$ABCD$ is a parallelogram, line $DF$ is drawn bisecting $BC$ at $E$ and meeting $AB$ (extended) at $F$ from vertex $C$. Line $CH$ is drawn bisecting side $AD$ at $G$ and meeting $AB$ (extended) at $H$. Lines $DF$ and $CH$ intersect at $I$. If the area of parallelogram $ABCD$ is $x$, find the area of triangle $HFI$ in terms of $x$.

Let $S$ be an infinite set of positive integers, such that there exist four pairwise distinct $a,b,c,d \in S$ with $\gcd(a,b) \neq \gcd(c,d)$. Prove that there exist three pairwise distinct $x,y,z \in S$ such that $\gcd(x,y)=\gcd(y,z) \neq \gcd(z,x)$.

Let $\triangle ABC$ be a triangle with circumcircle $\Gamma$, and let $I$ be the center of the incircle of $\triangle ABC$. The lines $AI$, $BI$ and $CI$ intersect $\Gamma$ in $D \ne A$, $E \ne B$ and $F \ne C$. The tangent lines to $\Gamma$ in $F$, $D$ and $E$ intersect the lines $AI$, $BI$ and $CI$ in $R$, $S$ and $T$, respectively. Prove that \[\vert AR\vert \cdot \vert BS\vert \cdot \vert CT\vert = \vert ID\vert \cdot \vert IE\vert \cdot \vert IF\vert.\]

A sequence $ \left( a_n \right)_{n\ge 1} $ has the property that it´s nondecreasing, nonconstant and, for every natural $ n, a_n\big| n^2. $ Show that at least one of the following affirmations are true. $ \text{(i)} $ There exists an index $ n_1 $ such that $ a_n=n, $ for all $ n\ge n_1. $ $ \text{(ii)} $ There exists an index $ n_2 $ such that $ a_n=n^2, $ for all $ n\ge n_2. $

In a triangle $ABC$, with $AB \ne BC$, $E$ is a point on the line $AC$ such that $BE$ is perpendicular to $AC$. A circle passing through $A$ and touching the line $BE$ at a point $P \ne B$ intersects the line $AB$ for the second time at $X$. Let $Q$ be a point on the line $PB$ different from $P$ such that $BQ = BP$. Let $Y$ be the point of intersection of the lines $CP$ and $AQ$. Prove that the points $C, X, Y, A$ are concyclic if and only if $CX$ is perpendicular to $AB$.

Let $ D$ represent a repeating decimal. If $ P$ denotes the $ r$ figures of $ D$ which do not repeat themselves, and $ Q$ denotes the $ s$ figures of $ D$ which do repeat themselves, then the incorrect expression is: $ \textbf{(A)}\ D \equal{} .PQQQ\ldots \qquad\textbf{(B)}\ 10^rD \equal{} P.QQQ\ldots$ $ \textbf{(C)}\ 10^{r \plus{} s}D \equal{} PQ.QQQ\ldots \qquad\textbf{(D)}\ 10^r(10^s \minus{} 1)D \equal{} Q(P \minus{} 1)$ $ \textbf{(E)}\ 10^r\cdot10^{2s}D \equal{} PQQ.QQQ\ldots$

Determine all even numbers $n$, $n \in \mathbb N$ such that \[{ \frac{1}{d_{1}}+\frac{1}{d_{2}}+ \cdots +\frac{1}{d_{k}}=\frac{1620}{1003}}, \] where $d_1, d_2, \ldots, d_k$ are all different divisors of $n$.

For a positive integer $k\ge 2$ define $\mathcal{T}_k=\{(x,y)\mid x,y=0,1,\ldots, k-1\}$ to be a collection of $k^2$ lattice points on the cartesian coordinate plane. Let $d_1(k)>d_2(k)>\cdots$ be the decreasing sequence of the distinct distances between any two points in $T_k$. Suppose $S_i(k)$ be the number of distances equal to $d_i(k)$. Prove that for any three positive integers $m>n>i$ we have $S_i(m)=S_i(n)$.

Show that \[ \frac{(b+c)(a^4-b^2c^2)}{ab+2bc+ca}+\frac{(c+a)(b^4-c^2a^2)}{bc+2ca+ab}+\frac{(a+b)(c^4-a^2b^2)}{ca+2ab+bc} \geq 0 \] for all positive real numbers $a, \: b , \: c.$

A person standing on the edge of a fire escape simultaneously launches two apples, one straight up with a speed of $\text{7 m/s}$ and the other straight down at the same speed. How far apart are the two apples $2$ seconds after they were thrown, assuming that neither has hit the ground? (A) $\text{14 m}$ (B) $\text{20 m}$ (C) $\text{28 m}$ (D) $\text{34 m}$ (E) $\text{56 m}$

Determine the value of the expression $$ (1 +\tan(1^o))(1 + \tan(2^o))...(1 + \tan(45^o)).$$

Determine the positive integers expressible in the form $\frac{x^2+y}{xy+1}$, for at least $2$ pairs $(x,y)$ of positive integers

A triangle $ABC$ is given. A circle $\Gamma$, passing through $A$, is tangent to side $BC$ at point $P$ and intersects sides $AB$ and $AC$ at $M$ and $N$ respectively. Prove that the smaller arcs $MP$ and $NP$ of $\Gamma$ are equal iff $\Gamma$ is tangent to the circumcircle of $\Delta ABC$ at $A$.

