[b]8.1[/b] In the parallelogram $ABCD$ , the diagonal $AC$ is greater than the diagonal $BD$. The point $M$ on the diagonal $AC$ is such that around the quadrilateral $BCDM$ one can circumscribe a circle. Prove that $BD$ is the common tangent of the circles circumscribed around the triangles $ABM$ and $ADM$. [img]https://cdn.artofproblemsolving.com/attachments/b/3/9f77ff1f2198c201e5c270ec5b091a9da4d0bc.png[/img] [b]8.2 [/b] $A$ is an odd integer, $x$ and $y$ are roots of equation $t^2+At-1=0$. Prove that $x^4 + y^4$ and $x^5+ y^5$ are coprime integer numbers. [b]8.3[/b] A regular triangle is reflected symmetrically relative to one of its sides. The new triangle is again reflected symmetrically about one of its sides. This is repeated several times. It turned out that the resulting triangle coincides with the original one. Prove that an even number of reflections were made. [b]8.4 /7.6[/b] Several circles are arbitrarily placed in a circle of radius $3$, the sum of their radii is $25$. Prove that there is a straight line that intersects at least $9$ of these circles. [b]8.5 [/b] All two-digit numbers that do not end in zero are written one after another so that each subsequent number begins with that the same digit with which the previous number ends. Prove that you can do this and find the sum of the largest and smallest of all multi-digit numbers that can be obtained in this way. [url=https://artofproblemsolving.com/community/c6h3390996p32049528]8,6*[/url] (asterisk problems in separate posts) PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988084_1968_leningrad_math_olympiad]here[/url].

Aisling and Brendan take alternate moves in the following game. Before the game starts, the number $x = 2023$ is written on a piece of paper. Aisling makes the first move. A move from a positive integer $x$ consists of replacing $x$ either with $x + 1$ or with $x/p$ where $p$ is a prime factor of $x$. The winner is the first player to write $x = 1$. Determine whether Aisling or Brendan has a winning strategy for this game.

Let be four functions $ f,g,s,i:\mathbb{N}\longrightarrow\mathbb{N} $ such that $ s(x)=\max (f(x),g(x)) $ and $ i(x)=\min (f(x),g(x)) , $ for any natural number $ x. $ Prove that $ f=g $ if $ s $ is surjective and $ i $ injective.

In a plane, 2014 lines are distributed in 3 groups. in every group all the lines are parallel between themselves. What is the maximum number of triangles that can be formed, such that every side of such triangle lie on one of the lines?

$16$ children are sitting at a round table. After the break, they sit down again on table. They find that each child is either sitting on its original [lace or in one of the two neighboring places. How many seating arrangements are possible in this way after the break?

Let $n$ be a positive integer and $a_1, a_2, \dots , a_{2n}$ mutually distinct integers. Find all integers $x$ satisfying \[(x - a_1) \cdot (x - a_2) \cdots (x - a_{2n}) = (-1)^n(n!)^2.\]

Which of the following methods of proving a geometric figure a locus is not correct? $ \textbf{(A)}\ \text{Every point of the locus satisfies the conditions and every point not on the locus does not satisfy the conditions.}$ $ \textbf{(B)}\ \text{Every point not satisfying the conditions is not on the locus and every point on the locus does satisfy the conditions.}$ $ \textbf{(C)}\ \text{Every point satisfying the conditions is on the locus and every point on the locus satisfies the conditions.}$ $ \textbf{(D)}\ \text{Every point not on the locus does not satisfy the conditions and every point not satisfying} \\ \text{the conditions is not on the locus.}$ $ \textbf{(E)}\ \text{Every point satisfying the conditions is on the locus and every point not satisfying the conditions is not on the locus.}$

Let $ S$ be the set of $25$ points $(x, y)$ with $0\le x, y \le 4$. A triangle whose three vertices are in $S$ is chosen at random. What is the expected value of the square of its area?

Point $A$ is outside of a given circle $\omega$. Let the tangents from $A$ to $\omega$ meet $\omega$ at $S, T$ points $X, Y$ are midpoints of $AT, AS$ let the tangent from $X$ to $\omega$ meet $\omega$ at $R\neq T$. points $P, Q$ are midpoints of $XT, XR$ let $XY\cap PQ=K, SX\cap TK=L$ prove that quadrilateral $KRLQ$ is cyclic.

If $a$, $b$ and $c$ are sides of triangle which perimeter equals $1$, prove that: $a^2+b^2+c^2+4abc<\frac{1}{2}$

Let $a$ be any integer. Define the sequence $x_0,x_1,\ldots$ by $x_0=a$, $x_1=3$, and for all $n>1$ \[x_n=2x_{n-1}-4x_{n-2}+3.\] Determine the largest integer $k_a$ for which there exists a prime $p$ such that $p^{k_a}$ divides $x_{2011}-1$.

