Determine all polynomials $P (x)$ with real coefficients that apply $P (x^2) + 2P (x) = P (x)^2 + 2$.

find all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ st $f(f(m)+f(n))=m+n \,\forall m,n\in\mathbb{N}$ related: https://artofproblemsolving.com/community/c6h381298

On the coordinate plane, the parabola $y = x^2$ and the points $A(x_1, x_1^2)$, $B(x_2, x_2^2)$ are set such that $x_1=-998$, $x_2 =1999$ The segments $BX_1$, $AX_2$, $BX_3$, $AX_4$,..., $BX_{1997}$, $AX_{1998}$ and $X_k$ are constructed succesively with $(x_k,0)$, $1 \le k \le 1998$ and $x_3$, $x_4$,..., $x_{1998}$ are abscissas of the points of intersection of the parabola with segments $BX_1$, $AX_2$, $BX_3$, $AX_4$,..., $BX_{1997}$, $AX_{1998}$. Find the value $\frac{1}{x_{1999}}+\frac{1}{x_{2000}}$

Assume that $x, y \ge 0$. Show that $x^2 + y^2 + 1 \le \sqrt{(x^3 + y + 1)(y^3 + x + 1)}$.

What is the largest possible value of $|a_1 - 1| + |a_2-2|+...+ |a_n- n|$ where $a_1, a_2,..., a_n$ is a permutation of $1,2,..., n$?

Let $H_k =\Sigma_{i=1}^k \frac{1}{i}$ for all positive integers $k$. Find an closed-form expression for $\Sigma_{i=1}^k H_i$ in terms of $n$ and $H_n$.

Let $ f(n)$ be a function defined on the set of all positive integers and having its values in the same set. Suppose that $ f(f(n) \plus{} f(m)) \equal{} m \plus{} n$ for all positive integers $ n,m.$ Find the possible value for $ f(1988).$

Prove that there is a function $f$ from the set of all natural numbers to itself such that for any natural number $n$, $f(f(n)) = n^2$.

Let $P(x)$ and $Q(x)$ be non-constant integer-coefficient polynomials such that for any integer $x\in \mathbb Z$, there exists integer $y\in \mathbb Z$ such that $P(x)=Q(y)$. Prove that the degree of $Q$ divides the degree of $P$.

Tatjana imagined a polynomial $P(x)$ with nonnegative integer coefficients. Danica is trying to guess the polynomial. In each step, she chooses an integer $k$ and Tatjana tells her the value of $P(k)$. Find the smallest number of steps Danica needs in order to find the polynomial Tatjana imagined.

Prove that in any set of $2000$ distinct real numbers there exist two pairs $a>b$ and $c>d$ with $a \neq c$ or $b \neq d $, such that \[ \left| \frac{a-b}{c-d} - 1 \right|< \frac{1}{100000}. \]

A real number $ to $ is randomly and uniformly chosen from the $ [- 3,4] $ interval. What is the probability that all roots of the polynomial $ x ^ 3 + ax ^ 2 + ax + 1 $ are real?

Compute \[\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}}}}\]

