(E)}\ \frac{1\plus{}\sqrt2}{4}$

Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.

Let $A$, $B$ and $C$ be $n \times n$ matrices with complex entries satisfying $$A^2=B^2=C^2 \text{ and } B^3 = ABC + 2I.$$ Prove that $A^6=I$.

One-round chess tournament involves $ 10 $ players from two countries. For a victory, one point is given, for a draw - half a point, for defeat - zero. All players scored a different number of points. Prove that one of the chess players scored in meetings with his countrymen less points, than meeting with players from another country.

Let $a_1, a_2, \cdots, a_n$ be a sequence of real numbers with $a_1+a_2+\cdots+a_n=0$. Define the score $S(\sigma)$ of a permutation $\sigma=(b_1, \cdots b_n)$ of $(a_1, \cdots a_n)$ to be the minima of the sum $$(x_1-b_1)^2+\cdots+(x_n-b_n)^2$$ over all real numbers $x_1\le \cdots \le x_n$. Prove that $S(\sigma)$ attains the maxima over all permutations $\sigma$, if and only if for all $1\le k\le n$, $$b_1+b_2+\cdots+b_k\ge 0.$$ [i]Proposed by Anzo Teh Zhao Yang[/i]

Prove that in the set $ \{1,2, \ldots, 1989\}$ can be expressed as the disjoint union of subsets $ A_i, \{i \equal{} 1,2, \ldots, 117\}$ such that [b]i.)[/b] each $ A_i$ contains 17 elements [b]ii.)[/b] the sum of all the elements in each $ A_i$ is the same.

Prove that if two medians in a triangle are equal in length, then the triangle is isosceles. (Note: A median in a triangle is a segment which connects a vertex of the triangle to the midpoint of the opposite side of the triangle.)

$AB$ is a chord of a circle with center $O$, $M$ is the midpoint of $AB$. A non-diameter chord is drawn through $M$ and intersects the circle at $C$ and $D$. The tangents of the circle from points $C$ and $D$ intersect line $AB$ at $P$ and $Q$, respectively. Prove that $PA$ = $QB$.

Show that for $p>1$ we have \[\lim_{n\rightarrow+\infty}\frac{1^p+2^p+...+(n-1)^p+n^p+(n-1)^p+...+2^p+1^p}{n^2} = +\infty\] Find the limit if $p=1$.

Prove that, among the positive numbers $$a,2a, ...,(n - 1)a.$$ there is one that differs from an integer by at most $1/n$.

Let $k,m,$ and $n$ be natural numbers such that $m+k+1$ is a prime greater than $n+1$. Let $c_{s}=s(s+1).$ Prove that the product \[(c_{m+1}-c_{k})(c_{m+2}-c_{k})\cdots (c_{m+n}-c_{k})\] is divisible by the product $c_{1}c_{2}\cdots c_{n}$.

How many non-empty subsets $ S$ of $ \{1, 2, 3, \ldots, 15\}$ have the following two properties? (1) No two consecutive integers belong to $ S$. (2) If $ S$ contains $ k$ elements, then $ S$ contains no number less than $ k$. $ \textbf{(A) } 277\qquad \textbf{(B) } 311\qquad \textbf{(C) } 376\qquad \textbf{(D) } 377\qquad \textbf{(E) } 405$

Let $ABC$ be an acute-angled triangle with $AB> AC$ and orthocenter $H$. Let $D$ the projection of $A$ on $BC$. Let $E$ be the reflection of $C$ wrt $D$. The lines $AE$ and $BH$ intersect at point $S$. Let $N$ be the midpoint of $AE$ and let $M$ be the midpoint of $BH$. Prove that $MN$ is perpendicular to $DS$.

Let $a_0,a_1,\cdots ,a_n$ be numbers from the interval $(0,\pi/2)$ such that \[ \tan (a_0-\frac{\pi}{4})+ \tan (a_1-\frac{\pi}{4})+\cdots +\tan (a_n-\frac{\pi}{4})\geq n-1. \] Prove that \[ \tan a_0\tan a_1 \cdots \tan a_n\geq n^{n+1}. \]

Let $n$ be an positive integer. Find the smallest integer $k$ with the following property; Given any real numbers $a_1 , \cdots , a_d $ such that $a_1 + a_2 + \cdots + a_d = n$ and $0 \le a_i \le 1$ for $i=1,2,\cdots ,d$, it is possible to partition these numbers into $k$ groups (some of which may be empty) such that the sum of the numbers in each group is at most $1$.

The fractions $ \frac{ax\plus{}b}{cx\plus{}d}$ and $ \frac{b}{d}$ are unequal if: $ \textbf{(A)}\ a\equal{}c\equal{}1, x\neq 0 \qquad \textbf{(B)}\ a\equal{}b\equal{}0 \qquad \textbf{(C)}\ a\equal{}c\equal{}0 \\ \textbf{(D)}\ x\equal{}0 \qquad \textbf{(E)}\ ad\equal{}bc$

Find, with argument, the integer solutions of the equation \[3z^2 = 2x^3 + 385x^2 + 256x - 58195.\]

Let $AD, BE$, and $CF$ denote the altitudes of triangle $\vartriangle ABC$. Points $E'$ and $F'$ are the reflections of $E$ and $F$ over $AD$, respectively. The lines $BF'$ and $CE'$ intersect at $X$, while the lines $BE'$ and $CF'$ intersect at the point $Y$. Prove that if $H$ is the orthocenter of $\vartriangle ABC$, then the lines $AX, YH$, and $BC$ are concurrent.