Prove that a regular polygon with an odd number of edges cannot be partitioned into four pieces with equal areas by two lines that pass through the center of polygon.

Find all positive integers $(x, n)$ such that $x^{n}+2^{n}+1$ divides $x^{n+1}+2^{n+1}+1$.

We say that two sequences $x,y \colon \mathbb{N} \to \mathbb{N}$ are [i]completely different[/i] if $x_n \neq y_n$ holds for all $n\in \mathbb{N}$. Let $F$ be a function assigning a natural number to every sequence of natural numbers such that $F(x)\neq F(y)$ for any pair of completely different sequences $x$, $y$, and for constant sequences we have $F \left((k,k,\dots)\right)=k$. Prove that there exists $n\in \mathbb{N}$ such that $F(x)=x_{n}$ for all sequences $x$.

The average (arithmetic mean) age of a group consisting of doctors and lawyers in 40. If the doctors average 35 and the lawyers 50 years old, then the ratio of the numbers of doctors to the number of lawyers is $ \textbf{(A)}\ 3: 2 \qquad \textbf{(B)}\ 3: 1 \qquad \textbf{(C)}\ 2: 3 \qquad \textbf{(D)}\ 2: 1 \qquad \textbf{(E)}\ 1: 2$

Let $S$ be a set of positive integers such that $\lfloor \sqrt{x}\rfloor =\lfloor \sqrt{y}\rfloor $ for all $x, y \in S$. Show that the products $xy$, where $x, y \in S$, are pairwise distinct.

Let $\mathcal{S}$ be a set consisting of $n \ge 3$ positive integers, none of which is a sum of two other distinct members of $\mathcal{S}$. Prove that the elements of $\mathcal{S}$ may be ordered as $a_1, a_2, \dots, a_n$ so that $a_i$ does not divide $a_{i - 1} + a_{i + 1}$ for all $i = 2, 3, \dots, n - 1$.

A natural number $A$ is given. One may add to it one of its divisors $d$ ($1 < d < A$). One may then repeat this operation with the new number $A + d$ and so on. Prove that starting from $A = 4$ one can get any composite number by these operations. (M Vyalyi)

