Solve the system of equations: $$\frac{x-1}{xy-3}=\frac{y-2}{xy-4}=\frac{3-x-y}{7-x^2-y^2}$$

For two quadratic trinomials $P(x)$ and $Q(x)$ there is a linear function $\ell(x)$ such that $P(x)=Q(\ell(x))$ for all real $x$. How many such linear functions $\ell(x)$ can exist? [i](A. Golovanov)[/i]

Let $t_1,t_2,\ldots,t_k$ be different straight lines in space, where $k>1$. Prove that points $P_i$ on $t_i$, $i=1,\ldots,k$, exist such that $P_{i+1}$ is the projection of $P_i$ on $t_{i+1}$ for $i=1,\ldots,k-1$, and $P_1$ is the projection of $P_k$ on $t_1$.

Let $P (x)$ be a polynomial of degree $2n$, such that $P (k) =\frac{k}{k + 1}$ for $k = 0,...,2n$. Determine $P (2n + 1)$.

Let $ p$ be a prime and let \[ l_k(x,y)\equal{}a_kx\plus{}b_ky \;(k\equal{}1,2,...,p^2)\ .\] be homogeneous linear polynomials with integral coefficients. Suppose that for every pair $ (\xi,\eta)$ of integers, not both divisible by $ p$, the values $ l_k(\xi,\eta), \;1\leq k\leq p^2 $, represent every residue class $ \textrm{mod} \;p$ exactly $ p$ times. Prove that the set of pairs $ \{(a_k,b_k): 1\leq k \leq p^2 \}$ is identical $ \textrm{mod} \;p$ with the set $ \{(m,n): 0\leq m,n \leq p\minus{}1 \}.$

A triplet of positive integers $(x, y, z)$ satisfying $x, y, z > 1$ and $x^3 - yz^3 = 2021$ is called [i]primary [/i] if at least two of the integers $x, y, z$ are prime numbers. a) Find at least one primary triplet. b) Show that there are infinitely many primary triplets.

What is the value of $$\left(\sum_{k=1}^{20} \log_{5^k} 3^{k^2}\right)\cdot\left(\sum_{k=1}^{100} \log_{9^k} 25^k\right)?$$ $\textbf{(A) }21 \qquad \textbf{(B) }100\log_5 3 \qquad \textbf{(C) }200\log_3 5 \qquad \textbf{(D) }2,200\qquad \textbf{(E) }21,000$

Let $\mathbb{N}^*=\{1,2,\ldots\}$. Find al the functions $f: \mathbb{N}^*\rightarrow \mathbb{N}^*$ such that: (1) If $x<y$ then $f(x)<f(y)$. (2) $f\left(yf(x)\right)=x^2f(xy)$ for all $x,y \in\mathbb{N}^*$.

Let $ABCD$ be a cyclic quadrilateral such that $\angle BAD \le \angle ADC$. Prove that $AC + CD \le AB + BD$.

There is a football championship with $6$ teams involved, such that for any $2$ teams $A$ and $B$, $A$ plays with $B$ and $B$ plays with $A$ ($2$ such games are distinct). After every match, the winning teams gains $3$ points, the loosing team gains $0$ points and if there is a draw, both teams gain $1$ point each.\\ \\ In the end, the team standing on the last place has $12$ points and there are no $2$ teams that scored the same amount of points.\\ \\ For all the remaining teams, find their final scores and provide an example with the outcomes of all matches for at least one of the possible final situations. $\textit{(Andrei Bâra)}$

Show that for any positive integer $n > 2$ we can find $n$ distinct positive integers such that the sum of their reciprocals is $1$.

Prove that there are infinitely many positive integers $ n$ such that $ n^{2} \plus{} 1$ has a prime divisor greater than $ 2n \plus{} \sqrt {2n}$. [i]Author: Kestutis Cesnavicius, Lithuania[/i]

In a family album, there are ten photos. On each of them, three people are pictured: in the middle stands a man, to the right of him stands his brother, and to the left of him stands his son. What is the least possible total number of people pictured, if all ten of the people standing in the middle of the ten pictures are different.

