The function $ƒ$ is defined at $[0,1]$, and $f\{f(x)\} = ƒ(x)$. $\exists _{c\in [0,1]} \left[f(c) =\frac12 \right]$ Determine $f\left(\frac12 \right).$ $\forall _{t\in [0,1]}\exists _{s\in [0,1]}[f(s) = t]$. Determine $f$. Prove that the function $g$, with $g(x) = x$,$0 \le x \le k$, $g(x) = k$, $k \le x \le 1$ satisfies the relation $g\{g(x)\} = g(x)$.

Prove that: the polynomial$$(x(x+1)(x+2)(x+3))^{2^{2023}}+1$$is irreducible in $\mathbb{Q}[x].$

Find all prime numbers $p$ and $q$ such that $2^2+p^2+q^2$ is also prime. Please remember to hide your solution. (by using the hide tags of course.. I don't literally mean that you should hide it :ninja: )

For some $k > 0$ the lines $50x + ky = 1240$ and $ky = 8x + 544$ intersect at right angles at the point $(m,n)$. Find $m + n$.

Find all functions $u:R\rightarrow{R}$ for which there exists a strictly monotonic function $f:R\rightarrow{R}$ such that $f(x+y)=f(x)u(y)+f(y)$ for all $x,y\in{\mathbb{R}}$

Let $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ be unit vectors in $R^3$. Prove that $$\sqrt{1-\overrightarrow{a}\cdot\overrightarrow{b}}\le \sqrt{1-\overrightarrow{a}\cdot\overrightarrow{c}}+\sqrt{1-\overrightarrow{c}\cdot\overrightarrow{b}}$$ (A.Mirotin)

If $a, b, c$ are positive integers such that \[0 < a^{2}+b^{2}-abc \le c,\] show that $a^{2}+b^{2}-abc$ is a perfect square.

Find all functions $f: R \rightarrow R$ such that \[f(x+yf(x))+f(xf(y)-y)=f(x)-f(y)+2xy\] for all $x,y \in R$

Find the maximum possible value of $k$ for which there exist distinct reals $x_1,x_2,\ldots ,x_k $ greater than $1$ such that for all $1 \leq i, j \leq k$, $$x_i^{\lfloor x_j \rfloor }= x_j^{\lfloor x_i\rfloor}.$$ [i]Proposed by Morteza Saghafian[/i]

[b]p1.[/b] A band of pirates unloaded some number of treasure chests from their ship. The number of pirates was between $60$ and $69$ (inclusive). Each pirate handled exactly $11$ treasure chests, and each treasure chest was handled by exactly $7$ pirates. Exactly how many treasure chests were there? Show that your answer is the only solution. [b]p2.[/b] Let $a$ and $b$ be the lengths of the legs of a right triangle, let $c$ be the length of the hypotenuse, and let $h$ be the length of the altitude drawn from the vertex of the right angle to the hypotenuse. Prove that $c+h>a+b$. [b]p3.[/b] Prove that $$\frac{1}{70}< \frac{1}{2} \frac{3}{4} \frac{5}{6} ... \frac{2001}{2002} < \frac{1}{40}$$ [b]p4.[/b] Given a positive integer $a_1$ we form a sequence $a_1 , a_2 , a _3,...$ as follows: $a_2$ is obtained from $a_1$ by adding together the digits of $a_1$ raised to the $2001$-st power; $a_3$ is obtained from $a_2$ using the same rule, and so on. For example, if $a_1 =25$, then $a_2 =2^{2001}+5^{2001}$, which is a $1399$-digit number containing $106$ $0$'s, $150$ $1$'s, 4124$ 42$'s, $157$ $3$'s, $148$ $4$'s, $141$ $5$'s, $128$ $6$'s, $1504 47$'s, $152$ $8$'s, $143$ $9$'s. So $a_3 = 106 \times 0^{2001}+ 150 \times 1^{2001}+ 124 \times 2^{2001}+ 157 \times 3^{2001}+ ...+ 143 \times 9^{2001}$ which is a $1912$-digit number, and so forth. Prove that if any positive integer $a_1$ is chosen to start the sequence, then there is a positive integer $M$ (which depends on $a_1$ ) that is so large that $a_n < M$ for all $n=1,2,3,...$ [b]p5.[/b] Let $P(x)$ be a polynomial with integer coefficients. Suppose that there are integers $a$, $b$, and $c$ such that $P(a)=0$, $P(b)=1$, and $P(c)=2$. Prove that there is at most one integer $n$ such that $P(n)=4$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Real numbers $a, b, c$ are such that $$a^2+c-bc = b^2+a-ca = c^2+b-ab$$ Does it follow that $a=b=c$? [i]Proposed by Mykhailo Shtandenko[/i]

Show that $ \pi$ is irrational.

For $n \ge 3$, let $S=a_1+a_2+\cdots+a_n$ and $T=b_1b_2\cdots b_n$ for positive real numbers $a_1,a_2,\ldots,a_n, b_1,b_2 ,\ldots,b_n$, where the numbers $b_i$ are pairwise distinct. (a) Find the number of distinct real zeroes of the polynomial \[f(x)=(x-b_1)(x-b_2)\cdots(x-b_n)\sum_{j=1}^n \frac{a_j}{x-b_j}\] (b) Prove the inequality \[\frac1{n-1}\sum_{j=1}^n \left(1-\frac{a_j}{S}\right)b_j > \left(\frac{T}{S}\sum_{j=1}^{n} \frac{a_j}{b_j}\right)^{\frac1{n-1}}\]

Suppose $x_1,x_2,...,x_{60}\in [-1,1]$ , find the maximum of $$ \sum_{i=1}^{60}x_i^2(x_{i+1}-x_{i-1}),$$ where $x_{i+60}=x_i$.

