Find all positive integers $n \geqslant 2$ for which there exist $n$ real numbers $a_1<\cdots<a_n$ and a real number $r>0$ such that the $\tfrac{1}{2}n(n-1)$ differences $a_j-a_i$ for $1 \leqslant i<j \leqslant n$ are equal, in some order, to the numbers $r^1,r^2,\ldots,r^{\frac{1}{2}n(n-1)}$.

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(x)f(y)=f(x+y)+xy$ for all $x,y\in \mathbb{R}$.

In a certain city, age is reckoned in terms of real numbers rather than integers. Every two citizens $x$ and $x'$ either know each other or do not know each other. Moreover, if they do not, then there exists a chain of citizens $x = x_0, x_1, \ldots, x_n = x'$ for some integer $n \geq 2$ such that $ x_{i-1}$ and $x_i$ know each other. In a census, all male citizens declare their ages, and there is at least one male citizen. Each female citizen provides only the information that her age is the average of the ages of all the citizens she knows. Prove that this is enough to determine uniquely the ages of all the female citizens.

The sequence an is defined by $a_0 = 4, a_1 = 1$ and the recurrence formula $a_{n+1} = a_n + 6a_{n-1}$. The sequence $b_n$ is given by \[b_n=\sum_{k=0}^n \binom nk a_k.\] Find the coefficients $\alpha,\beta$  so that $b_n$ satisfies the recurrence formula $b_{n+1} = \alpha b_n + \beta b_{n-1}$. Find the explicit form of $b_n$.

Let $b_0, b_1, b_2, \ldots$ be a sequence of pairwise distinct nonnegative integers such that $b_0=0$ and $b_n<2n$ for all positive integers $n$. Prove that for each nonnegative integer $m$ there exist nonnegative integers $k, \ell$ such that \begin{align*} b_k+b_{\ell}=m. \end{align*}

Solve a set of inequalities in the domain of integer numbers: $$3x^2 +2yz \le 1+y^2$$ $$3y^2 +2zx \le 1+z^2$$ $$3z^2 +2xy \le 1+x^2$$

Let $b \geq 2$ be a positive integer. (a) Show that for an integer $N$, written in base $b$, to be equal to the sum of the squares of its digits, it is necessary either that $N = 1$ or that $N$ have only two digits. (b) Give a complete list of all integers not exceeding $50$ that, relative to some base $b$, are equal to the sum of the squares of their digits. (c) Show that for any base b the number of two-digit integers that are equal to the sum of the squares of their digits is even. (d) Show that for any odd base $b$ there is an integer other than $1$ that is equal to the sum of the squares of its digits.

For arbitrary pairwise distinct positive real numbers $a, b, c$, prove the inequality $$\frac{(a^2- b^2)^3 + (b^2-c^2)^3+(c^2-a^2)^3}{(a- b)^3 + (b-c)^3+(c-a)^3}> 8abc$$

Determine the smallest possible value of the sum $S (a, b, c) = \frac{ab}{c}+\frac{bc}{a}+\frac{ca}{b}$ where $a, b, c$ are three positive real numbers with $a^2 + b^2 + c^2 = 1$

In a function $f (x) = (x^2 + ax + b )/ (x^2 + cx + d)$ , the quadratics $x^2 + ax + b$ and $x^2 + cx + d$ have no common roots. Prove that the next two statements are equivalent: (i) there is a numerical interval without any values of $f(x)$ , (ii) $f(x)$ can be represented in the form $f (x) = f_1 (f_2( ... f_{n-1} (f_n (x))... ))$ where each of the functions $f_j$ is o f one of the three forms $k_j x + b_j, 1/x, x^2$ . (A Kanel)

Let $c_1,c_2,\ldots ,c_n,b_1,b_2,\ldots ,b_n$ $(n\geq 2)$ be positive real numbers. Prove that the equation \[ \sum_{i=1}^nc_i\sqrt{x_i-b_i}=\frac{1}{2}\sum_{i=1}^nx_i\] has a unique solution $(x_1,\ldots ,x_n)$ if and only if $\sum_{i=1}^nc_i^2=\sum_{i=1}^nb_i$.

Given is a nonzero real number $\alpha$. Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that $$f(f(x+y))=f(x+y)+f(x)f(y)+\alpha xy$$ for all $x, y \in \mathbb{R}$.

Find all couples of polynomials $(P,Q)$ with real coefficients, such that for infinitely many $x\in\mathbb R$ the condition \[ \frac{P(x)}{Q(x)}-\frac{P(x+1)}{Q(x+1)}=\frac{1}{x(x+2)}\] Holds. [i] Nikolai Nikolov, Oleg Mushkarov[/i]

Find all non-zero real numbers $a, b, c$ such that the following polynomial has four (not necessarily distinct) positive real roots. $$P(x) = ax^4 - 8ax^3 + bx^2 - 32cx + 16c$$

A positive integer $n$ is [i]magical[/i] if $\lfloor \sqrt{\lceil \sqrt{n} \rceil} \rfloor=\lceil \sqrt{\lfloor \sqrt{n} \rfloor} \rceil$. Find the number of magical integers between $1$ and $10,000$ inclusive.