Consider a $100\times 100$ square unit lattice $\textbf{L}$ (hence $\textbf{L}$ has $10000$ points). Suppose $\mathcal{F}$ is a set of polygons such that all vertices of polygons in $\mathcal{F}$ lie in $\textbf{L}$ and every point in $\textbf{L}$ is the vertex of exactly one polygon in $\mathcal{F}.$ Find the maximum possible sum of the areas of the polygons in $\mathcal{F}.$ [i]Michael Ren and Ankan Bhattacharya, USA[/i]

[b]p1.[/b] Thirty players participate in a chess tournament. Every player plays one game with every other player. What maximal number of players can get exactly $5$ points? (any game adds $1$ point to the winner’s score, $0$ points to a loser’s score, in the case of a draw each player obtains $1/2$ point.) [b]p2.[/b] A father and his son returned from a fishing trip. To make their catches equal the father gave to his son some of his fish. If, instead, the son had given his father the same number of fish, then father would have had twice as many fish as his son. What percent more is the father's catch more than his son's? [b]p3.[/b] What is the maximal number of pieces of two shapes, [img]https://cdn.artofproblemsolving.com/attachments/a/5/6c567cf6a04b0aa9e998dbae3803b6eeb24a35.png[/img] and [img]https://cdn.artofproblemsolving.com/attachments/8/a/7a7754d0f2517c93c5bb931fb7b5ae8f5e3217.png[/img], that can be used to tile a $7\times 7$ square? [b]p4.[/b] Six shooters participate in a shooting competition. Every participant has $5$ shots. Each shot adds from 1 to $10$ points to shooter’s score. Every person can score totally for all five shots from $5$ to $50$ points. Each participant gets $7$ points for at least one of his shots. The scores of all participants are different. We enumerate the shooters $1$ to $6$ according to their scores, the person with maximal score obtains number $1$, the next one obtains number $2$, the person with minimal score obtains number $6$. What score does obtain the participant number 3? The total number of all obtained points is $264$. [b]p5.[/b] There are $2014$ stones in a pile. Two players play the following game. First, player $A$ takes some number of stones (from $1$ to $30$) from the pile, then player B takes $1$ or $2$ stones, then player $A$ takes $2$ or $3$ stones, then player $B$ takes $3$ or $4$ stones, then player A takes $4$ or $5$ stones, etc. The player who gets the last stone is the winner. If no player gets the last stone (there is at least one stone in the pile but the next move is not allowed) then the game results in a draw. Who wins the game using the right strategy? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

If $b$ men take $c$ days to lay $f$ bricks, then the number of days it will take $c$ men working at the same rate to lay $b$ bricks, is $\textbf{(A) }fb^2\qquad\textbf{(B) }b/f^2\qquad\textbf{(C) }f^2/b\qquad\textbf{(D) }b^2/f\qquad \textbf{(E) }f/b^2$

Find all triples $(a,b,c)$ of integers for which the equation \[x^3-a^2x^2+b^2x-ab+3c=0\] has three distinct integer roots $x_1,x_2,x_3$ which are pairwise coprime.

Find, with justification, all positive real numbers $a,b,c$ satisfying the system of equations: \[a\sqrt{b}=a+c,b\sqrt{c}=b+a,c\sqrt{a}=c+b.\]

Aurick has a cup, a right cone with a circular base of radius $\frac12$, filled with milk tea. The slant height of the cup is $1$, and the tea fills the cup $\frac12$ of the way up the cup’s side. Suppose that Aurick tips the cup just to the point of spilling, as shown in the diagram. The new slant height EA and the tilted tea surface’s major axis $ET$ form $\angle T EA$. Compute $\cos(\angle T EA)$. [img]https://cdn.artofproblemsolving.com/attachments/e/e/76e12ee31ce4ba8a5daaf0f5538b98726a0d37.png[/img]

A function f : $R$ $\rightarrow$ $R$ satisfies the property $f(x^2) - f^2(x) \geq 1/4$ for all x. Verify if the function is one-one.

$a)$ Let $a$, $b$ and $c$ be positive real numbers. Prove that for all positive integers $m$ holds: $$(a+b)^m+(b+c)^m+(c+a)^m \leq 2^m(a^m+b^m+c^m)$$ $b)$ Does inequality $a)$ holds for $1)$ arbitrary real numbers $a$, $b$ and $c$ $2)$ any integer $m$

Let $A_1A_2A_3A_4A_5A_6A_7A_8$ be a right octagon with center $O$ and $\lambda_1$,$\lambda_2$, $\lambda_3$, $\lambda_4$ be some rational numbers for which: $\lambda_1 \overrightarrow{OA_1}+\lambda_2 \overrightarrow{OA_2}+\lambda_3 \overrightarrow{OA_3}+\lambda_4 \overrightarrow{OA_4} =\overrightarrow{o}$. Prove that $\lambda_1=\lambda_2=\lambda_3=\lambda_4=0$.

If the points $(1,y_1)$ and $(-1,y_2)$ lie on the graph of $y=ax^2+bx+c$, and $y_1-y_2=-6$, then $b$ equals: $\textbf{(A) }-3\qquad \textbf{(B) }0\qquad \textbf{(C) }3\qquad \textbf{(D) }\sqrt{ac}\qquad \textbf{(E) }\dfrac{a+c}2$

Let $X$ be a non empty subset of $\mathbb{N} = \{1,2,\ldots \}$. Suppose that for all $x \in X$, $4x \in X$ and $\lfloor \sqrt{x} \rfloor \in X$. Prove that $X=\mathbb{N}$.

In triangle $ABC$, $D$ is a point on $AB$ between $A$ and $B$, $E$ is a point on $AC$ between $A$ and $C$, and $F$ is a point on $BC$ between $B$ and $C$ such that $AF$, $BE$, and $CD$ all meet inside $\triangle ABC$ at a point $G$. Given that the area of $\triangle ABC$ is $15$, the area of $\triangle ABE$ is $5$, and the area of $\triangle ACD$ is $10$, compute the area of $\triangle ABF$.