Found problems: 85335
Six teams participate in a hockey tournament. Each team plays once against every other team. In each game, a team is awarded $3$ points for a win, $1$ point for a draw, and none for a loss. After the tournament the teams are ranked by total points. No two teams have the same total points. Each team (except the bottom team) has $2$ points more than the team ranking one place lower. Prove that the team that finished fourth has won two games and lost three games.
Five teams participated in the commercial football tournament. Each had to play exactly one match with each other. Due to financial difficulties, the organizers canceled some games. In the end It turned out that all teams scored a different number of points and not a single team in the points column had a zero. What is the smallest number of games could be played in a tournament if three points were awarded for a win, for a draw - one, for a defeat - zero?
[b]5.[/b] Show that a ring $R$ is commutative if for every $x \in R$ the element $x^{2}-x$ belongs to the centre of $R$. [b](A. 18)[/b]
In the following diagram, let $ABCD$ be a square and let $M,N,P$ and $Q$ be the midpoints of its sides. Prove that
\[S_{A'B'C'D'} = \frac 15 S_{ABCD}.\]
[asy]
import graph; size(200); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen qqttzz = rgb(0,0.2,0.6); pen qqzzff = rgb(0,0.6,1);
draw((0,4)--(4,4),qqttzz+linewidth(1.6pt)); draw((4,4)--(4,0),qqttzz+linewidth(1.6pt)); draw((4,0)--(0,0),qqttzz+linewidth(1.6pt)); draw((0,0)--(0,4),qqttzz+linewidth(1.6pt)); draw((0,4)--(2,0),qqzzff+linewidth(1.2pt)); draw((2,4)--(4,0),qqzzff+linewidth(1.2pt)); draw((0,2)--(4,4),qqzzff+linewidth(1.2pt)); draw((0,0)--(4,2),qqzzff+linewidth(1.2pt));
dot((0,4),ds); label("$A$", (0.07,4.12), NE*lsf); dot((0,0),ds); label("$D$", (-0.27,-0.37), NE*lsf); dot((4,0),ds); label("$C$", (4.14,-0.39), NE*lsf); dot((4,4),ds); label("$B$", (4.08,4.12), NE*lsf); dot((2,4),ds); label("$M$", (2.08,4.12), NE*lsf); dot((4,2),ds); label("$N$", (4.2,1.98), NE*lsf); dot((2,0),ds); label("$P$", (1.99,-0.49), NE*lsf); dot((0,2),ds); label("$Q$", (-0.48,1.9), NE*lsf); dot((0.8,2.4),ds); label("$A'$", (0.81,2.61), NE*lsf); dot((2.4,3.2),ds); label("$B'$", (2.46,3.47), NE*lsf); dot((3.2,1.6),ds); label("$C'$", (3.22,1.9), NE*lsf); dot((1.6,0.8),ds); label("$D'$", (1.14,0.79), NE*lsf); clip((-4.44,-11.2)--(-4.44,6.41)--(16.48,6.41)--(16.48,-11.2)--cycle);
[/asy]
[$S_{X}$ denotes area of the $X.$]
On the attraction "Merry parking", the auto has only two position* of a steering wheel: "right", and "strongly right". So the auto can move along an arc with radius $r_1$ or $r_2$. The auto started from a point $A$ to the Nord, it covered the distance $\ell$ and rotated to the angle $a < 2\pi$. Find the locus of its possible endpoints.
The graph, $G$ of $y = \log_{10}x$ is rotated $90^{\circ}$ counter-clockwise about the origin to obtain a new graph $G'$. Which of the following is an equation for $G'$?
$ \textbf{(A)}\ y = \log_{10}\left(\frac{x + 90}{9}\right)\qquad\textbf{(B)}\ y = \log_{x}10\qquad\textbf{(C)}\ y = \frac{1}{x + 1}\qquad\textbf{(D)}\ y = 10^{-x}\qquad\textbf{(E)}\ y = 10^{x} $
The geometric series $ a \plus{} ar \plus{} ar^{2} \plus{} ...$ has a sum of $ 7$, and the terms involving odd powers of $ r$ have a sum of $ 3$. What is $ a \plus{} r$?
