A single section at a stadium can hold either $7$ adults or $11$ children. When $N$ sections are completely lled, an equal number of adults and children will be seated in them. What is the least possible value of $N$?

$(FRA 4)$ A right-angled triangle $OAB$ has its right angle at the point $B.$ An arbitrary circle with center on the line $OB$ is tangent to the line $OA.$ Let $AT$ be the tangent to the circle different from $OA$ ($T$ is the point of tangency). Prove that the median from $B$ of the triangle $OAB$ intersects $AT$ at a point $M$ such that $MB = MT.$

Each positive integer number $n \ ge 1$ is assigned the number $p_n$ which is the product of all its non-zero digits. For example, $p_6 = 6$, $p_ {32} = 6$, $p_ {203} = 6$. Let $S = p_1 + p_2 + p_3 + \dots + p_ {999}$. Find the largest prime that divides $S $.

For any positive integer $n$, let $\tau (n)$ denote the number of its positive divisors (including 1 and itself). Determine all positive integers $m$ for which there exists a positive integer $n$ such that $\frac{\tau (n^{2})}{\tau (n)}=m$.

What is the smallest possible average of four distinct positive even integers? $ \text{(A)}\ 3\qquad\text{(B)}\ 4\qquad\text{(C)}\ 5\qquad\text{(D)}\ 6\qquad\text{(E)}\ 7 $

[u]Round 5[/u] [b]p13.[/b] For a square pyramid whose base has side length $9$, a square is formed by connecting the centroids of the four triangular faces. What is the area of the square formed by the centroids? [b]p14.[/b] Farley picks a real number p uniformly at random in the range $\left( \frac13, \frac23 \right)$. She then creates a special coin that lands on heads with probability $p$ and tails with probability $1 - p$. She flips this coin, and it lands on heads. What is the probability that $p > \frac12$? [b]p15.[/b] Let $ABCD$ be a quadrilateral with $\angle A = \angle C = 90^o$. Extend $AB$ and $CD$ to meet at point $P$. Given that $P B = 3$, $BA = 21$, and $P C = 1$, find $BD^2$ [u]Round 6[/u] [b]p16.[/b] Three congruent, mutually tangent semicircles are inscribed in a larger semicircle, as shown in the diagram below. If the larger semicircle has a radius of $30$ units, what is the radius of one of the smaller semicircles? [img]https://cdn.artofproblemsolving.com/attachments/5/e/1b73791e95dc4ed6342f0151f3f63e1b31ae3c.png[/img] [b]p17.[/b] In isosceles trapezoid $ABCD$ with $BC \parallel AD$, the distances from $A$ and $B$ to line $CD$ are $3$ and $9$, respectively. If the distance between the two bases of trapezoid $ABCD$ is $5$, find the area of quadrilateral $ABCD$. [b]p18.[/b] How many ways are there to tile the “$E$” shape below with dominos? A domino covers two adjacent squares. [img]https://cdn.artofproblemsolving.com/attachments/b/b/82bdb8d8df8bc3d00b9aef9eb39e55358c4bc6.png[/img] [u]Round 7[/u] [b]p19.[/b] In isoceles triangle $ABC$, $AC = BC$ and $\angle ACB = 20^o$. Let $\Omega$ be the circumcircle of triangle $ABC$ with center $O$, and let $M$ be the midpoint of segment $BC$. Ray $\overrightarrow{OM}$ intersects $\Omega$ at $D$. Let $\omega$ be the circle with diameter $OD$. $AD$ intersects $\omega$ again at a point $X$ not equal to $D$. Given $OD = 2$, find the area of triangle $OXD$. [b]p20.[/b] Find the smallest odd prime factor of $2023^{2029} + 2026^{2029} - 1$. [b]p21.[/b] Achyuta, Alan, Andrew, Anish, and Ava are playing in the EMCC games. Each person starts with a paper with their name taped on their back. A person is eliminated from the game when anybody rips their paper off of their back. The game ends when one person remains. The remaining person then rips their paper off of their own back. At the end of the game, each person collects the papers that they ripped off. How many distinct ways can the papers be distributed at the end of the game? [u]Round 8[/u] [b]p22.[/b] Anthony has three random number generators, labelled $A$, $B$ and $C$. $\bullet$ Generator$ A$ returns a random number from the set $\{12, 24, 36, 48, 60\}$. $\bullet$ Generator $B$ returns a random number from the set $ \{15, 30, 45, 60\}$. $\bullet$ Generator $C$ returns a random number from the set $\{20, 40, 60\}$. He uses generator $A$, $B$, and then $C$ in succession, and then repeats this process indefinitely. Anthony keeps a running total of the sum of all previously generated numbers, writing down the new total every time he uses a generator. After he uses each machine $10 $ times, what is the average number of multiples of $60$ that Anthony will have written down? [b]p23.[/b] A laser is shot from one of the corners of a perfectly reflective room shaped like an equilateral triangle. The laser is reflected 2497 times without shining into a corner of the room, but after the 2497th reflection, it shines directly into the corner it started from. How many different angles could the laser have been initially pointed? [b]p24.[/b] We call a k-digit number blissful if the number of positive integers $n$ such that $n^n$ ends in that $k$-digit number happens to be nonzero and finite. What is the smallest value of $k$ such that there exists a blissful $k$-digit number? PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3131523p28369592]here[/url].. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