Let $ABC$ be an acute triangle with orthocenter $H$ and circumcircle $\Omega$. Let $M$ be the midpoint of side $BC$. Point $D$ is chosen from the minor arc $BC$ on $\Gamma$ such that $\angle BAD = \angle MAC$. Let $E$ be a point on $\Gamma$ such that $DE$ is perpendicular to $AM$, and $F$ be a point on line $BC$ such that $DF$ is perpendicular to $BC$. Lines $HF$ and $AM$ intersect at point $N$, and point $R$ is the reflection point of $H$ with respect to $N$. Prove that $\angle AER + \angle DFR = 180^\circ$. [i]Proposed by Li4.[/i]

The number of postive integers less than $1000$ divisible by neither $5$ nor $7$ is: $\text{(A)}\ 688 \qquad \text{(B)}\ 686\qquad \text{(C)}\ 684 \qquad \text{(D)}\ 658\qquad \text{(E)}\ 630$

Let $ABC$ be an acute, scalene triangle. Let $H$ be the orthocenter and $O$ be the circumcenter of triangle $ABC$, and let $P$ be a point interior to the segment $HO.$ The circle with center $P$ and radius $PA$ intersects the lines $AB$ and $AC$ again at $R$ and $S$, respectively. Denote by $Q$ the symmetric point of $P$ with respect to the perpendicular bisector of $BC$. Prove that points $P$, $Q$, $R$ and $S$ lie on the same circle.

A $\text{2.25 kg}$ mass undergoes an acceleration as shown below. How much work is done on the mass? [asy] // Code by riben size(350); // Axes draw((0,0)--(12,0),lightgray); draw((0,-3)--(0,5)); // Tick Marks draw((2,0)--(2,-0.2)); label("2",(2,-0.2),S*2); draw((4,0)--(4,-0.2)); label("4",(4,-0.2),S*2); draw((6,0)--(6,-0.2)); label("6",(6,-0.2),S*2); draw((8,0)--(8,-0.2)); label("8",(8,-0.2),S*2); draw((10,0)--(10,-0.2)); label("10",(10,-0.2),S*2); draw((12,0)--(12,-0.2)); label("12",(12,-0.2),S*2); draw((0,-2)--(-0.2,-2)); label("-2",(-0.2,-2),W); draw((0,0)--(-0.2,0),lightgray); label("0",(-0.2,0),W); draw((0,2)--(-0.2,2),lightgray); label("2",(-0.2,2),W); draw((0,4)--(-0.2,4)); label("4",(-0.2,4),W); // Dashed Lines draw((0,-2)--(12,-2),dashed); draw((0,2)--(12,2),dashed+lightgray); draw((0,4)--(12,4),dashed); draw((2,5)--(2,0.2),dashed); draw((4,5)--(4,0.2),dashed); draw((6,5)--(6,0.2),dashed); draw((8,5)--(8,0.2),dashed); draw((10,5)--(10,0.2),dashed); draw((12,5)--(12,0.2),dashed); draw((2,-1)--(2,-3),dashed); draw((4,-1)--(4,-3),dashed); draw((6,-1)--(6,-3),dashed); draw((8,-1)--(8,-3),dashed); draw((10,-1)--(10,-3),dashed); draw((12,-1)--(12,-3),dashed); // Path path A=(0,0)--(2,4)--(4,4)--(6,2)--(8,0)--(10,-2)--(12,0); draw(A,linewidth(1.5)); // Labels label(scale(0.85)*rotate(90)*"Acceleration (m/s/s)",(0,1),W*7); label(scale(0.75)*"Position (m)",(11,0),N); [/asy] (A) $\text{36 J}$ (B) $\text{22 J}$ (C) $\text{5 J}$ (D)$\text{-17 J}$ (E) $\text{-36 J}$

