Let $ABC$ be an equilateral triangle. The points $M$ and $K$ are located in different half-planes with respect to line $BC$, so that the point $M \in (AB)$ ¸and the triangle $MKC$ is equilateral. Prove that the lines $AC$ and $BK$ are parallel.

A total of $n$ people compete in a mathematical match which contains $15$ problems where $n>12$. For each problem, $1$ point is given for a right answer and $0$ is given for a wrong answer. Analysing each possible situation, we find that if the sum of points every group of $12$ people get is no less than $36$, then there are at least $3$ people that got the right answer of a certain problem, among the $n$ people. Find the least possible $n$.

Let $S$ be the set of natural numbers whose digits are different and belong to the set $\{1, 3, 5, 7\}$. Calculate the sum of the elements of $S$.

Let $m$ be a positive integer and $a$ a positive real number. Find the greatest value of $a^{m_1}+a^{m_2}+\ldots+a^{m_p}$ where $m_1+m_2+\ldots+m_p=m, m_i\in\mathbb{N},i=1,2,\ldots,p;$ $1\leq p\leq m, p\in\mathbb{N}$.

Let $a_0, a_1, \ldots, a_n, a_{n+1}$ be a sequence of real numbers satisfying the following conditions: \[a_0 = a_{n+1 }= 0,\]\[ |a_{k-1} - 2a_k + a_{k+1}| \leq 1 \quad (k = 1, 2,\ldots , n).\] Prove that $|a_k| \leq \frac{k(n+1-k)}{2} \quad (k = 0, 1,\ldots ,n + 1).$

There are arbitrary 7 points in the plane. Circles are drawn through every 4 possible concyclic points. Find the maximum number of circles that can be drawn.

Clarabelle wants to travel from $(0,0)$ to $(6,2)$ in the coordinate plane. She is able to move in one-unit steps up, down, or right, must stay between $y=0$ and $y=2$ (inclusive), and is not allowed to visit the same point twice. How many paths can she take? [i]Proposed by Connor Gordon[/i]

The shape of a dog kennel from above is an equilateral triangle with side length $1$ m and its corners in points $A, B$ and $C$, as shown in the picture. The chain of the dog is of length $6$ m and its end is fixed to the corner in point $A$. The dog himself is in point $K$ in a way that the chain is tight and points $K, A$ and $B$ are on the same straight line. The dog starts to move clockwise around the kennel, holding the chain tight all the time. How long is the walk of the dog until the moment when the chain is tied round the kennel at full? [img]https://cdn.artofproblemsolving.com/attachments/9/5/616f8adfe66e2eb60f1a6c3f26e652c45f3e27.png[/img]

Let $ f$ be a function defined on the set of non-negative integers and taking values in the same set. Given that (a) $ \displaystyle x \minus{} f(x) \equal{} 19\left[\frac{x}{19}\right] \minus{} 90\left[\frac{f(x)}{90}\right]$ for all non-negative integers $ x$; (b) $ 1900 < f(1990) < 2000$, find the possible values that $ f(1990)$ can take. (Notation : here $ [z]$ refers to largest integer that is $ \leq z$, e.g. $ [3.1415] \equal{} 3$).

Prove the following assertion: If $c_1,c_2,\ldots ,c_n\ (n\ge 2)$ are real numbers such that \[ (n-1)(c_1^2+c_2^2+\cdots +c_n^2)=(c_1+c_2+\cdots + c_n)^2,\] then either all these numbers are nonnegative or all these numbers are nonpositive.

Sunn and Tacks play a game alternately choosing a word among the following (German) words: ”bad”, ”binse”, ”kafig”, ”kosewort”, ”maitag”, ”name”, ”pol”, ”parade”, ”wolf”. Two words are said to compatible if they have exactly one consonant in common. In the first round, Sunn selects a word for herself and one for Tacks. In every consequent round, each player selects a word that is compatible with the one they chose in the previous round. Tacks wins the game if the two players successively select the same word. (a) Prove that Tacks can always win. How many rounds are necessary for that? (b) Upon Sunn’s desire, the word ”kafig” was replaced with the word ”feige”. Prove that Sunn can prevent Tacks from winning.