If the arithmetic mean of $a$ and $b$ is double their geometric mean, with $a>b>0$, then a possible value for the ratio $\frac{a}{b}$, to the nearest integer, is $\textbf{(A)}\ 5 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\ 14 \qquad\textbf{(E)}\ \text{none of these} $

A vintage tram departs a stop with a certain number of boys and girls on board. At the first stop, a third of the girls get out and their places are taken by boys. At the next stop, a third of the boys get out and their places are taken by girls. There are now two more girls than boys and as many boys as there originally were girls. How many boys and girls were there on board at the start?

For each integer $k\geq 2$, determine all infinite sequences of positive integers $a_1$, $a_2$, $\ldots$ for which there exists a polynomial $P$ of the form \[ P(x)=x^k+c_{k-1}x^{k-1}+\dots + c_1 x+c_0, \] where $c_0$, $c_1$, \dots, $c_{k-1}$ are non-negative integers, such that \[ P(a_n)=a_{n+1}a_{n+2}\cdots a_{n+k} \] for every integer $n\geq 1$.

A continuous function $f : [a,b] \to [a,b]$ is called piecewise monotone if $[a, b]$ can be subdivided into finitely many subintervals $$I_1 = [c_0,c_1], I_2 = [c_1,c_2], \dots , I_\ell = [ c_{\ell - 1},c_\ell ]$$ such that $f$ restricted to each interval $I_j$ is strictly monotone, either increasing or decreasing. Here we are assuming that $a = c_0 < c_1 < \cdots < c_{\ell - 1} < c_\ell = b$. We are also assuming that each $I_j$ is a maximal interval on which $f$ is strictly monotone. Such a maximal interval is called a lap of the function $f$, and the number $\ell = \ell (f)$ of distinct laps is called the lap number of $f$. If $f : [a,b] \to [a,b]$ is a continuous piecewise-monotone function, show that the sequence $( \sqrt[n]{\ell (f^n )})$ converges; here $f^n$ means $f$ composed with itself $n$-times, so $f^2 (x) = f(f(x))$ etc.

How many nonempty subsets of ${1, 2, \dots, 15}$ are there such that the sum of the squares of each subset is a multiple of $5$?

Show that the equation $x^2-7y^2 = 1$ has infinitely many solutions in natural numbers.

Given $A = \left\{1,2,3,5,8,13,21,34,55\right\}$, how many of the numbers between $3$ and $89$ cannot be written as the sum of two elements of set $A$? $ \mathrm{(A) \ } 34 \qquad \mathrm{(B) \ } 35 \qquad \mathrm {(C) \ } 43\qquad \mathrm{(D) \ } 51 \qquad \mathrm{(E) \ } 55$

Let $p(x)$ be a polynomial with integer coefficients, and let $a, b$ and $c$ be three distinct integers. Show that it is not possible to have $p(a) = b$, $p(b) = c$, and $p(c) = a$.

Let $r$ be a real number such that $\sqrt[3]{r} - \frac{1}{\sqrt[3]{r}}=2$. Find $r^3 - \frac{1}{r^3}$.

Find the area of the equilateral triangle that includes vertices at $\left(-3,5\right)$ and $\left(-5,9\right)$. $\text{(A) }3\sqrt{3}\qquad\text{(B) }10\sqrt{3}\qquad\text{(C) }\sqrt{30}\qquad\text{(D) }2\sqrt{15}\qquad\text{(E) }5\sqrt{3}$

Let $x$ be the least real number greater than $1$ such that $\sin(x)$ = $\sin(x^2)$, where the arguments are in degrees. What is $x$ rounded up to the closest integer? $\textbf{(A) } 10 \qquad \textbf{(B) } 13 \qquad \textbf{(C) } 14 \qquad \textbf{(D) } 19 \qquad \textbf{(E) } 20$

It is said that a positive integer $n > 1$ has the property ($p$) if in its prime factorization $n = p_1^{a_1} \cdot ... \cdot p_j^{a_j}$ at least one of the prime factors $p_1, ... , p_j$ has the exponent equal to $2$. a) Find the largest number $k$ for which there exist $k$ consecutive positive integers that do not have the property ($p$). b) Prove that there is an infinite number of positive integers $n$ such that $n, n + 1$ and $n + 2$ have the property ($p$).