A student took a rectangular piece of paper with length equal to one meter and width equal to five centimeters. The student brought the ends together, turning one end 180 degrees and gluing the surfaces to create a figure called a Möbius strip. On one side of this strip, the student placed a flea and an ant. It is known that if the flea and the ant move in different directions on the Möbius strip, they will meet each other in 2 minutes. However, if they move in the same direction, they will meet in 7 minutes. Given that the flea is faster than the ant and both move at constant speeds, determine the speed of the flea. [i]Proposed by Lia Chitishvili, Georgia[/i]

Let $f$ be a function defined on the set of positive rational numbers with the property that $f(a\cdot b)=f(a)+f(b)$ for all positive rational numbers $a$ and $b$. Suppose that $f$ also has the property that $f(p)=p$ for every prime number $p$. For which of the following numbers $x$ is $f(x)<0?$ $\textbf{(A) } \frac{17}{32} \qquad \textbf{(B) } \frac{11}{16} \qquad \textbf{(C) } \frac{7}{9} \qquad \textbf{(D) } \frac{7}{6} \qquad \textbf{(E) } \frac{25}{11}$

Justify if continuity can be affirmed, denied or cannot be decided in the point$ x = 0$ of a real function $f(x)$ of real variable, in each of the three (independent) cases . a) It is known only that for all natural $n$: $f\left( \frac{1}{2n}\right)= 1$ and $f\left( \frac{1}{2n+1}\right)= -1$. b) It is known that for all nonnegative real $x$ is $f(x) = x^2$ and for negative real $x$ is $f(x) = 0$. c) It is only known that for all natural $n$ it is $f\left( \frac{1}{n}\right)= 1$.

$\sqrt[3]{x+\sqrt[3]{x+\sqrt[3]{x+ \sqrt[3]{x ...}}}}= 8$. Find $x$.

Let $a_0=0<a_1<a_2<...<a_n$ be real numbers . Supppose $p(t)$ is a real valued polynomial of degree $n$ such that $\int_{a_j}^{a_{j+1}} p(t)\,dt = 0\ \ \forall \ 0\le j\le n-1$ Show that , for $0\le j\le n-1$ , the polynomial $p(t)$ has exactly one root in the interval $ (a_j,a_{j+1})$

The real value of $x$ such that $64^{x-1}$ divided by $4^{x-1}$ equals $256^{2x}$ is: $\textbf{(A)}\ -\dfrac{2}{3} \qquad \textbf{(B)}\ -\dfrac{1}{3} \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ \dfrac{1}{4} \qquad \textbf{(E)}\ \dfrac{3}{8} $

Find the sum of $$\frac{\sigma(n) \cdot d(n)}{ \phi (n)}$$ over all positive $n$ that divide $ 60$. Note: The function $d(i)$ outputs the number of divisors of $i$, $\sigma (i)$ outputs the sum of the factors of $i$, and $\phi (i)$ outputs the number of positive integers less than or equal to $i$ that are relatively prime to $i$.

Given is a $n \times n$ chessboard. With the same probability, we put six pawns on its six cells. Let $p_n$ denotes the probability that there exists a row or a column containing at least two pawns. Find $\lim_{n \to \infty} np_n$.

Let $\alpha$ be an arbitrary angle and let $x = cos\alpha, y = cosn\alpha$ ($n \in Z$). i) Prove that to each value $x \in [-1, 1]$ corresponds one and only one value of $y$. Thus we can write $y$ as a function of $x, y = T_n(x)$. Compute $T_1(x), T_2(x)$ and prove that $T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x)$. From this it follows that $T_n(x)$ is a polynomial of degree $n$. ii) Prove that the polynomial $T_n(x$) has $n$ distinct roots in $[-1, 1]$.

Let \[f(x)=\frac{(x-18)(x-72)(x-98)(x-k)}{x}.\] There exist exactly three positive real values of $k$ such that $f$ has a minimum at exactly two real values of $x$. Find the sum of these three values of $k$.

Let $ABC$ be an acute triangle where $AC > BC$. Let $T$ denote the foot of the altitude from vertex $C$, denote the circumcentre of the triangle by $O$. Show that quadrilaterals $ATOC$ and $BTOC$ have equal area.

The tourist has come to the Moscow by train. All-day-long he wandered randomly through the streets. Than he had a supper in the cafe on the square and decided to return to the station only through the streets that he has passed an odd number of times. Prove that he is always able to do that.

$\omega$ is the circumcirle of an acute triangle $ABC$. The tangent line passing through $A$ intersects the tangent lines passing through points $B$ and $C$ at points $K$ and $L$, respectively. The line parallel to $AB$ through $K$ and the line parallel to $AC$ through $L$ intersect at point $P$. Prove that $BP=CP$. (Author: P. Kozhevnikov)

Mr. DoBa has a bag of markers. There are 2 blue, 3 red, 4 green, and 5 yellow markers. Mr. DoBa randomly takes out two markers from the bag. The probability that these two markers are different colors can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Raina Yang[/i]