For a triangle $\triangle ABC$ with an obtuse angle $\angle A$ , let $E , F$ be feet of altitudes from $B , C$ on sides $AC , AB$ respectively. The tangents from $B , C$ to circumcircle of triangle $\triangle ABC$ intersect line $EF$ at points $K , L$ respectively and we know that $\angle CLB=135$. Point $R$ lies on segment $BK$ in such a way that $KR=KL$ and let $S$ be a point on line $BK$ such that $K$ is between $B , S$ and $\angle BLS=135$. Prove that the circle with diameter $RS$ is tangent to circumcircle of triangle $\triangle ABC$. [i]Proposed by Mehran Talaei[/i]

For any odd prime $p$ and any integer $n,$ let $d_p (n) \in \{ 0,1, \dots, p-1 \}$ denote the remainder when $n$ is divided by $p.$ We say that $(a_0, a_1, a_2, \dots)$ is a [i]p-sequence[/i], if $a_0$ is a positive integer coprime to $p,$ and $a_{n+1} =a_n + d_p (a_n)$ for $n \geqslant 0.$ (a) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_n >b_n$ for infinitely many $n,$ and $b_n > a_n$ for infinitely many $n?$ (b) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_0 <b_0,$ but $a_n >b_n$ for all $n \geqslant 1?$ [I]United Kingdom[/i]

Let $ n \geq 3$ be a positive integer and let $ m \geq 2^{n\minus{}1}\plus{}1$. Prove that for each family of nonzero distinct subsets $ (A_j)_{j \in \overline{1, m}}$ of $ \{1, 2, ..., n\}$ there exist $ i$, $ j$, $ k$ such that $ A_i \cup A_j \equal{} A_k$.

2 Points $ P\left(a,\ \frac{1}{a}\right),\ Q\left(2a,\ \frac{1}{2a}\right)\ (a > 0)$ are on the curve $ C: y \equal{}\frac{1}{x}$. Let $ l,\ m$ be the tangent lines at $ P,\ Q$ respectively. Find the area of the figure surrounded by $ l,\ m$ and $ C$.

The $1 \times 1$ cells located around the perimeter of a $4 \times 4$ square are filled with the numbers $1, 2, \ldots, 12$ so that the sums along each of the four sides are equal. In the upper left corner cell is the number $1$, in the upper right - the number $5$, and in the lower right - the number $11$. [img]https://i.ibb.co/PM0ry1D/Kyiv-City-MO-2021-Round-1-10-2.png[/img] Under these conditions, what number can be located in the last corner cell? [i]Proposed by Mariia Rozhkova[/i]

Prove that the points symmetric to the vertices of a triangle with respect to the opposite side are collinear if and only if the distance from the orthocenter to the circumcenter is twice the circumradius.

Let $D_n$ be the set of divisors of $n$. Find all natural $n$ such that it is possible to split $D_n$ into two disjoint sets $A$ and $G$, both containing at least three elements each, such that the elements in $A$ form an arithmetic progression while the elements in $G$ form a geometric progression.

A point $P$ is chosen uniformly at random in the interior of triangle $ABC$ with side lengths $AB = 5$, $BC = 12$, $CA = 13$. The probability that a circle with radius $\frac13$ centered at $P$ does not intersect the perimeter of $ABC$ can be written as $\frac{m}{n}$ where $m, n$ are relatively prime positive integers. Find $m + n$.

Two circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. Line $\ell$ is tangent to $\omega_1$ at $P$ and to $\omega_2$ at $Q$ so that $A$ is closer to $\ell$ than $B$. Let $X$ and $Y$ be points on major arcs $\overarc{PA}$ (on $\omega_1$) and $AQ$ (on $\omega_2$), respectively, such that $AX/PX = AY/QY = c$. Extend segments $PA$ and $QA$ through $A$ to $R$ and $S$, respectively, such that $AR = AS = c\cdot PQ$. Given that the circumcenter of triangle $ARS$ lies on line $XY$, prove that $\angle XPA = \angle AQY$.

Prove that among any $n$ positive integers one can always find some (at least one) whose sum is divisible by $n$.

Let $ n$ be a positive integer. Determine the number of triangles (non congruent) with integral side lengths and the longest side length is $ n$.

The positive integers \( a \) and \( b \) are coprime and such that there exist positive integers \( m_2 \) and \( m_5 \) for which \( am_2 + b \) is a perfect square of a positive integer, and \( am_5 + b \) is a perfect fifth power of a positive integer. Does there always exist a positive integer \( n \) for which \( an + b \) is a perfect \( k \)-th power of a positive integer, if: a) \( k = 7 \); b) \( k = 10 \)?