$ \textbf{(A)}\ \frac {4}{3}\qquad \textbf{(B)}\ \frac {12}{7}\qquad \textbf{(C)}\ \frac {3}{2}\qquad \textbf{(D)}\ \frac {7}{3}\qquad \textbf{(E)}\ \frac {5}{2}$
Let be a natural number $ n. $ Solve in the set of $ 2\times 2 $ complex matrices the equation
$$ \begin{pmatrix} -2& 2007\\ 0&-2 \end{pmatrix} =X^{3n}-3X^n. $$
[i]Petru Vlad[/i]
We have a $2\times n$ rectangle. We call each $1\times1$ square a room and we show the room in the $i^{th}$ row and $j^{th}$ column as $(i,j)$. There are some coins in some rooms of the rectangle. If there exist more than $1$ coin in each room, we can delete $2$ coins from it and add $1$ coin to its right adjacent room OR we can delete $2$ coins from it and add $1$ coin to its up adjacent room. Prove that there exists a finite configuration of allowable operations such that we can put a coin in the room $(1,n)$.
In the octagon below all sides have the length $1$ and all angles are equal.
Determine the distance between the corners $A$ and $B$.
[img]https://1.bp.blogspot.com/-i6TAFDvcQ8w/XzXCRhnV_kI/AAAAAAAAMVw/rKrQMfPYYJIaCwl8hhdVHdqO4fIn8O7cwCLcBGAsYHQ/s0/2011%2BMogh%2Bp2.png[/img]
Consider the parabola $ P:x-y^2-(p+3)y-p=0,p\in\mathbb{R}^*. $ Show that $ P $ intersects the coordonate axis at three points, and that the circle formed by these three points passes through a fixed point.
Find the range of $ f(A)=\frac{\sin A(3\cos^{2}A+\cos^{4}A+3\sin^{2}A+\sin^{2}A\cos^{2}A)}{\tan A (\sec A-\sin A\tan A)} $ if $A\neq \dfrac{n\pi}{2}$.
A particle is moving randomly around a plane. It starts at $(0,0)$. Every second, it moves one unit randomly in a direction parallel to the $x$ or $y$ axis. At some time in the first hour, the particle was at the point $(2023,23)$. After $4092$ seconds, the particle is at $(x,y)$. Find the expected value of $x+y$.
[i]Individual #3[/i]
Aaron the ant walks on the coordinate plane according to the following rules. He starts at the origin $p_0=(0,0)$ facing to the east and walks one unit, arriving at $p_1=(1,0)$. For $n=1,2,3,\dots$, right after arriving at the point $p_n$, if Aaron can turn $90^\circ$ left and walk one unit to an unvisited point $p_{n+1}$, he does that. Otherwise, he walks one unit straight ahead to reach $p_{n+1}$. Thus the sequence of points continues $p_2=(1,1), p_3=(0,1), p_4=(-1,1), p_5=(-1,0)$, and so on in a counterclockwise spiral pattern. What is $p_{2015}$?
$ \textbf{(A) } (-22,-13)\qquad\textbf{(B) } (-13,-22)\qquad\textbf{(C) } (-13,22)\qquad\textbf{(D) } (13,-22)\qquad\textbf{(E) } (22,-13) $
Given a four digit string $ k=\overline{abcd} $, $ a, b, c, d\in \{0, 1, \cdots, 9\} $, prove that there exist a $n<20000$ such that $2^n$ contains $k$ as a substring when written in base $10$.
[Extra: Can you give a better bound? Mine is $12517$]
The circle $\omega$ with center $O$ has given. From an arbitrary point $T$ outside of $\omega$ draw tangents $TB$ and $TC$ to it. $K$ and $H$ are on $TB$ and $TC$ respectively.
