There is a game. The magician let the participant think up a positive integer (at least two digits). For example, an integer $ \displaystyle\overline{a_1a_2 \cdots a_n}$ is rearranged as $ \overline{a_{i_1}a_{i_2} \cdots a_{i_n}}$, that is, $ i_1, i_2, \cdots, i_n$ is a permutation of $ 1,2, \cdots, n$. Then we get $ n!\minus{}1$ integers. The participant is asked to calculate the sum of the $ n!\minus{}1$ numbers, then tell the magician the sum $ S$. The magician claims to be able to know the original number when he is told the sum $ S$. Try to decide whether the magician can be successful or not.

[asy] size(140); import geometry; dot((0,0));label("$(0,0)$",(0,0),SW); dot((4,3)); dot((5,4));label("$(5,4)$",(5,4),NE); draw((0,0)--(7,0), EndArrow); draw((0,0)--(0,6), EndArrow); add(grid(5,4)); [/asy] A leprechaun wishes to travel from the origin to a pot of gold located at the coordinate point $(5,4)$. If she can only move upwards and rightwards along the unit grid, must pass a checkpoint at $(1,2)$, and must avoid an evil thief at $(4,3)$, how many distinct paths can she take? $\textbf{(A) }7\qquad\textbf{(B) }15\qquad\textbf{(C) }21\qquad\textbf{(D) }45\qquad\textbf{(E) }126$

Jenn randomly chooses a number $J$ from $1, 2, 3,\ldots, 19, 20$. Bela then randomly chooses a number $B$ from $1, 2, 3,\ldots, 19, 20$ distinct from $J$. The value of $B - J$ is at least $2$ with a probability that can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

The sum of the angles on the bigger base of a trapezoid is $90^o$. Prove that the line segment whose ends are the midpoints of the bases, is equal to the line segment whose ends are the midpoints of the diagonals.

Find all positive integers $n\geq1$ such that there exists a pair $(a,b)$ of positive integers, such that $a^2+b+3$ is not divisible by the cube of any prime, and $$n=\frac{ab+3b+8}{a^2+b+3}.$$

Let $n$ be a positive integer, and $x$ be a positive real number. Prove that $$\sum_{k=1}^{n} \left( x \left[\frac{k}{x}\right] - (x+1)\left[\frac{k}{x+1}\right]\right) \leq n,$$ where $[x]$ denotes the largest integer not exceeding $x$.

(A.Zaslavsky) Given two triangles $ ABC$, $ A'B'C'$. Denote by $ \alpha$ the angle between the altitude and the median from vertex $ A$ of triangle $ ABC$. Angles $ \beta$, $ \gamma$, $ \alpha'$, $ \beta'$, $ \gamma'$ are defined similarly. It is known that $ \alpha \equal{} \alpha'$, $ \beta \equal{} \beta'$, $ \gamma \equal{} \gamma'$. Can we conclude that the triangles are similar?

Given a sequence $a_1,a_2,\ldots $ of non-negative real numbers satisfying the conditions: 1. $a_n + a_{2n} \geq 3n$; 2. $a_{n+1}+n \leq 2\sqrt{a_n \left(n+1\right)}$ for all $n\in\mathbb N$ (where $\mathbb N=\left\{1,2,3,...\right\}$). (1) Prove that the inequality $a_n \geq n$ holds for every $n \in \mathbb N$. (2) Give an example of such a sequence.

Six different-sized cubes are glued together, one on top of the other. The bottom cube has edge length $8$. Each of the other cubes has four vertices at the midpoints of the edges of the cube below it as shown. The entire solid is then dipped in red paint. What is the total area of the red-painted surface on the solid? (will insert image here later) $\text{(A) }630\qquad\text{(B) }632\qquad\text{(C) }648\qquad\text{(D) }694\qquad\text{(E) }756$

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]

Initially, there are three different points on the plane. Every minute, three points are chosen, for example $A$, $B$ and $C$, and a new point $D$ is generated which is symmetric to $A$ with respect to the perpendicular bisector of line segment $BC$. 24 hours later, it turns out that among all the points that were generated, there exist three collinear points. Prove that the three initial points were also collinear. (Author: V. Shmarov)

Given is a permutation of $1, 2, \ldots, 2023, 2024$ and two positive integers $a, b$, such that for any two adjacent numbers, at least one of the following conditions hold: 1) their sum is $a$; 2) the absolute value of their difference is $b$. Find all possible values of $b$.

