$ABCDE$ is convex pentagon. $\angle BCA=\angle BEA = \frac{\angle BDA}{2}, \angle BDC =\angle EDA$. Prove, that $\angle DEB=\angle DAC$

A function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ has property $ \mathcal{F} , $ if for any real number $ a, $ there exists a $ b<a $ such that $ f(x)\le f(a), $ for all $ x\in (b,a) . $ [b]a)[/b] Give an example of a function with property $ \mathcal{F} $ that is not monotone on $ \mathbb{R} . $ [b]b)[/b] Prove that a continuous function that has property $ \mathcal{F} $ is nondecreasing.

Find all point $M$ lying into given acute-angled triangle $ABC$ and such that the area of the triangle with vertices on the feet of the perpendiculars drawn from $M$ to the lines $BC$, $CA$, $AB$ is maximal. [i]H. Lesov[/i]

Alexandra draws a letter A which stands on the $x$-axis. [list=a][*]The left side of the letter A lies along the line with equation $y=3x+6$. What is the $x$-intercept of the line with equation $y=3x+6$? [*]The right side of the letter A lies along the line $L_2$ and the leter is symmetric about the $y$-axis. What is the equation of line $L_2$? [*]Determine the are of the triangle formed by the $x$ axis and the left and right sides of the letter A. [*]Alexandra completes the letter A by adding to Figure 1. She draws the horizontal part of the letter A along the line $y=c$, as in Figure 2. The area of the shaded region inside the letter A and above the line with equation $y=c$ is $\frac49$ of the total area of the region above the $x$ axis and between the left and right sides. Determine the value of $c$.[/list] [b]Figure 1[/b] [asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(10cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -4.8408113739622465, xmax = 5.491811096383217, ymin = -3.0244242161812847, ymax = 8.241467380517944; /* image dimensions */ pen cqcqcq = rgb(0.7529411764705882,0.7529411764705882,0.7529411764705882); Label laxis; laxis.p = fontsize(10); xaxis(xmin, xmax, EndArrow(6), above = true); yaxis(ymin, ymax, EndArrow(8.25),above = true); /* draws axes; NoZero hides '0' label */ /* draw figures */ draw((0,6)--(-2,0), linewidth(2)); draw((-2,0)--(2,0), linewidth(2)); draw((2,0)--(0,6), linewidth(2)); label("$y=3x+6$",(-2.874280000573916,3.508459668295191),SE*labelscalefactor); label("$L_2$",(1.3754276283584919,3.5917872688624928),SE*labelscalefactor); label("$O$",(0,0),SW*labelscalefactor); /* dots and labels */ clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy] [b]Figure 2[/b] [asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(10cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -4.707487213054563, xmax = 5.6251352572909, ymin = -3.4577277391312538, ymax = 7.808163857567977; /* image dimensions */ pen cqcqcq = rgb(0.7529411764705882,0.7529411764705882,0.7529411764705882); draw((-1.114884596113444,2.6553462116596678)--(1.1148845961134441,2.6553462116596678)--(0,6)--cycle, linewidth(2)); Label laxis; laxis.p = fontsize(10); xaxis(xmin, xmax, EndArrow(6), above = true); yaxis(ymin, ymax, EndArrow(6), above = true); /* draws axes; NoZero hides '0' label */ /* draw figures */ draw((0,6)--(-2,0), linewidth(2)); draw((-2,0)--(2,0), linewidth(2)); draw((2,0)--(0,6), linewidth(2)); label("$O$",(0,0),SW*labelscalefactor); draw((-1.114884596113444,2.6553462116596678)--(1.1148845961134441,2.6553462116596678), linewidth(2)); draw((-1.114884596113444,2.6553462116596678)--(1.1148845961134441,2.6553462116596678), linewidth(2)); draw((1.1148845961134441,2.6553462116596678)--(0,6), linewidth(2)); draw((0,6)--(-1.114884596113444,2.6553462116596678), linewidth(2)); fill((0,6)--(-1.114884596113444,2.6553462116596678)--(1.1148845961134441,2.6553462116596678)--cycle,black); label("$y=c$",(1.4920862691527148,3.1251527056856054),SE*labelscalefactor); /* dots and labels */ clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* yes i used geogebra fight me*/ [/asy]

I couldn't solve this problem and the only solution I was able to find was very unnatural (it was an official solution, I think) and I couldn't be satisfied with it, so I ask you if you can find some different solutions. The problem is really great one! If $M$ is the centroid of a triangle $ABC$, prove that the following inequality holds: \[\sin\angle CAM+\sin\angle CBM\leq\frac{2}{\sqrt3}.\] The equality occurs in a very strange case, I don't remember it.