[b]p1.[/b] Let $A$ be the point $(-1, 0)$, $B$ be the point $(0, 1)$ and $C$ be the point $(1, 0)$ on the $xy$-plane. Assume that $P(x, y)$ is a point on the $xy$-plane that satisfies the following condition $$d_1 \cdot d_2 = (d_3)^2,$$ where $d_1$ is the distance from $P$ to the line $AB$, $d_2$ is the distance from $P$ to the line $BC$, and $d_3$ is the distance from $P$ to the line $AC$. Find the equation(s) that must be satisfied by the point $P(x, y)$. [b]p2.[/b] On Day $1$, Peter sends an email to a female friend and a male friend with the following instructions: $\bullet$ If you’re a male, send this email to $2$ female friends tomorrow, including the instructions. $\bullet$ If you’re a female, send this email to $1$ male friend tomorrow, including the instructions. Assuming that everyone checks their email daily and follows the instructions, how many emails will be sent from Day $1$ to Day $365$ (inclusive)? [b]p3.[/b] For every rational number $\frac{a}{b}$ in the interval $(0, 1]$, consider the interval of length $\frac{1}{2b^2}$ with $\frac{a}{b}$ as the center, that is, the interval $\left( \frac{a}{b}- \frac{1}{2b^2}, \frac{a}{b}+\frac{1}{2b^2}\right)$ . Show that $\frac{\sqrt2}{2}$ is not contained in any of these intervals. [b]p4.[/b] Let $a$ and $b$ be real numbers such that $0 < b < a < 1$ with the property that $$\log_a x + \log_b x = 4 \log_{ab} x - \left(\log_b (ab^{-1} - 1)\right)\left(\log_a (ab^{-1} - 1) + 2 log_a ab^{-1} \right)$$ for some positive real number $x \ne 1$. Find the value of $\frac{a}{b}$. [b]p5.[/b] Find the largest positive constant $\lambda$ such that $$\lambda a^2 b^2 (a - b)^2 \le (a^2 - ab + b^2)^3$$ is true for all real numbers $a$ and $b$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $\{ a_n \}_{n=0}^{\infty}$, $\{ b_n \}_{n=0}^{\infty}$, and $\{ c_n \}_{n=0}^{\infty}$ be sequences of real numbers such that for all $k\geq 1$, \begin{align*} a_k&=\left\lfloor \sqrt{2}+\frac{k-1}{2024} \right\rfloor+a_{k-1} \\ b_k+c_k&=1 \\ a_{k-1}b_k&=a_kc_k. \end{align*} Suppose that $a_0=1$, $b_0=2$, and $c_0=3$. Given that $\sqrt2\approx1.4142$, compute \[ \sum_{k=1}^{2024}(a_kb_k-a_{k-1}c_k). \]

Find a nonzero monic polynomial $P(x)$ with integer coefficients and minimal degree such that $P(1-\sqrt[3]2+\sqrt[3]4)=0$. (A polynomial is called $\textit{monic}$ if its leading coefficient is $1$.)

The football tournament was played in one round. $3$ points were given for a win, $1$ point for a draw, and $0$ points for a loss. Could it be that the first place team under the old scoring system (win - $2$ points, draw - $1$ point, loss - $0$) would be last?

Let $P(x)$ be a polynomial with real coefficients. Prove that there exist positive integers $n$ and $k$ such that $k$ has $n$ digits and more than $P(n)$ positive divisors.

Find the probability such that when a polynomial in $\mathbb Z_{2027}[x]$ having degree at most $2026$ is chosen uniformly at random, $$x^{2027}-x | P^{k}(x) - x \iff 2021 | k $$ (note that $2027$ is prime). Here $P^k(x)$ denotes $P$ composed with itself $k$ times. [i]Proposed by Grant Yu[/i]

Find all functions $ f:\mathbb{N}\longrightarrow\mathbb{N} $ that satisfy the following relation: $$ f(x)^2+y\vdots x^2+f(y) ,\quad\forall x,y\in\mathbb{N} . $$

Find the maximum of the expression $$||...||x_1 - x_2|- x_3| -... | - x_{2023}|,$$ where $x_1,x_2,..., x_{2023}$ are distinct natural numbers between $1$ and $2023$.

Find all strictly increasing functions $f : Z \to R$ such that for any $m, n \in Z$ there exists a $k \in Z$ such that $f(k) = f(m) - f(n)$. Nguyễn Duy Thái Sơn

The sum of all the roots of $ 4x^3\minus{}8x^2\minus{}63x\minus{}9\equal{}0$ is: $ \textbf{(A)}\ 8 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ \minus{}8 \qquad \textbf{(D)}\ \minus{}2 \qquad \textbf{(E)}\ 0$

$a, b, c$ are real positive numbers for which $a+b+c=3$. Prove that $a^{12}+b^{12}+c^{12}+8(ab+bc+ca) \geq 27$