Let $ABC$ be an acute-angled triangle with $A=60^{\circ}$. Let $E,F$ be the feet of altitudes through $B,C$ respectively. Prove that $CE-BF=\tfrac{3}{2}(AC-AB)$ [i]Proposed by Fatemeh Sajadi[/i]

A polygon with $2n+1$ vertices is given. Show that it is possible to assign numbers $1,2,\ldots ,4n+2$ to the vertices and midpoints of the sides of the polygon so that for each side the sum of the three numbers assigned to it is the same.

Let $x$ and $y$ be positive real numbers. 1. Prove: if $x^3 - y^3 \ge 4x$, then $x^2 > 2y$. 2. Prove: if $x^5 - y^3 \ge 2x$, then $x^3 \ge 2y$.

Let $m=\frac{-1+\sqrt{17}}{2}$. Let the polynomial $P(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$ is given, where $n$ is a positive integer, the coefficients $a_0,a_1,a_2,...,a_n$ are positive integers and $P(m) =2018$ . Prove that the sum $a_0+a_1+a_2+...+a_n$ is divisible by $2$ .

Let $P$ be a point outside a circumference $\Gamma$, and let $PA$ be one of the tangents from $P$ to $\Gamma$. Line $l$ passes through $P$ and intersects $\Gamma$ at $B$ and $C$, with $B$ between $P$ and $C$. Let $D$ be the symmetric of $B$ with respect to $P$. Let $\omega_1$ and $\omega_2$ be the circles circumscribed to the triangles $DAC$ and $PAB$ respectively. $\omega_1$ and $\omega _2$ intersect at $E \neq A$. Line $EB$ cuts back to $\omega _1 $ in $F$. Prove that $CF = AB$.

Find the differentiable function $f(x)$ with $f(0)\neq 0$ satisfying $f(x+y)=f(x)f'(y)+f'(x)f(y)$ for all real numbers $x,\ y$.

Let $n > 3$ be an integer. Prove that there exist positive integers $x_1,..., x_n$ in geometric progression and positive integers $y_1,..., y_n$ in arithmetic progression such that $x_1<y_1<x_2<y_2<...<x_n<y_n$

Jar #1 contains five red marbles, three blue marbles, and one green marble. Jar #2 contains five blue marbles, three green marbles, and one red marble. Jar #3 contains five green marbles, three red marbles, and one blue marble. You randomly select one marble from each jar. Given that you select one marble of each color, the probability that the red marble came from jar #1, the blue marble came from jar #2, and the green marble came from jar #3 can be expressed as $\frac{m}{n}$, where m and n are relatively prime positive integers. Find m + n.

Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.

Two sides of a regular polygon of $n$ sides when extended meet at $28$ degrees. What is smallest possible value of $n$

From a point within a triangle, line segments are drawn to the vertices. A necessary and sufficient condition that the three triangles thus formed have equal areas is that the point be: $ \textbf{(A)}\ \text{the center of the inscribed circle} \qquad \\ \textbf{(B)}\ \text{the center of the circumscribed circle}\qquad \\ \textbf{(C)}\ \text{such that the three angles fromed at the point each be }{120^\circ}\qquad \\ \textbf{(D)}\ \text{the intersection of the altitudes of the triangle}\qquad \\ \textbf{(E)}\ \text{the intersection of the medians of the triangle}$

We call a function $f: \mathbb{Q}^+ \to \mathbb{Q}^+$ [i]good[/i] if for all $x,y \in \mathbb{Q}^+$ we have: $$f(x)+f(y)\geq 4f(x+y).$$ a) Prove that for all good functions $f: \mathbb{Q}^+ \to \mathbb{Q}^+$ and $x,y,z \in \mathbb{Q}^+$ $$f(x)+f(y)+f(z) \geq 8f(x+y+z)$$ b) Does there exists a good functions $f: \mathbb{Q}^+ \to \mathbb{Q}^+$ and $x,y,z \in \mathbb{Q}^+$ such that $$f(x)+f(y)+f(z) < 9f(x+y+z) ?$$

Given a poistive integer $ m, $ determine the smallest integer $ n\ge 2 $ such that for any coloring of the $ n^2 $ unit squares of a $ n\times n $ square with $ m $ colors, there are, at least, two unit squares $ (i,j),(k,l) $ that share the same color, where $ 1\le i,j,k,l\le n,i\neq j,k\neq l. $ [i]American Mathematical Monthly[/i]

