Evaluate the limit: \[ \lim_{n \to \infty}{n \cdot \sin{1} \cdot \sin{2} \cdot \dots \cdot \sin{n}}.\] Proposed to "Unirea" Intercounty contest, grade 11, Romania

Prove there exist infinitely many pairs $(x,y)$ of integers $1<x<y$, such that $x^3+y \mid x+y^3$. ([i]Dan Schwarz[/i])

Given a connected graph with $n$ edges, where there are no parallel edges. For any two cycles $C,C'$ in the graph, define its [i]outer cycle[/i] to be \[C*C'=\{x|x\in (C-C')\cup (C'-C)\}.\] (1) Let $r$ be the largest postive integer so that we can choose $r$ cycles $C_1,C_2,\ldots,C_r$ and for all $1\leq k\leq r$ and $1\leq i$, $j_1,j_2,\ldots,j_k\leq r$, we have \[C_i\neq C_{j_1}*C_{j_2}*\cdots*C_{j_k}.\] (Remark: There should have been an extra condition that either $j_1\neq i$ or $k\neq 1$) (2) Let $s$ be the largest positive integer so that we can choose $s$ edges that do not form a cycle. (Remark: A more precise way of saying this is that any nonempty subset of these $s$ edges does not form a cycle) Show that $r+s=n$. Note: A cycle is a set of edges of the form $\{A_iA_{i+1},1\leq i\leq n\}$ where $n\geq 3$, $A_1,A_2,\ldots,A_n$ are distinct vertices, and $A_{n+1}=A_1$.

Anna and Bob are playing a game on a rectangular board with $i$ rows and $j$ columns. Anna and Bob alternate turns with Anna going first. On each turn, a player places a penny in a square and then all squares in the same row and column of that square are marked. A player cannot place a penny in any marked square. When a player cannot place a penny in any square, they lose and the other player wins. How many ordered pairs of integers $(i, j)$ with $1 \le i \le 2020, 1 \le j \le 2020$ are there such that Anna wins?

Give are the integers $a_{1}=11 , a_{2}=1111, a_{3}=111111, ... , a_{n}= 1111...111$( with $2n$ digits) with $n > 8$ . Let $q_{i}= \frac{a_{i}}{11} , i= 1,2,3, ... , n$ the remainder of the division of $a_{i}$ by$ 11$ . Prove that the sum of nine consecutive quotients: $s_{i}=q_{i}+q_{i+1}+q_{i+2}+ ... +q_{i+8}$ is a multiple of $9$ for any $i= 1,2,3, ... , (n-8)$

Reduced to lowest terms, $ \frac {a^2\minus{}b^2}{ab}\minus{} \frac {ab\minus{}b^2}{ab\minus{}a^2}$ is equal to: $\textbf{(A)}\ \dfrac{a}{b} \qquad \textbf{(B)}\ \dfrac{a^2-2b^2}{ab} \qquad \textbf{(C)}\ a^2 \qquad \textbf{(D)}\ a-2b \qquad \textbf{(E)}\ \text{None of these}$

Given any set $S$ of positive integers, show that at least one of the following two assertions holds: (1) There exist distinct finite subsets $F$ and $G$ of $S$ such that $\sum_{x\in F}1/x=\sum_{x\in G}1/x$; (2) There exists a positive rational number $r<1$ such that $\sum_{x\in F}1/x\neq r$ for all finite subsets $F$ of $S$.

Let $$a_n=\frac{n^2+1}{\sqrt{n^4+4}}, \quad n=1,2,3,\dots$$ and let $b_n$ be the product of $a_1,a_2,a_3,\dots ,a_n$. Prove that $$\frac{b_n}{\sqrt{2}}=\frac{\sqrt{n^2+1}}{\sqrt{n^2+2n+2}},$$ and deduce that $$\frac{1}{n^3+1}<\frac{b_n}{\sqrt{2}}-\frac{n}{n+1}<\frac{1}{n^3}$$ for all positive integers $n$.

Ravon, Oscar, Aditi, Tyrone, and Kim play a card game. Each person is given $2$ cards out of a set of $10$ cards numbered $1,2,3, \dots,10.$ The score of a player is the sum of the numbers of their cards. The scores of the players are as follows: Ravon--$11,$ Oscar--$4,$ Aditi--$7,$ Tyrone--$16,$ Kim--$17.$ Which of the following statements is true? $\textbf{(A) }\text{Ravon was given card 3.}$ $\textbf{(B) }\text{Aditi was given card 3.}$ $\textbf{(C) }\text{Ravon was given card 4.}$ $\textbf{(D) }\text{Aditi was given card 4.}$ $\textbf{(E) }\text{Tyrone was given card 7.}$

Find all positive integers $n$ such that $\varphi(n)$ divides $n^2 + 3$.

A non isosceles triangle $ABC$ is given, in which $\angle A = 120^o$. Let $AL$ be its angle bisector, $AK$ be it's median, drawn from vertex $A$, point $O$ be the center of the circumcircle of this triangle, $F$ be the point of intersection of the lines $OL$ and $AK$. Prove that $\angle BFC = 60^o$.