In cells of a $2022 \times 2022$ table numbers from $1$ to $2022^2$ are written, in each cell exactly one number, all numbers are used once. For every row Vlad marks the second biggest number in it, Dima does the same for every column. It turned out that boys marked $4044$ pairwise distinct numbers, and there are $k$ numbers marked by Vlad, each of which is less than all numbers marked by Dima. Find the maximum possible value of $k$

There are $ n$ points in $ \mathbb R^3$ such that every three form an acute angled triangle. Find maximum of $ n$.

The numbers $1, 2, \ldots, n$ are written on a blackboard. Each minute, a student goes up to the board, chooses two numbers $x$ and $y$, erases them, and writes the number $2x+2y$ on the board. This continues until only one number remains. Prove that this number is at least $\frac{4}{9}n^3$. [i]Brian Hamrick.[/i]

Let $P$ be a point in the interior of an acute triangle $ABC$, and let $Q$ be its isogonal conjugate. Denote by $\omega_P$ and $\omega_Q$ the circumcircles of triangles $BPC$ and $BQC$, respectively. Suppose the circle with diameter $\overline{AP}$ intersects $\omega_P$ again at $M$, and line $AM$ intersects $\omega_P$ again at $X$. Similarly, suppose the circle with diameter $\overline{AQ}$ intersects $\omega_Q$ again at $N$, and line $AN$ intersects $\omega_Q$ again at $Y$. Prove that lines $MN$ and $XY$ are parallel. (Here, the points $P$ and $Q$ are [i]isogonal conjugates[/i] with respect to $\triangle ABC$ if the internal angle bisectors of $\angle BAC$, $\angle CBA$, and $\angle ACB$ also bisect the angles $\angle PAQ$, $\angle PBQ$, and $\angle PCQ$, respectively. For example, the orthocenter is the isogonal conjugate of the circumcenter.) [i]Proposed by Sammy Luo[/i]

Solve a set of inequalities in the domain of integer numbers: $$3x^2 +2yz \le 1+y^2$$ $$3y^2 +2zx \le 1+z^2$$ $$3z^2 +2xy \le 1+x^2$$

We call polynomials $A(x) = a_n x^n +. . .+a_1 x+a_0$ and $B(x) = b_m x^m +. . .+b_1 x+b_0$ ($a_n b_m \neq 0$) similar if the following conditions hold: $(i)$ $n = m$; $(ii)$ There is a permutation $\pi$ of the set $\{ 0, 1, . . . , n\} $ such that $b_i = a_{\pi (i)}$ for each $i \in {0, 1, . . . , n}$. Let $P(x)$ and $Q(x)$ be similar polynomials with integer coefficients. Given that $P(16) = 3^{2012}$, find the smallest possible value of $|Q(3^{2012})|$. [i]Proposed by Milos Milosavljevic[/i]

Tangents from the point $A$ to the circle $\Gamma$ touche this circle at $C$ and $D$.Let $B$ be a point on $\Gamma$,different from $C$ and $D$. The circle $\omega$ that passes through points $A$ and $B$ intersect with lines $AC$ and $AD$ at $F$ and $E$,respectively.Prove that the circumcircles of triangles $ABC$ and $DEB$ are tangent if and only if the points $C,D,F$ and $E$ are cyclic.