The bisectors of the angles $\angle{A},\angle{B},\angle{C}$ of a triangle $\triangle{ABC}$ intersect with the circumcircle $c_1(O,R)$ of $\triangle{ABC}$ at $A_2,B_2,C_2$ respectively.The tangents of $c_1$ at $A_2,B_2,C_2$ intersect each other at $A_3,B_3,C_3$ (the points $A_3,A$ lie on the same side of $BC$,the points $B_3,B$ on the same side of $CA$,and $C_3,C$ on the same side of $AB$).The incircle $c_2(I,r)$ of $\triangle{ABC}$ is tangent to $BC,CA,AB$ at $A_1,B_1,C_1$ respectively.Prove that $A_1A_2,B_1B_2,C_1C_2,AA_3,BB_3,CC_3$ are concurrent. [hide=Diagram][asy]import graph; size(11cm); 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 = -9.26871978147865, xmax = 19.467150423463277, ymin = -6.150626456647122, ymax = 10.10782642246474; /* image dimensions */ pen aqaqaq = rgb(0.6274509803921569,0.6274509803921569,0.6274509803921569); pen uququq = rgb(0.25098039215686274,0.25098039215686274,0.25098039215686274); draw((1.0409487561836381,4.30054785243355)--(0.,0.)--(6.,0.)--cycle, aqaqaq); /* draw figures */ draw((1.0409487561836381,4.30054785243355)--(0.,0.), uququq); draw((0.,0.)--(6.,0.), uququq); draw((6.,0.)--(1.0409487561836381,4.30054785243355), uququq); draw(circle((3.,1.550104087253063), 3.376806580383107)); draw(circle((1.9303371951242874,1.5188413314630436), 1.5188413314630436)); draw((1.0226422135625703,7.734611112525813)--(1.0559139088339535,1.4932847901569466), linetype("2 2")); draw((-1.2916762981259242,-1.8267024931300444)--(1.0559139088339535,1.4932847901569466), linetype("2 2")); draw((-0.2820306621765219,2.344520485530311)--(1.0559139088339535,1.4932847901569466), linetype("2 2")); draw((1.0559139088339535,1.4932847901569466)--(5.212367857300808,4.101231513568902), linetype("2 2")); draw((1.0559139088339535,1.4932847901569466)--(3.,-1.8267024931300442), linetype("2 2")); draw((12.047991949367804,-1.8267024931300444)--(1.0559139088339535,1.4932847901569466), linetype("2 2")); draw((1.0226422135625703,7.734611112525813)--(-1.2916762981259242,-1.8267024931300444)); draw((-1.2916762981259242,-1.8267024931300444)--(12.047991949367804,-1.8267024931300444)); draw((12.047991949367804,-1.8267024931300444)--(1.0226422135625703,7.734611112525813)); /* dots and labels */ dot((1.0409487561836381,4.30054785243355),linewidth(3.pt) + dotstyle); label("$A$", (0.5889800538632699,4.463280489351154), NE * labelscalefactor); dot((0.,0.),linewidth(3.pt) + dotstyle); label("$B$", (-0.5723380089304358,-0.10096957139619551), NE * labelscalefactor); dot((6.,0.),linewidth(3.pt) + dotstyle); label("$C$", (6.233525986976863,0.06107480945873997), NE * labelscalefactor); label("$c_1$", (1.9663572911302232,5.111458012770896), NE * labelscalefactor); dot((3.,-1.8267024931300442),linewidth(3.pt) + dotstyle); label("$A_2$", (2.9386235762598374,-2.3155761097469805), NE * labelscalefactor); dot((5.212367857300808,4.101231513568902),linewidth(3.pt) + dotstyle); label("$B_2$", (5.315274495465561,4.274228711687063), NE * labelscalefactor); dot((-0.2820306621765219,2.344520485530311),linewidth(3.pt) + dotstyle); label("$C_2$", (-0.9234341674494632,2.6807922999468636), NE * labelscalefactor); dot((1.0226422135625703,7.734611112525813),linewidth(3.pt) + dotstyle); label("$A_3$", (1.1291279900463889,7.893219884113956), NE * labelscalefactor); dot((-1.2916762981259242,-1.8267024931300444),linewidth(3.pt) + dotstyle); label("$B_3$", (-1.8146782621516093,-1.4783468086631473), NE * labelscalefactor); dot((12.047991949367804,-1.8267024931300444),linewidth(3.pt) + dotstyle); label("$C_3$", (12.148145888182015,-1.6673985863272387), NE * labelscalefactor); dot((1.9303371951242874,1.5188413314630436),linewidth(3.pt) + dotstyle); label("$I$", (2.047379481557691,1.681518618008095), NE * labelscalefactor); dot((1.9303371951242878,0.),linewidth(3.pt) + dotstyle); label("$A_1$", (1.4532167517562602,-0.5600953171518461), NE * labelscalefactor); label("$c_2$", (1.5072315453745722,3.247947632939138), NE * labelscalefactor); dot((2.9254299438737803,2.666303492733126),linewidth(3.pt) + dotstyle); label("$B_1$", (2.8576013858323694,3.1129106488933584), NE * labelscalefactor); dot((0.45412477306806903,1.8761589424582812),linewidth(3.pt) + dotstyle); label("$C_1$", (0,2.3296961414278368), NE * labelscalefactor); dot((1.0559139088339535,1.4932847901569466),linewidth(3.pt) + dotstyle); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */[/asy][/hide]