$(a)$ Prove that for $a, b, c, d \in\mathbb{R}, m \in [1,+\infty)$ with $am + b =-cm + d = m$, \[(i)\sqrt{a^2 + b^2}+\sqrt{c^2 + d^2}+\sqrt{(a-c)^2 + (b-d)^2}\ge \frac{4m^2}{1+m^2},\text{ and}\] \[(ii) 2 \le \frac{4m^2}{1+m^2} < 4.\] $(b)$ Express $a, b, c, d$ as functions of $m$ so that there is equality in $(i).$

$ M$ is a subset of $ \{1, 2, 3, \ldots, 15\}$ such that the product of any three distinct elements of $ M$ is not a square. Determine the maximum number of elements in $ M.$

Two right triangles are placed next to each other to form a quadrilateral as shown. What is the perimeter of the quadrilateral? [asy] size(4cm); fill((-5,0)--(0,12)--(0,6)--(8,0)--cycle, gray+opacity(0.3)); draw((0,0)--(0,12)--(-5,0)--cycle); draw((0,0)--(8,0)--(0,6)); label("5", (-2.5,0), S); label("13", (-2.5,6), dir(140)); label("6", (0,3), E); label("8", (4,0), S); [/asy] [i]Proposed by Nathan Xiong[/i]

$A,B,C,D$ are points on a circle in that order. Prove that $|AB-CD|+|AD-BC| \ge 2|AC-BD|$.

$a)$ Prove that $2x^3-3x^2+1\geq 0,~(\forall)x\geq0.$ $b)$ Let $x,y,z\geq 0$ such that $\frac{2}{1+x^3}+\frac{2}{1+y^3}+\frac{2}{1+z^3}=3.$ Prove that $\frac{1-x}{1-x+x^2}+\frac{1-y}{1-y+y^2}+\frac{1-z}{1-z+z^2}\geq 0.$

Consider the following $3\times 2$ array formed by using the numbers $1,2,3,4,5,6$, $$\begin{pmatrix} a_{11}& a_{12}\\a_{21}& a_{22}\\ a_{31}& a_{32}\end{pmatrix}=\begin{pmatrix}1& 6\\2& 5\\ 3& 4\end{pmatrix}.$$ Observe that all row sums are equal, but the sum of the square of the squares is not the same for each row. Extend the above array to a $3\times k$ array $(a_{ij})_{3\times k}$ for a suitable $k$, adding more columns, using the numbers $7,8,9,\dots ,3k$ such that $$\sum_{j=1}^k a_{1j}=\sum_{j=1}^k a_{2j}=\sum_{j=1}^k a_{3j}~~\text{and}~~\sum_{j=1}^k (a_{1j})^2=\sum_{j=1}^k (a_{2j})^2=\sum_{j=1}^k (a_{3j})^2$$

Let $ M^n \subset \mathbb{R}^{n\plus{}1}$ be a complete, connected hypersurface embedded into the Euclidean space. Show that $ M^n$ as a Riemannian manifold decomposes to a nontrivial global metric direct product if and only if it is a real cylinder, that is, $ M^n$ can be decomposed to a direct product of the form $ M^n\equal{}M^k \times \mathbb{R}^{n\minus{}k} \;(k<n)$ as well, where $ M^k$ is a hypersurface in some $ (k\plus{}1)$-dimensional subspace $ E^{k\plus{}1} \subset \mathbb{R}^{n\plus{}1} , \mathbb{R}^{n\minus{}k}$ is the orthogonal complement of $ E^{k\plus{}1}$. [i]Z. Szabo[/i]

Triangle $ABC$ is inscribed in circle $\omega$. A circle with chord $BC$ intersects segments $AB$ and $AC$ again at $S$ and $R$, respectively. Segments $BR$ and $CS$ meet at $L$, and rays $LR$ and $LS$ intersect $\omega$ at $D$ and $E$, respectively. The internal angle bisector of $\angle BDE$ meets line $ER$ at $K$. Prove that if $BE = BR$, then $\angle ELK = \tfrac{1}{2} \angle BCD$. [i]Proposed by Evan Chen[/i]

Find all functions $f : R \to R$ defined for real numbers, take real values and for all real $x$ and $y$ the equality holds: $$f(2^x+2y) =2^yf(f(x))f(y).$$

Point $D$ is chosen on the side $AC$ of an acute-angled triangle $ABC$. The median $AM$ intersects the altitude $CH$ and the segment $BD$ at points $N$ and $K$ respectively. Prove that if $AK = BK$, then $AN = 2KM$.

Let $x_0,\dots,x_{2017}$ are positive integers and $x_{2017}\geq\dots\geq x_0=1$ such that $A=\{x_1,\dots,x_{2017}\}$ consists of exactly $25$ different numbers. Prove that $\sum_{i=2}^{2017}(x_i-x_{i-2})x_i\geq 623$, and find the number of sequences that holds the case of equality.

Find all functions $f : R \to R$ such that $f(x^2)-f(y^2) \le (f(x)+y) (x-f(y))$ for all $x, y \in R$.

What is the value of the expression $\sqrt{16\sqrt{8\sqrt{4}}}$? $\textbf{(A) }4\qquad\textbf{(B) }4\sqrt{2}\qquad\textbf{(C) }8\qquad\textbf{(D) }8\sqrt{2}\qquad\textbf{(E) }16$

Let $ a, b, c $ be positive real numbers less than or equal to $ \sqrt{2} $ such that $ abc = 2 $, prove that $$ \sqrt{2}\displaystyle\sum_{cyc}\frac{ab + 3c}{3ab + c} \ge a + b + c $$