Let $x, y, z$ be positive real numbers. Prove that: $\frac{x + 2y}{z + 2x + 3y}+\frac{y + 2z}{x + 2y + 3z}+\frac{z + 2x}{y + 2z + 3x} \le \frac{3}{2}$

Find the natural solutions of the equation $x^3 - y^3 = xy + 61$.

Let $A = \{a_1, \dots, a_{2024}\}$ be a set of $2024$ pairwise distinct real numbers. Assume that there exist positive integers $b_1, b_2,\dotsc,b_{2024}$ such that \[ a_1b_1 + a_2b_2 + \dots + a_{2024}b_{2024} = 0. \] Prove that one can choose $a_{2025}, a_{2026}, a_{2027}, \dots$ such that $a_k \in A$ for all $k \ge 2025$ and, for every positive integer $d$, there exist infinitely many positive integers $n$ satisfying \[ \sum_{k=1}^n a_k k^d = 0. \] [i]Daniel Zhu[/i]

Let $x_i,y_i$ be positive real numbers, $i = 1,2,...,n$. Prove that $$\left( \sum_{i=1}^n (x_i +y_i)^2\right)\left( \sum_{i=1}^n\frac{1}{x_iy_i}\right)\ge 4n^2$$

Determine all real $ x$ satisfying the equation \[ \sqrt[5]{x^3 \plus{} 2x} \equal{} \sqrt[3]{x^5\minus{}2x}.\] Odd roots for negative radicands shall be included in the discussion.

Find all he positive integers $x, y, z, t, w$, such as: $x+\frac{1}{y+\frac{1}{z+\frac{1}{t+\frac{1}{w}}}}=\frac{1998}{115}$

A game is played on a $n \times n$ chessboard. In the beginning Bars the cat occupies any cell according to his choice. The $d$ sparrows land on certain cells according to their choice (several sparrows may land in the same cell). Bars and the sparrows play in turns. In each turn of Bars, he moves to a cell adjacent by a side or a vertex (like a king in chess). In each turn of the sparrows, precisely one of the sparrows flies from its current cell to any other cell of his choice. The goal of Bars is to get to a cell containing a sparrow. Can Bars achieve his goal a) if $d=\lfloor \frac{3\cdot n^2}{25}\rfloor$, assuming $n$ is large enough? b) if $d=\lfloor \frac{3\cdot n^2}{19}\rfloor$, assuming $n$ is large enough? c) if $d=\lfloor \frac{3\cdot n^2}{14}\rfloor$, assuming $n$ is large enough?

A box contains 11 balls, numbered 1,2,3,....,11. If 6 balls are drawn simultaneously at random, what is the probability that the sum of the numbers on the balls drawn is odd? A. $\frac{100}{231}$ B. $\frac{115}{231}$ C. $\frac{1}{2}$ D. $\frac{118}{231}$ E. $\frac{6}{11}$

For some values of c, the equation $x^c + y^c = z^c$ can be illustrated geometrically. For example, the case $c = 2$ can be illustrated by a right-angled triangle. By this we mean that, x, y, z is a solution of the equation $x^2 + y^2 = z^2$ if and only if there exists a right-angled triangle with catheters $x$ and $y$ and hypotenuse $z$. In this problem we will look at the cases $c = -\frac{1}{2}$ and $c = - 1$. a) Let $x, y$ and $z$ be the radii of three circles intersecting each other and a line, as shown, in the figure. Show that, $x^{-\frac{1}{2}}+ y^{-\frac{1}{2}} = z^{-\frac{1}{2}}$ [img]https://cdn.artofproblemsolving.com/attachments/5/7/5315e33e1750a3a49ae11e1b5527311117ce70.png[/img] b) Draw a geometric figure that illustrates the case in a similar way, $c = - 1$. The figure must be able to be constructed with a compass and a ruler. Describe such a construction and prove that, in the figure, lines $x, y$ and $z$ satisfy $x^{-1}+ y^{-1} = z^{-1}$. (All positive solutions of this equation should be possible values for $x, y$, and $z$ on such a figure, but you don't have to prove that.)

Define the polynomials $P_0, P_1, P_2 \cdots$ by: \[ P_0(x)=x^3+213x^2-67x-2000 \] \[ P_n(x)=P_{n-1}(x-n), n \in N \] Find the coefficient of $x$ in $P_{21}(x)$.

Find all pairs of real numbers $(x,y)$ satisfying $$(2x+1)^2+y^2+(y-2x)^2=\frac13.$$

Convex equiangular hexagon $ABCDEF$ has $AB=CD=EF=1$ and $BC = DE = FA = 4$. Congruent and pairwise externally tangent circles $\gamma_1$, $\gamma_2$, and $\gamma_3$ are drawn such that $\gamma_1$ is tangent to side $\overline{AB}$ and side $\overline{BC}$, $\gamma_2$ is tangent to side $\overline{CD}$ and side $\overline{DE}$, and $\gamma_3$ is tangent to side $\overline{EF}$ and side $\overline{FA}$. Then the area of $\gamma_1$ can be expressed as $\frac{m\pi}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$. [i]Proposed by Sean Li[/i]

The perpendicular bisectors of the sides AB and BC of a triangle ABC meet the lines BC and AB at the points X and Z, respectively. The angle bisectors of the angles XAC and ZCA intersect at a point B'. Similarly, define two points C' and A'. Prove that the points A', B', C' lie on one line through the incenter I of triangle ABC. [i]Extension:[/i] Prove that the points A', B', C' lie on the line OI, where O is the circumcenter and I is the incenter of triangle ABC. Darij

