Prove that for any real numbers $x, y, z$ at least one of numbers $x^2 + y + \frac{1}{4}, y^2 + z + \frac{1}{4}, z^2 + x + \frac{1}{4}$ is nonnegative. [i]Proposed by Oleksii Masalitin[/i]

Find, with proof, all real numbers $x$ satisfying $x = 2\left( 2 \left( 2\left( 2\left( 2x-1 \right)-1 \right)-1 \right)-1 \right)-1$. [i]Proposed by Evan Chen[/i]

Find the largest prime factor of $45^5-1.$

Find all quadruples of real numbers $(a,b,c,d)$ satisfying the system of equations \[\begin{cases}(b+c+d)^{2010}=3a\\ (a+c+d)^{2010}=3b\\ (a+b+d)^{2010}=3c\\ (a+b+c)^{2010}=3d\end{cases}\]

Let $a_d$ be the number of non-negative integer solutions $(a, b)$ to $a + b = d$ where $a \equiv b$ (mod $n$) for a fixed $n \in Z^+$. Consider the generating function $M(t) = a_0 + a_1t + a_2t^2 + ...$ Consider $$P(n) = \lim_{t\to 1} \left( nM(t) - \frac{1}{(1 - t)^2} \right).$$ Then $P(n)$, $n \in Z^+$ is a polynomial in $n$, so we can extend its domain to include all real numbers while having it remain a polynomial. Find $P(0)$.

Let $\mathbb{R}^+$ be the set of all positive real numbers. Find all functions $f: \mathbb{R}^+\to\mathbb{R}^+$ such that \[ \frac{f(a)}{1+a+ca}+\frac{f(b)}{1+b+ab}+\frac{f(c)}{1+c+bc} = 1 \] for all $a,b,c \in \mathbb{R}^+$ that satisfy $abc=1$.

Find all real numbers $x_{1}, x_{2}, \cdots, x_{n}$ satisfying,$$\sqrt{x_{1}-1^{2}}+2 \sqrt{x_{2}-2^{2}}+\cdots+n \sqrt{x_{n}-n^{2}}=\frac{1}{2}\left(x_{1}+x_{2}+\cdots+x_{n}\right)$$

Prove that for any positive numbers $x,y$ it holds that $x^xy^y \ge x^yy^x$.

[b]p1.[/b] There are $32$ balls in a box: $6$ are blue, $8$ are red, $4$ are yellow, and $14$ are brown. If I pull out three balls at once, what is the probability that none of them are brown? [b]p2.[/b] Circles $A$ and $B$ are concentric, and the area of circle $A$ is exactly $20\%$ of the area of circle $B$. The circumference of circle $B$ is $10$. A square is inscribed in circle $A$. What is the area of that square? [b]p3.[/b] If $x^2 +y^2 = 1$ and $x, y \in R$, let $q$ be the largest possible value of $x+y$ and $p$ be the smallest possible value of $x + y$. Compute $pq$. [b]p4.[/b] Yizheng and Jennifer are playing a game of ping-pong. Ping-pong is played in a series of consecutive matches, where the winner of a match is given one point. In the scoring system that Yizheng and Jennifer use, if one person reaches $11$ points before the other person can reach $10$ points, then the person who reached $11$ points wins. If instead the score ends up being tied $10$-to-$10$, then the game will continue indefinitely until one person’s score is two more than the other person’s score, at which point the person with the higher score wins. The probability that Jennifer wins any one match is $70\%$ and the score is currently at $9$-to-$9$. What is the probability that Yizheng wins the game? [b]p5.[/b] The squares on an $8\times 8$ chessboard are numbered left-to-right and then from top-to-bottom (so that the top-left square is $\#1$, the top-right square is $\#8$, and the bottom-right square is $\#64$). $1$ grain of wheat is placed on square $\#1$, $2$ grains on square $\#2$, $4$ grains on square $\#3$, and so on, doubling each time until every square of the chessboard has some number of grains of wheat on it. What fraction of the grains of wheat on the chessboard are on the rightmost column? [b]p6.[/b] Let $f$ be any function that has the following property: For all real numbers $x$ other than $0$ and $1$, $$f \left( 1 - \frac{1}{x} \right) + 2f \left( \frac{1}{1 - x}\right)+ 3f(x) = x^2.$$ Compute $f(2)$. [b]p7.[/b] Find all solutions of: $$(x^2 + 7x + 6)^2 + 7(x^2 + 7x + 6)+ 6 = x.$$ [b]p8.[/b] Let $\vartriangle ABC$ be a triangle where $AB = 25$ and $AC = 29$. $C_1$ is a circle that has $AB$ as a diameter and $C_2$ is a circle that has $BC$ as a diameter. $D$ is a point on $C_1$ so that $BD = 15$ and $CD = 21$. $C_1$ and $C_2$ clearly intersect at $B$; let $E$ be the other point where $C_1$ and $C_2$ intersect. Find all possible values of $ED$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that for all real numbers $a,b,$ and $c$: (i) If $a+b+c\ge 0$ then $f(a^3)+f(b^3)+f(c^3)\ge 3f(abc).$ (ii) If $a+b+c\le 0$ then $f(a^3)+f(b^3)+f(c^3)\le 3f(abc).$ [i]Proposed by Ashwin Sah[/i]

