Let $x$ and $y$ be real numbers satisfying the conditions $x + y = 4$ and $xy = 3$. Compute the value of $(x - y)^2$. A. $0$ B. $1$ C. $4$ D. $9$ E.$ -1$

Let $P(x)$ be a monic polynomial of degree $n$ with nonnegative coefficients and the free term equal to $1$. Prove that if all the roots of $P(x)$ are real, then $P(x) \geq (x+1)^{n}$ holds for every $x \geq 0$.

Let be a nonnegative integer $ n $ such that $ \sqrt n $ is not integer. Show that the function $$ f:\{ a+b\sqrt n | a,b\in\{ 0\}\cup\mathbb{N} , a^2-nb^2=1 \}\longrightarrow\{ 0\}\cup\mathbb{N} , f(x) =\lfloor x \rfloor $$ is injective and non-surjective.

Prove that for all real numbers $x$ of the interval $0 < x <\pi$ the inequality $$\sin x +\frac12 \sin 2x +\frac13 \sin 3x > 0$$ holds.

Prove that for two non-zero polynomials $ f(x,y),g(x,y)$ with real coefficients the system: \[ \left\{\begin{array}{c}f(x,y)\equal{}0\\ g(x,y)\equal{}0\end{array}\right.\] has finitely many solutions in $ \mathbb C^{2}$ if and only if $ f(x,y)$ and $ g(x,y)$ are coprime.

let $a_1,a_2,...a_n$ a sequence of real numbers such that $a_1+....+a_n=0$. define $b_i=a_1+a_2+....a_i$ for all $1 \leq i \leq n$ .suppose $b_i(a_{j+1}-a_{i+1}) \geq 0$ for all $1 \leq i \leq j \leq n-1$. Show that $$\max_{1 \leq l \leq n} |a_l| \geq \max_{1 \leq m \leq n} |b_m|$$

Find all functions $f:\mathbb{N}\to\mathbb{N}$ such that \[f(f(n))=n+2015\] where $n\in \mathbb{N}.$

Find all pairs of positive integers $(a, b)$ such that $$11ab \le a^3 - b^3 \le 12ab.$$

Do there exist infinitely many positive integers $a, b$ such that $$(a^2+1)(b^2+1)((a+b)^2+1)$$ is a perfect square? [i]Proposed Ivan Chan Guan Yu[/i]

For each integer $n \ge 2$, find all integer solutions of the following system of equations: \[x_1 = (x_2 + x_3 + x_4 + ... + x_n)^{2018}\] \[x_2 = (x_1 + x_3 + x_4 + ... + x_n)^{2018}\] \[\vdots\] \[x_n = (x_1 + x_2 + x_3 + ... + x_{n - 1})^{2018}\]

[b]p1.[/b] Steven has just learned about polynomials and he is struggling with the following problem: expand $(1-2x)^7$ as $a_0 +a_1x+...+a_7x^7$ . Help Steven solve this problem by telling him what $a_1 +a_2 +...+a_7$ is. [b]p2.[/b] Each element of the set ${2, 3, 4, ..., 100}$ is colored. A number has the same color as any divisor of it. What is the maximum number of colors? [b]p3.[/b] Fuchsia is selecting $24$ balls out of $3$ boxes. One box contains blue balls, one red balls and one yellow balls. They each have a hundred balls. It is required that she takes at least one ball from each box and that the numbers of balls selected from each box are distinct. In how many ways can she select the $24$ balls? [b]p4.[/b] Find the perfect square that can be written in the form $\overline{abcd} - \overline{dcba}$ where $a, b, c, d$ are non zero digits and $b < c$. $\overline{abcd}$ is the number in base $10$ with digits $a, b, c, d$ written in this order. [b]p5.[/b] Steven has $100$ boxes labeled from $ 1$ to $100$. Every box contains at most $10$ balls. The number of balls in boxes labeled with consecutive numbers differ by $ 1$. The boxes labeled $1,4,7,10,...,100$ have a total of $301$ balls. What is the maximum number of balls Steven can have? [b]p6.[/b] In acute $\vartriangle ABC$, $AB=4$. Let $D$ be the point on $BC$ such that $\angle BAD = \angle CAD$. Let $AD$ intersect the circumcircle of $\vartriangle ABC$ at $X$. Let $\Gamma$ be the circle through $D$ and $X$ that is tangent to $AB$ at $P$. If $AP = 6$, compute $AC$. [b]p7.[/b] Consider a $15\times 15$ square decomposed into unit squares. Consider a coloring of the vertices of the unit squares into two colors, red and blue such that there are $133$ red vertices. Out of these $133$, two vertices are vertices of the big square and $32$ of them are located on the sides of the big square. The sides of the unit squares are colored into three colors. If both endpoints of a side are colored red then the side is colored red. If both endpoints of a side are colored blue then the side is colored blue. Otherwise the side is colored green. If we have $196$ green sides, how many blue sides do we have? [b]p8.[/b] Carl has $10$ piles of rocks, each pile with a different number of rocks. He notices that he can redistribute the rocks in any pile to the other $9$ piles to make the other $9$ piles have the same number of rocks. What is the minimum number of rocks in the biggest pile? [b]p9.[/b] Suppose that Tony picks a random integer between $1$ and $6$ inclusive such that the probability that he picks a number is directly proportional to the the number itself. Danny picks a number between $1$ and $7$ inclusive using the same rule as Tony. What is the probability that Tony’s number is greater than Danny’s number? [b]p10.[/b] Mike wrote on the board the numbers $1, 2, ..., n$. At every step, he chooses two of these numbers, deletes them and replaces them with the least prime factor of their sum. He does this until he is left with the number $101$ on the board. What is the minimum value of $n$ for which this is possible? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all functions $f: \mathbb{Z} \to \mathbb{Z}$, such that $$f(x+y)((f(x) - f(y))^2+f(xy))=f(x^3)+f(y^3)$$ for all integers $x, y$.

