In $\triangle$ABC, points D, M lie on side BC and AB respectively, point P lies on segment AD. Line DM intersects segments BP, AC (extended part), PC (extended part) at E, F and N respectively. Show that if DE=DF, then DM=DN.

Let $n$ be a positive integer. Show that \begin{align*}&\quad\,\,\frac{1}{\binom{n}{1}}+\frac{1}{2\binom{n}{2}}+\frac{1}{3\binom{n}{3}}+\cdots+\frac{1}{n\binom{n}{n}}\\&=\frac{1}{2^{n-1}}+\frac{1}{2\cdot2^{n-2}}+\frac{1}{3\cdot2^{n-3}}+\cdots+\frac{1}{n\cdot2^0}.\end{align*}

$n$ being a given integer, find all functions $f\colon \mathbb{Z} \to \mathbb{Z}$, such that for all integers $x,y$ we have $f\left( {x + y + f(y)} \right) = f(x) + ny$.

There is $n\times n$ chessboard. Each square has a number between $0$ and $k$. There is a button for each row and column, which increases the number of $n$ numbers of the row or column the button represents(if the number of the square is $k$, then it becomes $0$). If certain button is pressed, call it 'operation.' And we have a chessboard which is filled with 0(for all squares). After some 'operation's, the numbers of squares are different now. Prove that we can make all of the number $0$ within $kn$ 'operation's.

Let $n$ be a positive integer and $P,Q$ be polynomials with real coefficients with $P(x)=x^nQ(\frac{1}{x})$ and $P(x) \geq Q(x)$ for all real numbers $x$. Prove that $P(x)=Q(x)$ for all real number $x$.

At Port Aventura there are $16$ secret agents, each of whom is watching one or more other agents. It is known that if agent $A$ is watching agent $B$, then $B$ is not watching $A$. Moreover, any $10$ agents can be ordered so that the first is watching the second, the second is watching the third, etc, the last is watching the first. Show that any $11$ agents can also be so ordered.

An acute triangle $ABC$ with $AB<AC<BC$ is inscribed in a circle $c(O,R)$. The circle $c_1(A,AC)$ intersects the circle $c$ at point $D$ and intersects $CB$ at $E$. If the line $AE$ intersects $c$ at $F$ and $G$ lies in $BC$ such that $EB=BG$, prove that $F,E,D,G$ are concyclic.

On a Cartesian plane 101 planes are drawn and all points of intersection are labeled. Is it possible, that for every line, 50 of the points have positive coordinates and 50 of the points have negative coordinates [I]Proposed by S. Ivanov[/i]

Let $ f, g$ and $ a$ be polynomials with real coefficients, $ f$ and $ g$ in one variable and $ a$ in two variables. Suppose \[ f(x) \minus{} f(y) \equal{} a(x, y)(g(x) \minus{} g(y)) \forall x,y \in \mathbb{R}\] Prove that there exists a polynomial $ h$ with $ f(x) \equal{} h(g(x)) \text{ } \forall x \in \mathbb{R}.$

For which integers $n \ge 2$ can we arrange numbers $1,2, \ldots, n$ in a row, such that for all integers $1 \le k \le n$ the sum of the first $k$ numbers in the row is divisible by $k$?

Indians find those sequences of non-negative real numbers $x_0, x_1,...$ [i]mystical [/i]t hat satisfy $x_0 < 2021$, $x_{i+1} = \lfloor x_i \rfloor \{x_i\}$ for every $i \ge 0$, furthermore the sequence contains an integer different from $0$. How many sequences are mystical according to the Indians?

The glass gauge on a cylindrical coffee maker shows that there are 45 cups left when the coffee maker is $36\%$ full. How many cups of coffee does it hold when it is full? [asy] draw((5,0)..(0,-1.3)..(-5,0)); draw((5,0)--(5,10)); draw((-5,0)--(-5,10)); draw(ellipse((0,10),5,1.3)); draw(circle((.3,1.3),.4)); draw((-.1,1.7)--(-.1,7.9)--(.7,7.9)--(.7,1.7)--cycle); fill((-.1,1.7)--(-.1,4)--(.7,4)--(.7,1.7)--cycle,black); draw((-2,11.3)--(2,11.3)..(2.6,11.9)..(2,12.2)--(-2,12.2)..(-2.6,11.9)..cycle);[/asy] $ \text{(A)}\ 80\qquad\text{(B)}\ 100\qquad\text{(C)}\ 125\qquad\text{(D)}\ 130\qquad\text{(E)}\ 262 $

How many [b]distinguishable[/b] rearrangements of the letters in CONTEST have both the vowels first? (For instance, OETCNST is a one such arrangements but OTETSNC is not.) $ \textbf{(A)}\ 60\qquad \textbf{(B)}\ 120\qquad \textbf{(C)}\ 240\qquad \textbf{(D)}\ 720\qquad \textbf{(E)}\ 2520$

Find all prime numbers p such that $2^p+p^2 $ is also a prime number.