For a real coefficient cubic polynomial $f(x)=ax^3+bx^2+cx+d$, denote three roots of the equation $f(x)=0$ by $\alpha,\beta,\gamma$. Prove that the three roots $\alpha,\beta,\gamma$ are distinct real numbers iff the real symmetric matrix $$\begin{pmatrix} 3 & p_1 & p_2 \\ p_1 & p_2 & p_3 \\ p_2 & p_3 & p_4 \end{pmatrix},\quad p_i = \alpha^i + \beta^i + \gamma^i$$ is positive definite.

Given $0 < a \le x_1\le x_2\le ... \le x_n \le b$. Prove that $$(x_1+x_2+...+x_n)\left ( \frac{1}{x_1}+ \frac{1}{x_2}+...+ \frac{1}{x_n}\right)\le \frac{(a+b)^2}{4ab}n^2$$

For each positive integer $k$, let $a_k$ be the greatest integer not exceeding $\sqrt{k}$ and let $b_k$ be the greatest integer not exceeding $\sqrt[3]{k}$. Calculate $$\sum_{k=1}^{2003} (a_k-b_k).$$

a) Does there exist a real number $x$ such that $x+\sqrt{3}$ and $x^2+\sqrt{3}$ are both rationals? b) Does there exist a real number $y$ such that $y+\sqrt{3}$ and $y^3+\sqrt{3}$ are both rationals?

Let $a_1, a_2,...,a_{2018}$ be a sequence of numbers such that all its elements are elements of a set $\{-1,1\}$. Sum $$S=\sum \limits_{1 \leq i < j \leq 2018} a_i a_j$$ can be negative and can also be positive. Find the minimal value of this sum

Let $a$ be a positive real number and let $b= \sqrt[3] {a+ \sqrt {a^{2}+1}} + \sqrt[3] {a- \sqrt {a^{2}+1}}$. Prove that $b$ is a positive integer if, and only if, $a$ is a positive integer of the form $\frac{1}{2} n(n^{2}+3)$, for some positive integer $n$.