A triangle in the coordinate plane has vertices $A(\log_21,\log_22)$, $B(\log_23,\log_24)$, and $C(\log_27,\log_28)$. What is the area of $\triangle ABC$? $ \textbf{(A) }\log_2\frac{\sqrt3}7\qquad \textbf{(B) }\log_2\frac3{\sqrt7}\qquad \textbf{(C) }\log_2\frac7{\sqrt3}\qquad \textbf{(D) }\log_2\frac{11}{\sqrt7}\qquad \textbf{(E) }\log_2\frac{11}{\sqrt3}\qquad $

For real numbers $ x_1, x_2, \dots, x_n $ and $ y_1, y_2, \dots, y_n $ , the inequalities hold $ x_1 \geq x_2 \geq \ldots \geq x_n> 0 $ and $$ y_1 \geq x_1, ~ y_1y_2 \geq x_1x_2, ~ \dots, ~ y_1y_2 \dots y_n \geq x_1x_2 \dots x_n. $$ Prove that $ ny_1 + (n-1) y_2 + \dots + y_n \geq x_1 + 2x_2 + \dots + nx_n $.

Given a positive integer $A$ written in decimal expansion: $(a_{n},a_{n-1}, \ldots, a_{0})$ and let $f(A)$ denote $\sum^{n}_{k=0} 2^{n-k}\cdot a_k$. Define $A_1=f(A), A_2=f(A_1)$. Prove that: [b]I.[/b] There exists positive integer $k$ for which $A_{k+1}=A_k$. [b]II.[/b] Find such $A_k$ for $19^{86}.$

Let be a natural number $ n. $ Prove that there is a polynomial $ P\in\mathbb{Z} [X,Y] $ such that $ a+b+c=0 $ implies $$ a^{2n+1}+b^{2n+1}+c^{2n+1}=abc\left( P(a,b)+P(b,c)+P(c,a)\right) $$ [i]Dan Brânzei[/i]

For some positive integer $n$, the number $110n^3$ has $110$ positive integer divisors, including $1$ and the number $110n^3$. How many positive integer divisors does the number $81n^4$ have? $\textbf{(A) }110 \qquad \textbf{(B) } 191 \qquad \textbf{(C) } 261 \qquad \textbf{(D) } 325 \qquad \textbf{(E) } 425$

A circle radius $320$ is tangent to the inside of a circle radius $1000$. The smaller circle is tangent to a diameter of the larger circle at a point $P$. How far is the point $P$ from the outside of the larger circle?

Let $p > 7$ be a prime which leaves residue 1 when divided by 6. Let $m=2^p-1,$ then prove $2^{m-1}-1$ can be divided by $127m$ without residue.

A convex polygon $A_1A_2 . . .A_n$ of $n$ sides and inscribed in a circle, has its sides that satisfy the inequalities $$A_nA_1 > A_1A_2 > A_2A_3 >...> A_{n-1}A_n$$ Show that its interior angles satisfy the inequalities $$\angle A_1 < \angle A_2 < \angle A_3 < ... < \angle A_{n-1}, \angle A_{n-1} > \angle A_n> \angle A_1.$$

Consider a composition of functions $\sin, \cos, \tan, \cot, \arcsin, \arccos, \arctan, \arccos$, applied to the number $1$. Each function may be applied arbitrarily many times and in any order. (ex: $\sin \cos \arcsin \cos \sin\cdots 1$). Can one obtain the number $2010$ in this way?

A weird calculator has a numerical display and only two buttons, $\boxed{D\sharp}$ and $\boxed{D\flat}$. The first button doubles the displayed number and then adds $1$. The second button doubles the displayed number and then subtracts $1$. For example, if the display is showing $5$, then pressing the $\boxed{D\sharp}$ produces $11$. If the display shows $5$ and we press $\boxed{D\flat}$, we get $9$. If the display shows $5$ and we press the sequence $\boxed{D\sharp}$, $\boxed{D\flat}$, $\boxed{D\sharp}$, $\boxed{D\sharp}$, we get a display of $87$. [list=i] [*] Suppose the initial displayed number is $1$. Give a sequence of exactly eight button presses that will result in a display of $313$. [*] Suppose the initial displayed number is $1$, and we then perform exactly eight button presses. Describe all the numbers that can possibly result? Prove your answer by explaining how all these numbers can be produced and that no other numbers can be produced. [/list]

Let $ABC$ be an acute triangle with $\displaystyle{AB<AC<BC}$ inscribed in circle $ \displaystyle{c(O,R)}$.The excircle $\displaystyle{(c_A)}$ has center $\displaystyle{I}$ and touches the sides $\displaystyle{BC,AC,AB}$ of the triangle $ABC$ at $\displaystyle{D,E,Z} $ respectively.$ \displaystyle{AI}$ cuts $\displaystyle{(c)}$ at point $M$ and the circumcircle $\displaystyle{(c_1)}$ of triangle $\displaystyle{AZE}$ cuts $\displaystyle{(c)}$ at $K$.The circumcircle $\displaystyle{(c_2)}$ of the triangle $\displaystyle{OKM}$ cuts $\displaystyle{(c_1)} $ at point $N$.Prove that the point of intersection of the lines $AN,KI$ lies on $ \displaystyle{(c)}$.

Let triangle $ABC$ satisfy $2BC = AB+AC$ and have incenter $I$ and circumcircle $\omega$. Let $D$ be the intersection of $AI$ and $\omega$ (with $A, D$ distinct). Prove that $I$ is the midpoint of $AD$.