We have $2^m$ sheets of paper, with the number $1$ written on each of them. We perform the following operation. In every step we choose two distinct sheets; if the numbers on the two sheets are $a$ and $b$, then we erase these numbers and write the number $a + b$ on both sheets. Prove that after $m2^{m -1}$ steps, the sum of the numbers on all the sheets is at least $4^m$ . [i]Proposed by Abbas Mehrabian, Iran[/i]

Let $ABCDEF$ be a regular hexagon of side length 1. Compute the area of the intersection of the circle centered at $A$ passing through $C$ and the circle centered at $D$ passing through $E$. [i]Proposed by Robert Trosten[/i]

We have a square of side $1$ and a number $\ell$ such that $0 <\ell <\sqrt2$. Two players $A$ and $B$, in turn, draw in the square an open segment (without its two ends) of length $\ell $, starts A. Each segment after the first cannot have points in common with the previously drawn segments. He loses the player who cannot make his play. Determine if either player has a winning strategy.

Let $K, T$ be the points of tangency of inscribed and exscribed circles to the side $BC$ triangle $ABC$, $M$ is the midpoint of the side $BC$. Using a compass and a ruler, construct triangle ABC given rays $AK$ and $AT$ (points $K, T$ are not marked on them) and point $M$.

I am not a spammer, at least, this is the way I use to think about myself, and thus I will not open a new thread for the following problem from today's DeMO exam: Let Q(n) denote the sum of the digits of a positive integer n. Prove that $Q\left(Q\left(Q\left(2005^{2005}\right)\right)\right)=7$. [[b]EDIT:[/b] Since this post was split into a new thread, I comment: The problem is completely analogous to the problem posted at http://www.mathlinks.ro/Forum/viewtopic.php?t=31409 , with the only difference that you have to consider the number $2005^{2005}$ instead of $4444^{4444}$.] Darij

Let $f, g:\mathbb{Z}\rightarrow [0,\infty )$ be two functions such that $f(n)=g(n)=0$ with the exception of finitely many integers $n$. Define $h:\mathbb{Z}\rightarrow [0,\infty )$ by \[h(n)=\max \{f(n-k)g(k): k\in\mathbb{Z}\}.\] Let $p$ and $q$ be two positive reals such that $1/p+1/q=1$. Prove that \[ \sum_{n\in\mathbb{Z}}h(n)\geq \Bigg(\sum_{n\in\mathbb{Z}}f(n)^p\Bigg)^{1/p}\Bigg(\sum_{n\in\mathbb{Z}}g(n)^q\Bigg)^{1/q}.\]

An infinite table whose rows and columns are numbered with positive integers, is given. For a sequence of functions $f_1(x), f_2(x), \ldots $ let us place the number $f_i(j)$ into the cell $(i,j)$ of the table (for all $i, j\in \mathbb{N}$). A sequence $f_1(x), f_2(x), \ldots $ is said to be {\it nice}, if all the numbers in the table are positive integers, and each positive integer appears exactly once. Determine if there exists a nice sequence of functions $f_1(x), f_2(x), \ldots $, such that each $f_i(x)$ is a polynomial of degree 101 with integer coefficients and its leading coefficient equals to 1.

A sequence of numbers $a_1,a_2,...,a_{1999}$ is given. In each move it is allowed to choose two of the numbers, say $a_m,a_n$, and replace them by the numbers $$\frac{a_n^2}{a_m^2}-\frac{n}{m}\left(\frac{a_m^2}{a_n}-a_m\right), \frac{a_m^2}{a_n^2}-\frac{m}{n}\left(\frac{a_n^2}{a_m}-a_n\right) $$ respectively. Starting with the sequence $a_i = 1$ for $20 \nmid i$ and $a_i =\frac{1}{5}$ for $20 \mid i$, is it possible to obtain a sequence whose all terms are integers?