Let \(a,b,c\) be three distinct positive real numbers such that \(abc=1\). Prove that $$\dfrac{a^3}{(a-b)(a-c)}+\dfrac{b^3}{(b-c)(b-a)}+\dfrac{c^3}{(c-a)(c-b)} \ge 3$$

Let $\mathbb{N}$ be the set of positive integers. Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a function such that for every positive integer $n \in \mathbb{N}$ $$ f(n) -n<2021 \quad \text{and} \quad f^{f(n)}(n) =n$$ Prove that $f(n)=n$ for infinitely many $n \in \mathbb{N}$

Let $AD$ be a median of $\triangle ABC$ such that $m(\widehat{ADB})=45^{\circ}$ and $m(\widehat{ACB})=30^{\circ}$. What is the measure of $\widehat{ABC}$ in degrees? $ \textbf{(A)}\ 75 \qquad\textbf{(B)}\ 90 \qquad\textbf{(C)}\ 105 \qquad\textbf{(D)}\ 120 \qquad\textbf{(E)}\ 135 $

It is possible to place positive integers into the vacant twenty-one squares of the $5 \times 5$ square shown below so that the numbers in each row and column form arithmetic sequences. Find the number that must occupy the vacant square marked by the asterisk (*). [asy] int i; for(i=1; i<5; i=i+1) { draw((0,2*i)--(10,2*i)); draw((2*i,0)--(2*i,10)); } string[] no={"0", "74", "103", "*", "186"}; pair[] yes={(1,1), (3,7), (5,3), (7,9), (9,5)}; for(i=0; i<5; i=i+1) { label(no[i], yes[i]); } draw(origin--(10,0)--(10,10)--(0,10)--cycle, linewidth(2));[/asy]

Each of the twenty dots on the graph below represents one of Sarah's classmates. Classmates who are friends are connected with a line segment. For her birthday party, Sarah is inviting only the following: all of her friends and all of those classmates who are friends with at least one of her friends. How many classmates will not be invited to Sarah's party? [asy]/* AMC8 2003 #18 Problem */ pair a=(102,256), b=(68,131), c=(162,101), d=(134,150); pair e=(269,105), f=(359,104), g=(303,12), h=(579,211); pair i=(534, 342), j=(442,432), k=(374,484), l=(278,501); pair m=(282,411), n=(147,451), o=(103,437), p=(31,373); pair q=(419,175), r=(462,209), s=(477,288), t=(443,358); pair oval=(282,303); draw(l--m--n--cycle); draw(p--oval); draw(o--oval); draw(b--d--oval); draw(c--d--e--oval); draw(e--f--g--h--i--j--oval); draw(k--oval); draw(q--oval); draw(s--oval); draw(r--s--t--oval); dot(a); dot(b); dot(c); dot(d); dot(e); dot(f); dot(g); dot(h); dot(i); dot(j); dot(k); dot(l); dot(m); dot(n); dot(o); dot(p); dot(q); dot(r); dot(s); dot(t); filldraw(yscale(.5)*Circle((282,606),80),white,black); label(scale(0.75)*"Sarah", oval);[/asy] $ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 7$

Find all quadruples \((a,b,c,d)\) of positive integers such that \(\displaystyle \frac{ac+bd}{a+c}\) and \(\displaystyle \frac{bc-ad}{b-d}\) are equal to the prime number \(90121\).

Suppose $a$ and $b$ are single-digit positive integers chosen independently and at random. What is the probability that the point $(a,b)$ lies above the parabola $y=ax^2-bx$? $ \textbf{(A)}\ \frac{11}{81} \qquad \textbf{(B)}\ \frac{13}{81} \qquad \textbf{(C)}\ \frac{5}{27} \qquad \textbf{(D)}\ \frac{17}{81} \qquad \textbf{(E)}\ \frac{19}{81} $

Two countries ($A$ and $B$) organize a conference, and they send an equal number of participants. Some of them have known each other from a previous conference. Prove that one can choose a nonempty subset $C$ of the participants from $A$ such that one of the following holds: [list][*]the participants from $B$ each know an even number of people in $C$, [*]the participants from $B$ each know an odd number of participants in $C$.[/list]

Let $p$ be a prime. We arrange the numbers in ${\{1,2,\ldots ,p^2} \}$ as a $p \times p$ matrix $A = ( a_{ij} )$. Next we can select any row or column and add $1$ to every number in it, or subtract $1$ from every number in it. We call the arrangement [i]good[/i] if we can change every number of the matrix to $0$ in a finite number of such moves. How many good arrangements are there?

The sequence $\{u_n\}$ is defined by $u_1 = 1, u_2 = 1, u_n = u_{n-1} + 2u_{n-2} for n \geq 3$. Prove that for any positive integers $n, p \ (p > 1), u_{n+p} = u_{n+1}u_{p} + 2u_nu_{p-1}$. Also find the greatest common divisor of $u_n$ and $u_{n+3}.$

Find the largest value for $\frac{y}{x}$ for pairs of real numbers $(x,y)$ which satisfy \[(x-3)^2 + (y-3)^2 = 6.\] $\textbf{(A) }3 + 2 \sqrt 2\qquad \textbf{(B) } 2 + \sqrt 3\qquad \textbf{(C ) }3 \sqrt 3\qquad \textbf{(D) }6\qquad \textbf{(E) }6 + 2 \sqrt 3$