Let $ABC$ be a triangle inscribed in the circle $K$ and consider a point $M$ on the arc $BC$ that do not contain $A$. The tangents from $M$ to the incircle of $ABC$ intersect the circle $K$ at the points $N$ and $P$. Prove that if $\angle BAC = \angle NMP$, then triangles $ABC$ and $MNP$ are congruent. Valentin Vornicu [hide= about Romania JBMO TST 2004 in aops]I found the Romania JBMO TST 2004 links [url=https://artofproblemsolving.com/community/c6h5462p17656]here [/url] but they were inactive. So I am asking for solution for the only geo I couldn't find using search. The problems were found [url=https://artofproblemsolving.com/community/c6h5135p16284]here[/url].[/hide]

There is a $19\times19$ board. Is it possible to mark some $1\times 1$ squares so that each of $10\times 10$ squares contain different number of marked squares?

As evidence that the correct answer does not mean the correctness of the proof, the teacher cited next example. Let's take the fraction $\frac{19}{95}$. After crossing out $9$ in the numerator and denominator (“reduction” by $9$), we get $\frac{1}{5}$ which is the correct answer. In the same way, a fraction $\frac{1999}{9995}$ can be “reduced” by three nines (cross out $999$ in the numerator and denominator). Is it possible that as a result of such a “reduction” we also get the correct answer, equal to $\frac13$ ? (We consider fractions of the form $\frac{1a}{a3}$. Here, with the letter $a$ we denote several numbers that follow in the same order in the numerator after $1$, and in the denominator before $3$. “Reduce” by $a$.)

Let $a,\ b>0$ be real numbers, $n\geq 2$ be integers. Evaluate $I_n=\int_{-\infty}^{\infty} \frac{exp(ia(x-ib))}{(x-ib)^n}dx.$

Let $ f: \mathbb R \rightarrow \mathbb R$ be a function such that $ f(tx_1\plus{}(1\minus{}t)x_2)\leq tf(x_1)\plus{}(1\minus{}t)f(x_2)$ for all $ x_1 , x_2 \in \mathbb R$ and $ t\in (0,1)$. Show that $ \sum_{k\equal{}1}^{2003}f(a_{k\plus{}1})a_k \geq \sum_{k\equal{}1}^{2003}f(a_k)a_{k\plus{}1}$ for all real numbers $ a_1,a_2,...,a_{2004}$ such that $ a_1\geq a_2\geq ... \geq a_{2003}$ and $ a_{2004}\equal{}a_1$

It is known that the graph of a quadratic trinomial $y = x^2 + px + q$ touches the graph of a straight line $y = 2x + p$. Prove that all such quadratic trinomials have the same minimum value. Find this smallest value.

A $\pm 1$-[i]sequence[/i] is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$

Let $ABC$ be an acute triangle with $AB \neq AC$ . Also let $M$ be the midpoint of the side $BC$ , $H$ the orthocenter of the triangle $ABC$ , $O_1$ the midpoint of the segment $AH$ and $O_2$ the center of the circumscribed circle of the triangle $BCH$ . Prove that $O_1AMO_2$ is a parallelogram .

Let $m$ be a positive integer, and let $a_0, a_1,\ldots,a_m$ be a sequence of reals such that $a_0=37$, $a_1=72$, $a_m=0$, and \[a_{k+1}=a_{k-1}-\frac{3}{a_k}\] for $k=1,2, \dots, m-1$. Find $m$.

Given a triangle $ABC$ in which the angle $B$ is equal to $60^\circ$. A circle inscribed in a triangle with a center $I$ touches the side $AC$ at point $K$. A line passing through the points of touching of this circle with the other sides of the triangle intersects the its circumcircle at points $M$ and $N$. Prove that the ray $KI$ divides the arc $MN$ in half.

Show that there exists the maximum value of the function $f(x,\ y)=(3xy+1)e^{-(x^2+y^2)}$ on $\mathbb{R}^2$, then find the value.

The $ 8\times 18$ rectangle $ ABCD$ is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is $ y$? [asy] unitsize(2mm); defaultpen(fontsize(10pt)+linewidth(.8pt)); dotfactor=4; draw((0,4)--(18,4)--(18,-4)--(0,-4)--cycle); draw((6,4)--(6,0)--(12,0)--(12,-4)); label("$D$",(0,4),NW); label("$C$",(18,4),NE); label("$B$",(18,-4),SE); label("$A$",(0,-4),SW); label("$y$",(9,1)); [/asy]$ \textbf{(A) } 6\qquad \textbf{(B) } 7\qquad \textbf{(C) } 8\qquad \textbf{(D) } 9\qquad \textbf{(E) } 10$