A beetle crawls along each of two intersecting straight lines at constant speeds, without changing direction. It is known that projections of the beetles on the $OX$ axis never coincide (neither in the past nor in the future). Prove that the projections of the beetles on the $OY$ axis will necessarily coincide or have coincided before. [hide=oroginal wording] По каждой из двух пересекающихся прямых с постоянными скоростями, не меняя направления, ползет по жуку. Известно, что проекции жуков на ось OX никогда не совпадают (ни в прошлом, ни в будущем). Докажите, что проекции жуков на ось OY обязательно совпадут или совпадали раньше.[/hide]

A finite sequence of natural numbers $a_1, a_2, \dots, a_n$ is given. A sub-sequence $a_{k+1}, a_{k+2}, \dots, a_l$ will be called a [i]repetition[/i] if there exists a natural number $p\leq \frac{l-k}2$ such that $a_i=a_{i+p}$ for $k+1\leq i\leq l-p$, but $a_i\neq a_{i+p}$ for $i=k$ (if $k>0$) and $i=l-p+1$ (if $l<n$). Show that the sequence contains less than $n$ repetitions.

Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths 3 and 4 units. In the corner where those sides meet at a right angle, he leaves a small unplanted square $S$ so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from $S$ to the hypotenuse is 2 units. What fraction of the field is planted? [asy] draw((0,0)--(4,0)--(0,3)--(0,0)); draw((0,0)--(0.3,0)--(0.3,0.3)--(0,0.3)--(0,0)); fill(origin--(0.3,0)--(0.3,0.3)--(0,0.3)--cycle, gray); label("$4$", (2,0), N); label("$3$", (0,1.5), E); label("$2$", (.8,1), E); label("$S$", (0,0), NE); draw((0.3,0.3)--(1.4,1.9), dashed); [/asy] $\textbf{(A) } \frac{25}{27} \qquad \textbf{(B) } \frac{26}{27} \qquad \textbf{(C) } \frac{73}{75} \qquad \textbf{(D) } \frac{145}{147} \qquad \textbf{(E) } \frac{74}{75} $

Consider an acute-angled triangle $ABC$ with $AB<AC$ and let $\omega$ be its circumscribed circle. Let $t_B$ and $t_C$ be the tangents to the circle $\omega$ at points $B$ and $C$, respectively, and let $L$ be their intersection. The straight line passing through the point $B$ and parallel to $AC$ intersects $t_C$ in point $D$. The straight line passing through the point $C$ and parallel to $AB$ intersects $t_B$ in point $E$. The circumcircle of the triangle $BDC$ intersects $AC$ in $T$, where $T$ is located between $A$ and $C$. The circumcircle of the triangle $BEC$ intersects the line $AB$ (or its extension) in $S$, where $B$ is located between $S$ and $A$. Prove that $ST$, $AL$, and $BC$ are concurrent. $\text{Vangelis Psychas and Silouanos Brazitikos}$

Find all positive integer solution: $$k^m+m^n=k^n+1$$ [I]Proposed by V. Frank, F. Petrov[/i]

Consider all polynomials of a complex variable, $P(z)=4z^4+az^3+bz^2+cz+d$, where $a, b, c$ and $d$ are integers, $0 \le d \le c \le b \le a \le 4$, and the polynomial has a zero $z_0$ with $|z_0|=1$. What is the sum of all values $P(1)$ over all the polynomials with these properties? $ \textbf{(A)}\ 84\qquad\textbf{(B)}\ 92\qquad\textbf{(C)}\ 100\qquad\textbf{(D)}\ 108 \qquad\textbf{(E)}\ 120 $

Let $M$ be the set of five points in space, none of which four do not lie in a plane. Let $R$ be a set of seven planes with properties: a) Each plane from the set $R$ contains at least one point of the set$ M$. b) None of the points of the set M lie in the five planes of the set $R$. Prove that there are also two distinct points $P$, $Q$, $ P \in M$, $Q \in M$, that the line $PQ$ is not the intersection of any two planes from the set $R$.