Let $A$ denote the set of all integers $n$ such that $1 \le n \le 10000$, and moreover the sum of the decimal digits of $n$ is $2$. Find the sum of the squares of the elements of $A$.

Find all real $x, y$, satisfying $$(x+1)^2(y+1)^2=27xy$$ and $$(x^2+1)(y^2+1)=10xy.$$

We have a pile of $2016$ cards and a hat. We take out one card, put it in the hat and then divide the remaining cards into two arbitrary non empty piles. In the next step, we choose one of the two piles, we move one card from this pile to the hat and then divide this pile into two arbitrary non empty piles. This procedure is repeated several times : in the $k$-th step $(k>1)$ we move one card from one of the piles existing after the step $(k-1)$ to the hat and then divide this pile into two non empty piles. Is it possible that after some number of steps we get all piles containing three cards each?

Given a regular dodecagon (a convex polygon with 12 congruent sides and angles) with area 1, there are two possible ways to dissect this polygon into 12 equilateral triangles and 6 squares. Let $T_1$ denote the union of all triangles in the first dissection, and $S_1$ the union of all squares. Define $T_2$ and $S_2$ similarly for the second dissection. Let $S$ and $T$ denote the areas of $S_1 \cap S_2$ and $T_1 \cap T_2$, respectively. If $\frac{S}{T} = \frac{a+b\sqrt{3}}{c}$ where $a$ and $b$ are integers, $c$ is a positive integer, and $\gcd(a,c)=1$, compute $10000a+100b+c$. [i]Proposed by Lewis Chen[/i]

Let $f(x)=x+\cfrac{1}{2x+\cfrac{1}{2x+\cfrac{1}{2x+\cdots}}}$. Find $f(99)f^\prime (99)$.

Two thousand points are given on a circle. Label one of the points 1. From this point, count 2 points in the clockwise direction and label this point 2. From the point labeled 2, count 3 points in the clockwise direction and label this point 3. (See figure.) Continue this process until the labels $1, 2, 3, \dots, 1993$ are all used. Some of the points on the circle will have more than one label and some points will not have a label. What is the smallest integer that labels the same point as 1993? [asy] int x=101, y=3*floor(x/4); draw(Arc(origin, 1, 360*(y-3)/x, 360*(y+4)/x)); int i; for(i=y-2; i<y+4; i=i+1) { dot(dir(360*i/x)); } label("3", dir(360*(y-2)/x), dir(360*(y-2)/x)); label("2", dir(360*(y+1)/x), dir(360*(y+1)/x)); label("1", dir(360*(y+3)/x), dir(360*(y+3)/x));[/asy]

For every $a \in \mathbb N$ denote by $M(a)$ the number of elements of the set \[ \{ b \in \mathbb N | a + b \text{ is a divisor of } ab \}.\] Find $\max_{a\leq 1983} M(a).$

An assignment of either a $ 0$ or a $ 1$ to each unit square of an $ m$x$ n$ chessboard is called $ fair$ if the total numbers of $ 0$s and $ 1$s are equal. A real number $ a$ is called $ beautiful$ if there are positive integers $ m,n$ and a fair assignment for the $ m$x$ n$ chessboard such that for each of the $ m$ rows and $ n$ columns , the percentage of $ 1$s on that row or column is not less than $ a$ or greater than $ 100\minus{}a$. Find the largest beautiful number.

Calculate $ \begin{pmatrix}1&0&0& \ldots &0\\\binom{1}{0} &\binom{1}{1} &0& \ldots & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ \binom{n}{0} &\binom{n}{1} & \binom{n}{2} & \ldots & \binom{n}{n}\end{pmatrix}^{-1} . $

Evaluate the definite integral $$\int_{-\pi}^{\pi}\frac{\sin nx}{(1+2^{x})\sin x} dx,$$ where $n$ is a natural number.

Show that the solution set of the inequality \[ \sum^{70}_{k \equal{} 1} \frac {k}{x \minus{} k} \geq \frac {5}{4} \] is a union of disjoint intervals, the sum of whose length is 1988.