The number of points with positive rational coordinates selected from the set of points in the xy-plane such that $x+y\leq 5$, is: $\text{(A)} \ 9 \qquad \text{(B)} \ 10 \qquad \text{(C)} \ 14 \qquad \text{(D)} \ 15 \qquad \text{(E)} \ \text{infinite}$

Integers $x$ and $y$ with $x>y>0$ satisfy $x+y+xy=80$. What is $x$? $\textbf{(A) }8\qquad\textbf{(B) }10\qquad\textbf{(C) }15\qquad\textbf{(D) }18\qquad\textbf{(E) }26$

For positive integers $m, n$ define the function $f_n(m) = 1^{2n} + 2^{2n} + 3^{2n} + \ldots +m^{2n}$. Prove that there are only finitely many pairs of positive integers $(a, b)$ such that $f_n(a) + f_n(b)$ is a prime number. [i]Proposed by Nazar Serdyuk[/i]

Prove the inequality $$\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\ldots+\frac{1}{2022!}>\frac{1^2}{2!}+\frac{2^2}{3!}+\frac{3^2}{4!}+\ldots+\frac{2022^2}{2023!}$$.

A music streaming service proposes songs classified in $10$ musical genres, so that each song belong to one and only one gender. The songs are played one after the other: the first $17$ are chosen by the user, but starting from the eighteenth the service automatically determines which song to play. Elisabetta has noticed that, if one makes the classification of which genres they appear several times during the last $17$ songs played, the new song always belongs to the genre at the top of the ranking or, in case of same merit, at one of the first genres. Prove that, however, the first $17$ tracks are chosen, from a certain point onwards the songs proposed are all of the same kind.

Let $ABCD$ be a parallelogram and let $E$ be a point on the side $AD$, such that $\frac{AE}{ED} = m$. Let $F$ be a point on $CE$, such that $BF \perp CE$, and the point $G$ is symmetrical to $F$ with respect to $AB$. If point $A$ is the circumcenter of triangle $BFG$, find the value of $m$.

Define two sets: $M=\{ (x,y)|y\geq x^2\} ,N=\{ (x,y)|x^2+(y-a)^2\leq 1\}$. If $M\cup N=N$, then the range value of $a$ is $\text{(A)}a\geq 1\frac{1}{4}\qquad\text{(B)}a=1\frac{1}{4}\qquad\text{(C)}a\geq 1\qquad\text{(D)}0<a<1$

Let $ABC$ be an obtuse triangle with $\angle A > \angle B > \angle C$, and let $M$ be a midpoint of the side $BC$. Let $D$ be a point on the arc $AB$ of the circumcircle of triangle $ABC$ not containing $C$. Suppose that the circle tangent to $BD$ at $D$ and passing through $A$ meets the circumcircle of triangle $ABM$ again at $E$ and $\overline{BD}=\overline{BE}$. $\omega$, the circumcircle of triangle $ADE$, meets $EM$ again at $F$. Prove that lines $BD$ and $AE$ meet on the line tangent to $\omega$ at $F$.

Find all functions $f : Z \to Z$ satisfying $\bullet$ $ f(p) > 0$ for all prime numbers $p$, $\bullet$ $p| (f(x) + f(p))^{f(p)}- x$ for all $x \in Z$ and all prime numbers $p$.

Construct an uncountable Hausdorff space in which the complement of the closure of any nonempty, open set is countable. [i]A. Hajnal, I. Juhasz[/i]

Given is a positive integer $k$. There are $n$ points chosen on a line, such the distance between any two adjacent points is the same. The points are colored in $k$ colors. For each pair of monochromatic points such that there are no points of the same color between them, we record the distance between these two points. If all distances are distinct, find the largest possible $n$.

Baron Münchausen boasts that he knows a remarkable quadratic triniomial with positive coefficients. The trinomial has an integral root; if all of its coefficients are increased by $1$, the resulting trinomial also has an integral root; and if all of its coefficients are also increased by $1$, the new trinomial, too, has an integral root. Can this be true?

[b](a)[/b] A $2 \times n$-table (with $n > 2$) is filled with numbers so that the sums in all the columns are different. Prove that it is possible to permute the numbers in the table so that the sums in the columns would still be different and the sums in the rows would also be different. [i]($2$ points)[/i] [b](b)[/b] A $100 \times 100$-table is filled with numbers such that the sums in all the columns are different. Is it always possible to permute the numbers in the table so that the sums in the columns would still be different and the sums in the rows would also be different? [i]($6$ points)[/i]