Let $x, y, z$ be positive reals. Prove that $$\frac{xz}{x^2 + xy + y^2 + 6z^2} + \frac{zx}{z^2 + zy + y^2 + 6x^2} + \frac{xy}{x^2 + xz + z^2 + 6y^2} \le \frac13$$

Let $a, b$, and $c$ be the lengths of the sides of an arbitrary triangle, and let $\alpha,\beta$, and $\gamma$ be the radian measures of its corresponding angles. Prove that $$ \frac{\pi}{3}\le \frac{\alpha a +\beta b + \gamma c}{a+b+c} < \frac{\pi}{2}.$$ Suggest spatial analogues of this inequality.

[b]p1.[/b] Let the series an be defined as $a_1 = 1$ and $a_n =\sum^{n-1}_{i=1} a_ia_{n-i}$ for all positive integers $n$. Evaluate $\sum^{\infty}_{i=1} \left(\frac14\right)^ia_i$. [b]p2.[/b] $a, b, c$ and $d$ are distinct real numbers such that $$a + \frac{1}{b}= b +\frac{1}{c}= c +\frac{1}{d}= d +\frac{1}{a}= x$$ Find |x|. [b]p3.[/b] Find all ordered tuples $(w, x, y, z)$ of complex numbers satisfying $$x + y + z + xy + yz + zx + xyz = -w$$ $$y + z + w + yz + zw + wy + yzw = -x$$ $$z + w + x + zw + wx + xz + zwx = -y$$ $$w + x + y + wx + xy + yw + wxy = -z$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a,b$ be positive integers. Find, with proof, the maximum possible value of $a\lceil b\lambda \rceil - b \lfloor a \lambda \rfloor$ for irrational $\lambda$.

Find all arcs $\theta$ such that $\frac{1}{\sin ^2 \theta}, \frac{1}{\cos ^2 \theta} $ are integer numbers and roots of equation $$x^2-ax+a=0.$$

Let $A_n$ be the number of arranged n-tuples of natural numbers $(a_1,a_2…a_n)$, such that $\frac{1}{a_1} +\frac{1}{a_2} +...+\frac{1}{a_n} =1$. Find the parity of $A_{68}$.

Nuts are placed in boxes. The mean value of the number of nuts in a box is $10$, and the mean value of the squares of the numbers of nuts in the boxes is less than $1000$. Prove that at least $10\%$ of the boxes are not empty. (AY Belov)

The expression below has six empty boxes. Each box is to be filled in with a number from $1$ to $6$, where all six numbers are used exactly once, and then the expression is evaluated. What is the maximum possible final result that can be achieved? $$\dfrac{\frac{\square}{\square}+\frac{\square}{\square}}{\frac{\square}{\square}}$$

Solve the equation, $$\sqrt{x+5}+\sqrt{16-x^2}=x^2-25$$

Find the maximum and minimum values of $x^2 + 2y^2 + 3z^2$ for real $x, y, z$ satisfying $x^2 + y^2 + z^2 = 1$.

If $f$ is a polynomial, and $f(-2)=3$, $f(-1)=-3=f(1)$, $f(2)=6$, and $f(3)=5$, then what is the minimum possible degree of $f$?

If $a_i \ (i = 1, 2, \ldots, n)$ are distinct non-zero real numbers, prove that the equation \[\frac{a_1}{a_1-x} + \frac{a_2}{a_2-x}+\cdots+\frac{a_n}{a_n-x} = n\] has at least $n - 1$ real roots.

Two positive valued sequences $\{ a_{n}\}$ and $\{ b_{n}\}$ satisfy: (a): $a_{0}=1 \geq a_{1}$, $a_{n}(b_{n+1}+b_{n-1})=a_{n-1}b_{n-1}+a_{n+1}b_{n+1}$, $n \geq 1$. (b): $\sum_{i=1}^{n}b_{i}\leq n^{\frac{3}{2}}$, $n \geq 1$. Find the general term of $\{ a_{n}\}$.