[b]p1.[/b] Sarah needed a ride home to the farm from town. She telephoned for her father to come and get her with the pickup truck. Being eager to get home, she began walking toward the farm as soon as she hung up the phone. However, her father had to finish milking the cows, so could not leave to get her until fifteen minutes after she called. He drove rapidly to make up for lost time. They met on the road, turned right around and drove back to the farm at two-thirds of the speed her father drove coming. They got to the farm two hours after she had called. She walked and he drove both ways at constant rates of speed. How many minutes did she spend walking? [b]p2.[/b] Let $A = (a,b)$ be any point in a coordinate plane distinct from the origin $O$. Let $M$ be the midpoint of $OA$, and let $P$ be a point such that $MP$ is perpendicular to $OA$ and the lengths $\overline{MP}$ and $\overline{OM}$ are equal. Determine the coordinates $(x,y)$ of $P$ in terms of $a$ and $b$. Give all possible solutions. [b]p3.[/b] Determine the exact sum of the series $$\frac{1}{1 \cdot 2\cdot 3} + \frac{1}{2\cdot 3\cdot 4} + \frac{1}{3\cdot 4\cdot 5} + ... + \frac{1}{98\cdot 99\cdot 100}$$ [b]p4.[/b] A six pound weight is attached to a four foot nylon cord that is looped over two pegs in the manner shown in the drawing. At $B$ the cord passes through a small loop in its end. The two pegs $A$ and $C$ are one foot apart and are on the same level. When the weight is released the system obtains an equilibrium position. Determine angle $ABC$ for this equilibrium position, and verify your answer. (Your verification should assume that friction and the weight of the cord are both negligible, and that the tension throughout the cord is a constant six pounds.) [img]https://cdn.artofproblemsolving.com/attachments/a/1/620c59e678185f01ca8743c39423234d5ba04d.png[/img] [b]p5.[/b] The four corners of a rectangle have the property that when they are taken three at a time, they determine triangles all of which have the same perimeter. We will consider whether a set of five points can have this property. Let $S = \{p_1, p_2, p_3, p_4, p_5\}$ be a set of five points. For each $i$ and $j$, let $d_{ij}$ denote the distance from $p_i$ to $p_j$. Suppose that $S$ has the property that all triangles with vertices in $S$ have the same perimeter. (a) Prove that $d$ must be the same for every pair $(i,j)$ with $i \ne j$. (b) Can such a five-element set be found in three dimensional space? Justify your answer. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$(a_i),(b_i)$ are nonconstant arithmetic and geometric progressions. $a_1=b_1,a_2/b_2=2,a_4/b_4=8$ Find $a_3/b_3$.

Let $p$ be a prime number gretater then $3$. What is the number of pairs $(m, n)$ of integers with $0 <m <n <p$, for which the polynomial $x^p + px^n + px^m +1$ is not a product of two non-constant polynomials with integer coefficients can be written?

For an integer $n > 0$, denote by $\mathcal F(n)$ the set of integers $m > 0$ for which the polynomial $p(x) = x^2 + mx + n$ has an integer root. [list=a] [*] Let $S$ denote the set of integers $n > 0$ for which $\mathcal F(n)$ contains two consecutive integers. Show that $S$ is infinite but \[ \sum_{n \in S} \frac 1n \le 1. \] [*] Prove that there are infinitely many positive integers $n$ such that $\mathcal F(n)$ contains three consecutive integers. [/list] [i]Ivan Borsenco[/i]

Find all polynomials $P(x)$ with real coefficients that satisfy the relation $$1+P(x)=\frac{P(x-1)+P(x+1)}2.$$

A function $f : \Bbb{N}_0 \to \Bbb{R}$ satisfies $f(1) = 3$ and \[f(m + n) + f(m - n) - m + n - 1 =\frac{f(2m) + f(2n)}{2},\] for any non-negative integers $m$ and $n$ with $m \geq n.$ Find all such functions $f$.

Let $p$ be an odd prime number. Let $S$ be the sum of all primitive roots modulo $p$. Show that if $p-1$ isn't squarefree (i. e., if there exist integers $k$ and $m$ with $k>1$ and $p-1=k^2m$), then $S \equiv 0 \mod p$. If not, then what is $S$ congruent to $\mod p$ ?

Assume : $x,y,z>0$ , $ x^2+y^2+z^2 = 3 $ . Prove the following inequality : $${\frac{x^{2009}-2008(x-1)}{y+z}+\frac{y^{2009}-2008(y-1)}{x+z}+\frac{z^{2009}-2008(z-1)}{x+y}\ge\frac{1}{2}(x+y+z)}$$

Find all the functions $f:R \to R$ satisfying $$(x + y)(f (x)-f (y)) = f (x^2) - f (y^2),\, \forall x, y \in R$$