[b]p1.[/b] Let $S$ be the sum of the $99$ terms: $$(\sqrt1 + \sqrt2)^{-1},(\sqrt2 + \sqrt3)^{-1}, (\sqrt3 + \sqrt4)^{-1},..., (\sqrt{99} + \sqrt{100})^{-1}.$$ Prove that $S$ is an integer. [b]p2.[/b] Determine all pairs of positive integers $x$ and $y$ for which $N=x^4+4y^4$ is a prime. (Your work should indicate why no other solutions are possible.) [b]p3.[/b] Let $w,x,y,z$ be arbitrary positive real numbers. Prove each inequality: (a) $xy \le \left(\frac{x+y}{2}\right)^2$ (b) $wxyz \le \left(\frac{w+x+y+z}{4}\right)^4$ (c) $xyz \le \left(\frac{x+y+z}{3}\right)^3$ [b]p4.[/b] Twelve points $P_1$,$P_2$, $...$,$P_{12}$ are equally spaaed on a circle, as shown. Prove: that the chords $\overline{P_1P_9}$, $\overline{P_4P_{12}}$ and $\overline{P_2P_{11}}$ have a point in common. [img]https://cdn.artofproblemsolving.com/attachments/d/4/2eb343fd1f9238ebcc6137f7c84a5f621eb277.png[/img] [b]p5.[/b] Two very busy men, $A$ and $B$, who wish to confer, agree to appear at a designated place on a certain day, but no earlier than noon and no later than $12:15$ p.m. If necessary, $A$ will wait $6$ minutes for $B$ to arrive, while $B$ will wait $9$ minutes for $A$ to arrive but neither can stay past $12:15$ p.m. Express as a percent their chance of meeting. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Determine the minimum value of $a^{2} + b^{2}$ when $(a,b)$ traverses all the pairs of real numbers for which the equation \[ x^{4} + ax^{3} + bx^{2} + ax + 1 = 0 \] has at least one real root.

Two jokers are added to a $52$ card deck and the entire stack of $54$ cards is shuffled randomly. What is the expected number of cards that will be strictly between the two jokers?

If $\mathrm{MATH} + \mathrm{WITH} = \mathrm{GIRLS}$, compute the smallest possible value of $\mathrm{GIRLS}$. Here $\mathrm{MATH}$ and $\mathrm{WITH}$ are 4-digit numbers and $\mathrm{GIRLS}$ is a 5-digit number (all with nonzero leading digits). Different letters represent different digits.

Two equal squares, one with red sides, another with blue ones, give an octagon in intersection. Prove that the sum of red octagon sides lengths is equal to the sum of blue octagon sides lengths.

[b]p1[/b]. Is it possible to find $n$ positive numbers such that their sum is equal to $1$ and the sum of their squares is less than $\frac{1}{10}$? [b]p2.[/b] In the country of Sepulia, there are several towns with airports. Each town has a certain number of scheduled, round-trip connecting flights with other towns. Prove that there are two towns that have connecting flights with the same number of towns. [b]p3.[/b] A $4 \times 4$ magic square is a $4 \times 4$ table filled with numbers $1, 2, 3,..., 16$ - with each number appearing exactly once - in such a way that the sum of the numbers in each row, in each column, and in each diagonal is the same. Is it possible to complete $\begin{bmatrix} 2 & 3 & * & * \\ 4 & * & * & *\\ * & * & * & *\\ * & * & * & * \end{bmatrix}$ to a magic square? (That is, can you replace the stars with remaining numbers $1, 5, 6,..., 16$, to obtain a magic square?) [b]p4.[/b] Is it possible to label the edges of a cube with the numbers $1, 2, 3, ... , 12$ in such a way that the sum of the numbers labelling the three edges coming into a vertex is the same for all vertices? [b]p5.[/b] (Bonus) Several ants are crawling along a circle with equal constant velocities (not necessarily in the same direction). If two ants collide, both immediately reverse direction and crawl with the same velocity. Prove that, no matter how many ants and what their initial positions are, they will, at some time, all simultaneously return to the initial positions. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A group of persons is called [i]good[/i] if its members can be distributed to several rooms so that nobody is acquainted with any person in the same room but it is possible to choose a person from each room so that all the chosen persons are acquainted with each other. A group is called [i]perfect[/i] if it is good and every set of its members is also good. A perfect group planned a party. However one of its members, Alice, brought here acquaintance Bob, who was not originally expected, and introduced him to all her other acquaintances. Prove that the new group is also perfect. [i]Author: C. Berge[/i]