A straight line tangent to a circle circumscribed about an isosceles triangle $ABC$ ($AB = AC$) at point $B$ intersects straight line $AC$ at point $P$, $E$ is the midpoint of $AB$ (fig.). What is the projection of $DE$ onto $AB$ if $PA = a$? [img]https://cdn.artofproblemsolving.com/attachments/e/3/59c67e8f5eb3d399656d86613bc699c8baf1c1.png[/img]

Let $n$ be a positive integer. (1) For a positive integer $k$ such that $1\leq k\leq n$, Show that : \[\int_{\frac{k-1}{2n}\pi}^{\frac{k}{2n}\pi} \sin 2nt\cos t\ dt=(-1)^{k+1}\frac{2n}{4n^2-1}(\cos \frac{k}{2n}\pi +\cos \frac{k-1}{2n}\pi).\] (2) Find the area $S_n$ of the part expressed by a parameterized curve $C_n: x=\sin t,\ y=\sin 2nt\ (0\leq t\leq \pi).$ If necessary, you may use ${\sum_{k=1}^{n-1} \cos \frac{k}{2n}\pi =\frac 12(\frac{1}{\tan \frac{\pi}{4n}}-1})\ (n\geq 2).$ (3) Find $\lim_{n\to\infty} S_n.$

Problems $15, 16$, and $17$ all refer to the following: In the very center of the Irenic Sea lie the beautiful Nisos Isles. In $1998$ the number of people on these islands is only 200, but the population triples every $25$ years. Queen Irene has decreed that there must be at least $1.5$ square miles for every person living in the Isles. The total area of the Nisos Isles is $24,900$ square miles. 15. Estimate the population of Nisos in the year $2050$. $ \text{(A)}\ 600\qquad\text{(B)}\ 800\qquad\text{(C)}\ 1000\qquad\text{(D)}\ 2000\qquad\text{(E)}\ 3000 $

Let $I,H,O$ be the incenter, centroid, and circumcenter of the nonisosceles triangle $ABC$. Prove that $AI \parallel HO$ if and only if $\angle BAC =120^{\circ}$.

Let $a_1=1$ and $a_n=n(a_{n-1}+1)$ for all $n\ge 2$ . Define : $P_n=\left(1+\frac{1}{a_1}\right)...\left(1+\frac{1}{a_n}\right)$ Compute $\lim_{n\to \infty} P_n$

Suppose $w,x,y,z$ satisfy \begin{align*}w+x+y+z&=25,\\wx+wy+wz+xy+xz+yz&=2y+2z+193\end{align*} The largest possible value of $w$ can be expressed in lowest terms as $w_1/w_2$ for some integers $w_1,w_2>0$. Find $w_1+w_2$.

$(GBR 1)$ The polynomial $P(x) = a_0x^k + a_1x^{k-1} + \cdots + a_k$, where $a_0,\cdots, a_k$ are integers, is said to be divisible by an integer $m$ if $P(x)$ is a multiple of $m$ for every integral value of $x$. Show that if $P(x)$ is divisible by $m$, then $a_0 \cdot k!$ is a multiple of $m$. Also prove that if $a, k,m$ are positive integers such that $ak!$ is a multiple of $m$, then a polynomial $P(x)$ with leading term $ax^k$can be found that is divisible by $m.$

Determine the value of $ \lim_{n\rightarrow\infty}\sum_{k \equal{} 0}^n\binom{n}{k}^{ \minus{} 1}$.

Find all continuous functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f(x+y)=f(x+f(y))$ for all $x,y\in\mathbb{R}$.

For how many primes $p$ less than $15$, there exists integer triples $(m,n,k)$ such that \[ \begin{array}{rcl} m+n+k &\equiv& 0 \pmod p \\ mn+mk+nk &\equiv& 1 \pmod p \\ mnk &\equiv& 2 \pmod p. \end{array} \] $ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ 6 $

Let $p(x) = x^7 +x^6 + b_5 x^5 + \cdots +b_0 $ and $q(x) = x^5 + c_4 x^4 + \cdots +c_0$ . If $p(i)=q(i)$ for $i=1,2,3,\cdots,6$ . Show that there exists a negative integer r such that $p(r)=q(r)$ .

Let $n>3$ be an integer. If $x_1<x_2<\ldots<x_{n+2}$ are reals with $x_1=0$, $x_2=1$ and $x_3>2$, what is the maximal value of $$(\frac{x_{n+1}+x_{n+2}-1}{x_{n+1}(x_{n+2}-1)})\cdot (\sum_{i=1}^{n}\frac{(x_{i+2}-x_{i+1})(x_{i+1}-x_i)}{x_{i+2}-x_i})?$$

A trapezium $ ABCD$, in which $ AB$ is parallel to $ CD$, is inscribed in a circle with centre $ O$. Suppose the diagonals $ AC$ and $ BD$ of the trapezium intersect at $ M$, and $ OM \equal{} 2$. [b](a)[/b] If $ \angle AMB$ is $ 60^\circ ,$ find, with proof, the difference between the lengths of the parallel sides. [b](b)[/b] If $ \angle AMD$ is $ 60^\circ ,$ find, with proof, the difference between the lengths of the parallel sides. [b][Weightage 17/100][/b]