Triangle $ABC$ has $AB=25$, $AC=29$, and $BC=36$. Additionally, $\Omega$ and $\omega$ are the circumcircle and incircle of $\triangle ABC$. Point $D$ is situated on $\Omega$ such that $AD$ is a diameter of $\Omega$, and line $AD$ intersects $\omega$ in two distinct points $X$ and $Y$. Compute $XY^2$. [i]Proposed by David Altizio[/i]

The following grid represents a mountain range; the number in each cell represents the height of the mountain located there. Moving from a mountain of height $a$ to a mountain of height $b$ takes $(b - a)^2$ time. Suppose that you start on the mountain of height $1$ and that you can move up, down, left, or right to get from one mountain to the next. What is the minimum amount of time you need to get to the mountain of height $49$? [img]https://cdn.artofproblemsolving.com/attachments/0/6/10b07a2b2ae4ba750cfffc3dc678880333c2de.png[/img]

Let $ABCD$ be a cyclic quadrilateral wih both diagonals perpendicular to each other and intersecting at point $O$. Let $E,F,G,H$ be the orthogonal projections of $O$ on sides $AB,BC,CD,DA$ respectively. a. Prove that $\angle EFG + \angle GHE = 180^o$ b. Prove that $OE$ bisects angle $\angle FEH$ .

Let $K$ and $L$ be the centers of the excircles of a non-isosceles triangle $ABC$ opposite $B$ and $C$ respectively. Let $M$ and $N$ be points in the plane of the triangle such that $BM$ bisects $AC$ and $CN$ bisects $AB$. Prove that the lines $KM$ and $NK$ meet on $BC$. [hide=Note]The problem in its current formulation is trivially wrong. No possible rectification is known to OP / was sent to the participants.[/hide]

Hannusya, Petrus and Mykolka drew independently one isosceles triangle $ABC$, all angles of which are measured as a integer number of degrees. It turned out that the bases $AC$ of these triangles are equals and for each of them on the ray $BC$ there is a point $E$ such that $BE=AC$, and the angle $AEC$ is also measured by an integer number of degrees. Is it in necessary that: a) all three drawn triangles are equal to each other? b) among them there are at least two equal triangles?

Let $ a>0$ and $ f: [0,\infty) \to [0,a]$ be a continuous function on $ (0,\infty)$ and having Darboux property on $ [0,\infty)$. Prove that if $ f(0)\equal{}0$ and for all nonnegative $ x$ we have \[ xf(x) \geq \int^x_0 f(t) dt ,\] then $ f$ admits primitives on $ [0,\infty)$.

Let $n \ge 4$ be a natural number. Let $A_1A_2
\cdots
A_n$ be a regular polygon and $X = \{ 1,2,3....,n \} $. A subset $\{ i_1, i_2,\cdots, i_k \} $ of $X$, with $k \ge 3$ and $i_1 < i_2 <
\cdots < i_k$, is called a good subset if the angles of the polygon $A_{i_1}A_{i_2}\cdots
A_{i_k}$ , when arranged in the increasing order, are in an arithmetic progression. If $n$ is a prime, show that a proper good subset of $X$ contains exactly four elements.

Simplify \[\frac{bx(a^2x^2 + 2a^2y^2 + b^2y^2) + ay(a^2x^2 + 2b^2x^2 + b^2y^2)}{bx + ay}.\] $ \textbf{(A)}\ a^2x^2 + b^2y^2\qquad\textbf{(B)}\ (ax + by)^2\qquad\textbf{(C)}\ (ax + by)(bx + ay)\qquad\textbf{(D)}\ 2(a^2x^2 + b^2y^2)\qquad\textbf{(E)}\ (bx + ay)^2 $

Find the value of $\frac12+\frac{4}{2^2} +\frac{9}{2^3} +\frac{16}{2^4} + ...$

Find all real solutions to the equation $$\lfloor x \rfloor ^2 + \lfloor x \rfloor= x^2-\frac14.$$

For a given positive integer $n\ge 2$, suppose positive integers $a_i$ where $1\le i\le n$ satisfy $a_1<a_2<\ldots <a_n$ and $\sum_{i=1}^n \frac{1}{a_i}\le 1$. Prove that, for any real number $x$, the following inequality holds \[\left(\sum_{i=1}^n\frac{1}{a_i^2+x^2}\right)^2\le\frac{1}{2}\cdot\frac{1}{a_1(a_1-1)+x^2} \] [i]Li Shenghong[/i]

In a certain Zoom meeting, there are $4$ students. How many ways are there to split them into any number of distinguishable breakout rooms, each with at least $ 1$ student?

Divide the upper right quadrant of the plane into square cells with side length $1$. In this quadrant, $n^2$ cells are colored, show that there’re at least $n^2+n$ cells (possibly including the colored ones) that at least one of its neighbors are colored.

A [i]hedgehog[/i] is a circle without its boundaries. The diameter of the hedgehog is the diameter of the corresponding circle. We say that the hedgehog sits at the at the point where the center of the circle is located. We are given a triangle with sides $a, b, c$, with hedgehogs sitting at its vertices. It is known that inside the triangle there is a point from which you can reach any side of the triangle by walking along a straight line without hitting any hedgehog. What is the largest possible sum of the diameters of these hedgehogs? [i]Proposed by Oleksiy Masalitin[/i]