In a sign pyramid a cell gets a "+" if the two cells below it have the same sign, and it gets a "-" if the two cells below it have different signs. The diagram below illustrates a sign pyramid with four levels. How many possible ways are there to fill the four cells in the bottom row to produce a "+" at the top of the pyramid? [asy] unitsize(2cm); path box = (-0.5,-0.2)--(-0.5,0.2)--(0.5,0.2)--(0.5,-0.2)--cycle; draw(box); label("$+$",(0,0)); draw(shift(1,0)*box); label("$-$",(1,0)); draw(shift(2,0)*box); label("$+$",(2,0)); draw(shift(3,0)*box); label("$-$",(3,0)); draw(shift(0.5,0.4)*box); label("$-$",(0.5,0.4)); draw(shift(1.5,0.4)*box); label("$-$",(1.5,0.4)); draw(shift(2.5,0.4)*box); label("$-$",(2.5,0.4)); draw(shift(1,0.8)*box); label("$+$",(1,0.8)); draw(shift(2,0.8)*box); label("$+$",(2,0.8)); draw(shift(1.5,1.2)*box); label("$+$",(1.5,1.2)); [/asy] $\textbf{(A) } 2 \qquad \textbf{(B) } 4 \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 12 \qquad \textbf{(E) } 16$

Solve system of equations $$\begin{cases} x+\dfrac{x+y}{x^2+y^2}=1 \\ x+\dfrac{x-y}{x^2+y^2}=2 \end{cases}$$

A triangle $ABC$ is inscribed in a circle. Let $A_1$ be the point diametrically opposed to $A$, $A_0$ be the midpoint of the side $BC$ and $A_2$ be the point symmetric to $A_1$ with respect to $A_0$; the points $B_2$ and $C_2$ are defined in a similar way starting from $B$ and $C$. Prove that the three points $A_2$, $B_2$ and $C_2$ coincide. (A Jagubjanz)

Let $A, B \in M_2(\mathbb{R})$ such as $det(AB+BA)\leq 0$. Prove that $$det(A^2+B^2)\geq 0$$

[b]Q .[/b]Prove that $$\left(\sum_\text{cyc}(a-x)^4\right)\ +\ 2\left(\sum_\text{sym}x^3y\right)\ +\ 4\left(\sum_\text{cyc}x^2y^2\right)\ +\ 8xyza \geqslant \left(\sum_\text{cyc}(a-x)^2(a^2-x^2)\right)$$where $a=x+y+z$ and $x,y,z \in \mathbb{R}.$ [i]Proposed by srijonrick[/i]

If $0\le a\le b\le c$ real numbers, prove that $(a+3b)(b+4c)(c+2a)\ge 60abc$.

A segment $AB$ is fixed on the plane. Consider all acute-angled triangles with side $AB$. Find the locus of а) the vertices of their greatest angles, b) their incenters.

What is the value of ${ \sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+5\sqrt{1+\cdots }}}}}} $?

Let $k$ be the incircle of triangle $ABC$. It touches $AB=c, BC=a, AC=b$ at $C_1, A_1, B_1$, respectively. Suppose that $KC_1$ is a diameter of the incircle. Let $C_1A_1$ intersect $KB_1$ at $N$ and $C_1B_1$ intersect $KA_1$ at $M$. Find the length of $MN$.

Let $n$ be a fixed natural number. Find all $n$ tuples of natural pairwise distinct and coprime numbers like $a_1,a_2,\ldots,a_n$ such that for $1\leq i\leq n$ we have \[ a_1+a_2+\ldots+a_n|a_1^i+a_2^i+\ldots+a_n^i \]