[u]Round 5 [/u] [b]p13.[/b] Suppose $\vartriangle ABC$ is an isosceles triangle with $\overline{AB} = \overline{BC}$, and $X$ is a point in the interior of $\vartriangle ABC$. If $m \angle ABC = 94^o$, $m\angle ABX = 17^o$, and $m\angle BAX = 13^o$, then what is $m\angle BXC$ (in degrees)? [b]p14.[/b] Find the remainder when $\sum^{2019}_{n=1} 1 + 2n + 4n^2 + 8n^3$ is divided by $2019$. [b]p15.[/b] How many ways can you assign the integers $1$ through $10$ to the variables $a, b, c, d, e, f, g, h, i$, and $j$ in some order such that $a < b < c < d < e, f < g < h < i$, $a < g, b < h, c < i$, $f < b, g < c$, and $h < d$? [u]Round 6 [/u] [b]p16.[/b] Call an integer $n$ equi-powerful if $n$ and $n^2$ leave the same remainder when divided by 1320. How many integers between $1$ and $1320$ (inclusive) are equi-powerful? [b]p17.[/b] There exists a unique positive integer $j \le 10$ and unique positive integers $n_j$ , $n_{j+1}$, $...$, $n_{10}$ such that $$j \le n_j < n_{j+1} < ... < n_{10}$$ and $${n_{10} \choose 10}+ {n_9 \choose 9}+ ... + {n_j \choose j}= 2019.$$ Find $n_j + n_{j+1} + ... + n_{10}$. [b]p18.[/b] If $n$ is a randomly chosen integer between $1$ and $390$ (inclusive), what is the probability that $26n$ has more positive factors than $6n$? [u]Round 7[/u] [b]p19.[/b] Suppose $S$ is an $n$-element subset of $\{1, 2, 3, ..., 2019\}$. What is the largest possible value of $n$ such that for every $2 \le k \le n$, $k$ divides exactly $n - 1$ of the elements of $S$? [b]p20.[/b] For each positive integer $n$, let $f(n)$ be the fewest number of terms needed to write $n$ as a sum of factorials. For example, $f(28) = 3$ because $4! + 2! + 2! = 28$ and 28 cannot be written as the sum of fewer than $3$ factorials. Evaluate $f(1) + f(2) + ... + f(720)$. [b]p21.[/b] Evaluate $\sum_{n=1}^{\infty}\frac{\phi (n)}{101^n-1}$ , where $\phi (n)$ is the number of positive integers less than or equal to n that are relatively prime to $n$. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c4h2788993p24519281]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the largest possible integer $k$, such that the following statement is true: Let $2009$ arbitrary non-degenerated triangles be given. In every triangle the three sides are coloured, such that one is blue, one is red and one is white. Now, for every colour separately, let us sort the lengths of the sides. We obtain \[ \left. \begin{array}{rcl} & b_1 \leq b_2\leq\ldots\leq b_{2009} & \textrm{the lengths of the blue sides }\\ & r_1 \leq r_2\leq\ldots\leq r_{2009} & \textrm{the lengths of the red sides }\\ \textrm{and } & w_1 \leq w_2\leq\ldots\leq w_{2009} & \textrm{the lengths of the white sides }\\ \end{array}\right.\] Then there exist $k$ indices $j$ such that we can form a non-degenerated triangle with side lengths $b_j$, $r_j$, $w_j$. [i]Proposed by Michal Rolinek, Czech Republic[/i]

$\boxed{A2}$ Find the maximum value of $z+x$ if $x,y,z$ are satisfying the given conditions.$x^2+y^2=4$ $z^2+t^2=9$ $xt+yz\geq 6$

Let $I$ and $J$ be intervals. Let $\varphi,\psi:I\to\mathbb{R}$ be strictly increasing continuous functions and let $\Phi,\Psi:J\to\mathbb{R}$ be continuous functions. Suppose that $\varphi(x)+\psi(x)=x$ and $\Phi(u)+\Psi(u)=u$ holds for all $x\in I$ and $u\in J$. Show that if $f:I\to J$ is a continuous solution of the functional inequality $$f\big(\varphi(x)+\psi(y)\big)\le \Phi\big(f(x)\big)+\Psi\big(f(y)\big)\qquad (x,y\in I),$$then $\Phi\circ f\circ \varphi^{-1}$ and $\Psi\circ f\circ \psi^{-1}$ are convex functions.

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]

Let $\alpha,\beta,\gamma$ measures of angles of opposite to the sides of triangle with measures $a,b,c$ respectively.Prove that $$2(cos^2\alpha+cos^2\beta+cos^2\gamma)\geq \frac{a^2}{b^2+c^2}+\frac{b^2}{a^2+c^2}+\frac{c^2}{a^2+b^2}$$

Denote by $m(a,b)$ the arithmetic mean of positive real numbers $a,b$. Given a positive real function $g$ having positive derivatives of the first and second order, define $\mu (a,b)$ the mean value of $a$ and $b$ with respect to $g$ by $2g(\mu (a,b)) = g(a)+g(b)$. Decide which of the two mean values $m$ and $\mu$ is larger.

Let $k\ge2$ be an integer. Find the smallest integer $n \ge k+1$ with the property that there exists a set of $n$ distinct real numbers such that each of its elements can be written as a sum of $k$ other distinct elements of the set.

Let $ P $ be a point in the interior of a triangle $ ABC $, and let $ D, E, F $ be the point of intersection of the line $ AP $ and the side $ BC $ of the triangle, of the line $ BP $ and the side $ CA $, and of the line $ CP $ and the side $ AB $, respectively. Prove that the area of the triangle $ ABC $ must be $ 6 $ if the area of each of the triangles $ PFA, PDB $ and $ PEC $ is $ 1 $.

Let $f, g$ be functions $\mathbb{R} \rightarrow \mathbb{R}$ such that for all reals $x,y$, $$f(g(x) + y) = g(x + y)$$ Prove that either $f$ is the identity function or $g$ is periodic. [i]Proposed by Pranjal Srivastava[/i]

Evaluate $$2023 \cdot \frac{2023^6 +27}{(2023^2 +3)(2024^3 -1)}-2023^2.$$ [i]Proposed by Evin Liang[/i]

For integers $a$ and $b$, $a + b$ is a root of $x^2 + ax + b = 0$. Compute the smallest possible value of $ab$.