Let $a, b, c$ and $d$ be real numbers with $a < b < c < d$. Sort the numbers $x = a \cdot b + c \cdot d, y = b \cdot c + a \cdot d$ and $z = c \cdot a + b \cdot d$ in ascending\order and prove the correctness of your result. (R. Henner, Vienna)

Find all functions $ f: \mathbb{R}^{ \plus{} }\to\mathbb{R}^{ \plus{} }$ satisfying $ f\left(x \plus{} f\left(y\right)\right) \equal{} f\left(x \plus{} y\right) \plus{} f\left(y\right)$ for all pairs of positive reals $ x$ and $ y$. Here, $ \mathbb{R}^{ \plus{} }$ denotes the set of all positive reals. [i]Proposed by Paisan Nakmahachalasint, Thailand[/i]

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

Given a prime $p$, show that \[\left(1+p\sum_{k=1}^{p-1}k^{-1}\right)^2 \equiv 1-p^2\sum_{k=1}^{p-1}k^{-2} \pmod{p^4}.\] [i]Timothy Chu.[/i]

Suppose that $ a$ and $ b$ are real numbers such that the quadratic polynomial $ f(x)\equal{}x^2\plus{}ax\plus{}b$ has no nonnegative real roots. Prove that there exist two polynomials $ g,h$ whose coefficients are nonnegative real numbers such that: $ f(x)\equal{}\frac{g(x)}{h(x)}$ for all real numbers $ x$.

[b]p1.[/b] Prove that $x = 2$ is the only real number satisfying $3^x + 4^x = 5^x$. [b]p2.[/b] Show that $\sqrt{9 + 4\sqrt5} -\sqrt{9 - 4\sqrt5}$ is an integer. [b]p3.[/b] Two players $A$ and $B$ play a game on a round table. Each time they take turn placing a round coin on the table. The coin has a uniform size, and this size is at least $10$ times smaller than the table size. They cannot place the coin on top of any part of other coins, and the whole coin must be on the table. If a player cannot place a coin, he loses. Suppose $A$ starts first. If both of them plan their moves wisely, there will be one person who will always win. Determine whether $A$ or $B$ will win, and then determine his winning strategy. [b]p4.[/b] Suppose you are given $4$ pegs arranged in a square on a board. A “move” consists of picking up a peg, reflecting it through any other peg, and placing it down on the board. For how many integers $1 \le n \le 2013$ is it possible to arrange the $4$ pegs into a [i]larger [/i] square using exactly $n$ moves? Justify your answers. [b]p5.[/b] Find smallest positive integer that has a remainder of $1$ when divided by $2$, a remainder of $2$ when divided by $3$, a remainder of $3$ when divided by $5$, and a remainder of $5$ when divided by $7$. [b]p6.[/b] Find the value of $$\sum_{m|496,m>0} \frac{1}{m},$$ where $m|496$ means $496$ is divisible by $m$. [b]p7.[/b] What is the value of $${100 \choose 0}+{100 \choose 4}+{100 \choose 8}+ ... +{100 \choose 100}?$$ [b]p8.[/b] An $n$-term sequence $a_0, a_1, ...,a_n$ will be called [i]sweet [/i] if, for each $0 \le i \le n -1$, $a_i$ is the number of times that the number $i$ appears in the sequence. For example, $1, 2, 1,0$ is a sweet sequence with $4$ terms. Given that $a_0$, $a_1$, $...$, $a_{2013}$ is a sweet sequence, find the value of $a^2_0+ a^2_1+ ... + a^2_{2013}.$ PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

a) Calculate the product $$\left(1+\frac{1}{2}\right) \left(1+\frac{1}{3}\right) \left(1+\frac{1}{4}\right)... \left(1+\frac{1}{2006}\right) \left(1+\frac{1}{2007}\right)$$ b) Let the set $$A =\left\{\frac{1}{2}, \frac{1}{3},\frac{1}{4}, ...,\frac{1}{2006}, \frac{1}{2007}\right\}$$ Determine the sum of all products of $2$, of $4$, of $6$,... , of $2004$ ¸and of $ 2006$ different elements of the set $A$.

Given an acute angled triangle $ABC$ with $M$ being the mid-point of $AB$ and $P$ and $Q$ are the feet of heights from $A$ to $BC$ and $B$ to $AC$ respectively. Show that if the line $AC$ is tangent to the circumcircle of $BMP$ then the line $BC$ is tangent to the circumcircle of $AMQ$.

The roots of the equation \[ x^3-3ax^2+bx+18c=0 \] form a non-constant arithmetic progression and the roots of the equation \[ x^3+bx^2+x-c^3=0 \] form a non-constant geometric progression. Given that $a,b,c$ are real numbers, find all positive integral values $a$ and $b$.

Find all functions $ f: \mathbb {R} ^ + \to \mathbb {R} ^ + $ such that for any $ x, y \in \mathbb {R} ^ + $ the following equality holds: \[f (x) f (y) = f \left (\frac {xy} {x f (x) + y} \right). \] $ \mathbb {R} ^ + $ denotes the set of positive real numbers.