The picture shows $10$ equal regular pentagons where each two neighbouring pentagons have a common side. The smaller circle is tangent to one side of each pentagon and the larger circle passes through the opposite vertices of these sides. Find the area of the larger circle if the area of the smaller circle is $1$. [img]https://cdn.artofproblemsolving.com/attachments/0/6/84fe98370868a5cf28d92d4b207ccb00e6eaa3.png[/img]

Let numbers $1,2,3,...,2010$ stand in a row at random. Consider row, obtain by next rule: For any number we sum it and it's number in a row (For example for row $( 2,7,4)$ we consider a row $(2+1;7+2;4+3)=(3;9;7)$ ); Proved, that in resulting row we can found two equals numbers, or two numbers, which is differ by $2010$

A graph $G$ is called $\textit{divisibility graph}$ if the vertices can be assigned distinct positive integers such that between two vertices assigned $u, v$ there is an edge iff $\frac{u} {v}$ or $\frac{v} {u}$ is a positive integer. Show that for any positive integer $n$ and $0 \leq e \leq \frac{n(n-1)}{2}$, there is a $\textit{divisibility graph}$ with $n$ vertices and $e$ edges. [hide=Remark on source of 10.3] It appears to be Kvant 2022 Issue 10 M2719, so it will not be posted; the same problem was also used as 9.4.

How many 7-element subsets of $\{1, 2, 3,\ldots , 14\}$ are there, the sum of whose elements is divisible by $14$?

Triangle $ABC$ has incircle $\omega$ and $A$-excircle $\omega_A.$ Circle $\gamma_B$ passes through $B$ and is externally tangent to $\omega$ and $\omega_A.$ Circle $\gamma_C$ passes through $C$ and is externally tangent to $\omega$ and $\omega_A.$ If $\gamma_B$ intersects line $BC$ again at $D,$ and $\gamma_C$ intersects line $BC$ again at $E,$ prove that $BD=EC.$

Elisa adds the digits of her year of birth and observes that the result coincides with the last two digits of the year her grandfather was born. Furthermore, the last two digits of the year she was born are precisely the current age of her grandfather. Find the year Elisa was born and the year her grandfather was born.

Let $n$ be a positive integer and let $d_{1},d_{2},,\ldots ,d_{k}$ be its divisors, such that $1=d_{1}<d_{2}<\ldots <d_{k}=n$. Find all values of $n$ for which $k\geq 4$ and $n=d_{1}^{2}+d_{2}^{2}+d_{3}^{2}+d_{4}^{2}$.

Let $\omega$ be the circumcircle of triangle $ABC$, $O$ be its center, $A'$ be the point of $\omega$ opposite to $A$, and $D$ be a point on a minor arc $BC$ of $\omega$. A point $D'$ is the reflection of $D$ about $BC$. The line $A'D'$ meets for the second time at point $E$. The perpendicular bisector to $D'E$ meets $AB$ and $AC$ at points $F$ and $G$ respectively. Prove that $\angle FOG = 180^\circ - 2\angle BAC$.

Let be given a triangle $ABC$ and points $D,M,N$ belong to $BC,AB,AC$, respectively. Suppose that $MD$ is parallel to $AC$ and $ND$ is parallel to $AB$. If $S_{\vartriangle BMD} = 9$ cm $^2, S_{\vartriangle DNC} = 25$ cm$^2$, compute $S_{\vartriangle AMN}$?

1) Cells of $8 \times 8$ table contain pairwise distinct positive integers. Each integer is prime or a product of two primes. It is known that for any integer $a$ from the table there exists integer written in the same row or in the same column such that it is not relatively prime with $a$. Find maximum possible number of prime integers in the table. 2) Cells of $2n \times 2n$ table, $n \ge 2,$ contain pairwise distinct positive integers. Each integer is prime or a product of two primes. It is known that for any integer $a$ from the table there exist integers written in the same row and in the same column such that they are not relatively prime with $a$. Find maximum possible number of prime integers in the table.

Determine the polynomial $P\in \mathbb{R}[x]$ for which there exists $n\in \mathbb{Z}_{>0}$ such that for all $x\in \mathbb{Q}$ we have: \[P\left(x+\frac1n\right)+P\left(x-\frac1n\right)=2P(x).\] [i]Dumitru Bușneag[/i]

Country $A$ uses a currency known as the shell. The nation uses only two coins, each worth a whole number of shells. The largest amount of shell not obtainable using a combination of these two coins is $215$. Find the number of possible pairs of values these two coins could have. (a value of $15$ and $4$ is the same as having a $4$ and $15$) $\textbf{(A) } 6\qquad\textbf{(B) } 7\qquad\textbf{(C) } 8\qquad\textbf{(D) } 9\qquad\textbf{(E) } 10$

Given that $\sum_{k=1}^{35}\sin 5k=\tan \frac mn,$ where angles are measured in degrees, and $m$ and $n$ are relatively prime positive integers that satisfy $\frac mn<90,$ find $m+n.$

Let $ABCD$ be a cyclic quadrilateral such that the triangles $BCD$ and $CDA$ are not equilateral. Prove that if the Simson line of $A$ with respect to $\triangle BCD$ is perpendicular to the Euler line of $BCD$, then the Simson line of $B$ with respect to $\triangle ACD$ is perpendicular to the Euler line of $\triangle ACD$.