[b]a)[/b] $B'$ and $C'$ are the second intersection point of $OB$ and $OC$ with $\omega$ respectively. $K'$ and $H'$ are on angle bisectors of $\angle BCO$ and $\angle CBO$ respectively such that $KK' \bot BC$ and $HH'\bot BC$. Prove that $K,H',B'$ are collinear if and only if $H,K',C'$ are collinear.
[b]b)[/b] Consider there exist two circle in $TBC$ such that they are tangent two each other at $J$ and both of them are tangent to $\omega$.and one of them is tangent to $TB$ at $K$ and other one is tangent to $TC$ at $H$. Prove that two quadrilateral $BKJI$ and $CHJI$ are cyclic ($I$ is incenter of triangle $OBC$).
Initially the equation
$$\star \frac{1}{x-1} \star \frac{1}{x-2} \star \frac{1}{x-4} ... \star \frac{1}{x-2^{2023}}=0$$
is written on the board. In each turn Aslı and Zehra deletes one of the stars in the equation and writes $+$ or $-$ instead. The first move is performed by Aslı and continues in order. What is the maximum number of real solutions Aslı can guarantee after all the stars have been replaced by signs?
Let $(R,+,\cdot)$ be a ring with center $Z=\{a\in\mathbb{R}:ar=ra,\forall r\in\mathbb{R}\}$ with the property that the group $U=U(R)$ of its invertible elements is finite. Given that $G$ is the group of automorphisms of the additive group $(R,+),$ prove that \[|G|\geq\frac{|U|^2}{|Z\cap U|}.\][i]Dragoș Crișan[/i]
Scalene triangle $ABC$ has incenter $I$ and circumcircle $\Omega$ with center $O$. $H$ is the orthocenter of triangle $BIC$, and $T$ is a point on $\Omega$ for which $\angle ATI=90^\circ$. Circle $(AIO)$ intersects line $IH$ again at $X$. Show that the lines $AX, HT$ intersect on $\Omega$.
Let $d$ be a diameter of a circle $k$, and let $A$ be an arbitrary point on this diameter $d$ in the interior of $k$. Further, let $P$ be a point in the exterior of $k$. The circle with diameter $PA$ meets the circle $k$ at the points $M$ and $N$.
Find all points $B$ on the diameter $d$ in the interior of $k$ such that
\[\measuredangle MPA = \measuredangle BPN \quad \text{and} \quad PA \leq PB.\]
(i. e. give an explicit description of these points without using the points $M$ and $N$).
In a zoo, there are five more foxes than rabbits, and three more foxes than pandas. Are there more pandas or rabbits, and by how much more?
$\textbf{(A) }\text{Pandas, }1\qquad\textbf{(B) }\text{Pandas, }2\qquad\textbf{(C) }\text{Rabbits, }1\qquad\textbf{(D) }\text{Rabbits, }2\qquad\textbf{(E) }\text{Rabbits, }3$
Two circles $U,V$ have distinct radii,tangent to each other externally at $T$.$A,B$ are points on $U,V$ respectively,both distinct from $T$,such that $\angle ATB=90$.
(1)Prove that line $AB$ passes through a fixed point;
(2)Find the locus of the midpoint of $AB$.
It is known that a cells square can be cut into $n$ equal figures of $k$ cells.
Prove that it is possible to cut it into $k$ equal figures of $n$ cells.
Let $P(x)$ be a polynomial with real coefficients so that equation $P(m) + P(n) = 0$ has infi nitely many pairs of integer solutions $(m,n)$. Prove that graph of $y = P(x)$ has a center of symmetry.
In a quadrilateral $ABCD$ with an incircle, $AB = CD; BC < AD$ and $BC$ is parallel to $AD$. Prove that the bisector of $\angle C$ bisects the area of $ABCD$.