For given positive integer $n$ find all quartets $(x_1,x_2,x_3,x_4)$ such that $x_1^2+x_2^2+x_3^2+x_4^2=4^n$

Let $e > 0$ be a given real number. Find the least value of $f(e)$ (in terms of $e$ only) such that the inequality $a^{3}+ b^{3}+ c^{3}+ d^{3} \leq e^{2}(a^{2}+b^{2}+c^{2}+d^{2}) + f(e)(a^{4}+b^{4}+c^{4}+d^{4})$ holds for all real numbers $a, b, c, d$.

Two circles $O_1$ and $O_2$ intersect at $M,N$. The common tangent line nearer to $M$ of the two circles touches $O_1,O_2$ at $A,B$ respectively. Let $C,D$ be the symmetric points of $A,B$ with respect to $M$ respectively. The circumcircle of triangle $DCM$ intersects circles $O_1$ and $O_2$ at points $E,F$ respectively which are distinct from $M$. Prove that the circumradii of the triangles $MEF$ and $NEF$ are equal.

You are given a segment $AB$. Find the locus of the vertices $C$ of acute-angled triangles $ABC$.

In how many ways can the numbers $0,1,2,\dots , 9$ be arranged in such a way that the odd numbers form an increasing sequence, also the even numbers form an increasing sequence? $ \textbf{(A)}\ 126 \qquad\textbf{(B)}\ 189 \qquad\textbf{(C)}\ 252 \qquad\textbf{(D)}\ 315 \qquad\textbf{(E)}\ \text{None} $

The lateral sides and diagonals of a trapezoid intersect a line $l$, determining three equal segments on it. Must $l$ be parallel to the bases of the trapezoid?

$a,b,c$ are positive real numbers such that $\max\{\frac{a(b+c)}{a^2+bc},\frac{b(c+a)}{b^2+ca},\frac{c(a+b)}{c^2+ab}\}\le \frac{5}{2}$. Prove inequality $$\frac{a(b+c)}{a^2+bc}+\frac{b(c+a)}{b^2+ca}+\frac{c(a+b)}{c^2+ab}\le 3$$

Find all integer $n$ such that the equation $2x^2 + 5xy + 2y^2 = n$ has integer solution for $x$ and $y$.

Let $LOV ER$ be a convex pentagon such that $LOV E$ is a rectangle. Given that $OV = 20$ and $LO =V E = RE = RL = 23$, compute the radius of the circle passing through $R$, $O$, and $V$ .

In the following list of numbers, the integer $n$ appears $n$ times in the list for $1 \leq n \leq 200.$ $$1, 2, 2, 3, 3, 3, 4, 4, 4, 4, …, 200, 200, …, 200$$ What is the median of the numbers in this list? $\textbf{(A)}\ 100.5 \qquad\textbf{(B)}\ 134 \qquad\textbf{(C)}\ 142 \qquad\textbf{(D)}\ 150.5 \qquad\textbf{(E)}\ 167$

Using projective transformations prove the Pascal theorem (also find where the Pascal line intersects the circle).

$ABCD$ is a convex quadrilateral such that $AB<AD$. The diagonal $\overline{AC}$ bisects $\angle BAD$, and $m\angle ABD=130^\circ$. Let $E$ be a point on the interior of $\overline{AD}$, and $m\angle BAD=40^\circ$. Given that $BC=CD=DE$, determine $m\angle ACE$ in degrees.

Call a pair of integers $a$ and $b$ square makers , if $ab+1$ is a perfect square. Determine for which $n$ is it possible to divide the set $\{1,2, \dots , 2n\}$ into $n$ pairs of square makers.