Let $a, b, c$ be positive real numbers. Prove that $$8(a+b+c) \left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a} \right) \le 9 \left(1+\frac{a}{b} \right)\left(1+\frac{b}{c} \right)\left(1+\frac{c}{a} \right)$$

Find all real solutions of the system $$\begin{cases} x_1 +x_2 +...+x_{2000} = 2000 \\ x_1^4 +x_2^4 +...+x_{2000}^4= x_1^3 +x_2^3 +...+x_{2000}^3\end{cases}$$

For every positive integer $M \geq 2$, find the smallest real number $C_M$ such that for any integers $a_1, a_2,\ldots , a_{2023}$, there always exist some integer $1 \leq k < M$ such that  \[\left\{\frac{ka_1}{M}\right\}+\left\{\frac{ka_2}{M}\right\}+\cdots+\left\{\frac{ka_{2023}}{M}\right\}\leq C_M.\] Here, $\{x\}$ is the unique number in the interval $[0, 1)$ such that $x - \{x\}$ is an integer. [i] Proposed by usjl[/i]

Given a positive real $C \geq 1$ and a sequence $a_1, a_2, a_3, \cdots$ satisfying for any positive integer $n,$ $a_n \geq 0$ and for any real $x \geq 1$, $$\left|x\lg x-\sum_{k=1}^{[x]}\left[\frac{x}{k}\right]a_k \right| \leq Cx,$$ where $[x]$ is defined as the largest integer that does not exceed $x$. Prove that for any real $y \geq 1$, $$\sum_{k=1}^{[y]}a_k < 3Cy.$$

For $\forall$ $m\in \mathbb{N}$ with $\pi (m)$ we denote the number of prime numbers that are no bigger than $m$. Find all pairs of natural numbers $(a,b)$ for which there exist polynomials $P,Q\in \mathbb{Z}[x]$ so that for $\forall$ $n\in \mathbb{N}$ the following equation is true: $\frac{\pi (an)}{\pi (bn)} =\frac{P(n)}{Q(n)}$.

Let $\alpha>1$ and consider the function $f(x)=x^{\alpha}$ for $x \ge 0$. For $t>0$, define $M(t)$ as the largest area that a triangle with vertices $(0, 0), (s, f(s)), (t, f(t))$ could reach, for $s \in (0,t)$. Let $A(t)$ be the area of the region bounded by the segment with endpoints $(0, 0)$ ,$(t, f(t))$ and the graph of $y =f(x)$. (a) Show that $A(t)/M(t)$ does not depend on $t$. We denote this value by $c(\alpha)$. Find $c(\alpha)$. (b) Determine the range of values of $c(\alpha)$ when $\alpha$ varies in the interval $(1, +\infty)$. [hide=Note]Google translated from [url=http://ciim.uan.edu.co/ciim-2020-pruebas-virtuales/pruebas-virtuales]http://ciim.uan.edu.co/ciim-2020-pruebas-virtuales/pruebas-virtuales[/url][/hide]

Determine all nonempty finite sets of positive integers $\{a_1, \dots, a_n\}$ such that $a_1 \cdots a_n$ divides $(x + a_1) \cdots (x + a_n)$ for every positive integer $x$. [i]Proposed by Ankan Bhattacharya[/i]

Let the sequence $\{a_n\}_{n \geq 1}$ be defined by \[ a_1 = 1, \quad a_{n+1} = a_n + \frac{1}{\sqrt[2024]{a_n}} \quad \text{for } n \geq 1, \, n \in \mathbb{N} \] Prove that \[ a_n^{2025} >n^{2024} \] for all positive integers $n \geq 2$. $\textbf{Proposed by Prajit Adhikari, Nepal.}$

For a given arithmetic series the sum of the first $50$ terms is $200$, and the sum of the next $50$ terms is $2700$. The first term in the series is: ${{ \textbf{(A)}\ -1221 \qquad\textbf{(B)}\ -21.5 \qquad\textbf{(C)}\ -20.5 \qquad\textbf{(D)}\ 3 }\qquad\textbf{(E)}\ 3.5 } $

If $a,b,c,d$ are four positive reals such that $abcd= 1$ , prove that $(1+a) (1+b) (1 +c ) (1 +d ) \geq 16.$

Determine all functions $f:\mathbb{C}\rightarrow\mathbb{C}$ such that $$|wf(z)+zf(w)|=2|zw|$$ for all $w,z\in\mathbb{C}$.

Let $P$ denote the set of all ordered pairs $ \left(p,q\right)$ of nonnegative integers. Find all functions $f: P \rightarrow \mathbb{R}$ satisfying \[ f(p,q) \equal{} \begin{cases} 0 & \text{if} \; pq \equal{} 0, \\ 1 \plus{} \frac{1}{2} f(p+1,q-1) \plus{} \frac{1}{2} f(p-1,q+1) & \text{otherwise} \end{cases} \] Compare IMO shortlist problem 2001, algebra A1 for the three-variable case.

Consider a triangular table of positive integers \[ \begin{matrix} & & & a_{11} & a_{12} & a_{13} & & & \\ & & a_{21} & a_{22} & a_{23} & a_{24} & a_{25} & & \\ & a_{31} & a_{32} & a_{33} & a_{34} & a_{35} & a_{36} & a_{37} & \\ \iddots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{matrix} \] The first row consists of odd numbers only. For $i>1,j\ge1$ we have \[a_{ij}=a_{i-1,j-2}+a_{i-1,j-1}+a_{i-1,j}.\] If we get out of range with the second index, we consider such $a$ to be zero (eg. $a_{22}=0+a_{11}+a_{12}$ and $a_{37}=a_{25}+0+0$). Show that for every $i>1$ there is $j\in\{1,\ldots,2i+1\}$ such that $a_{ij}$ is even.

A train travels a certain distance at a constant speed. Whose speed is increased by $10$ kilometers per hour, the trip can be made $40$ minutes faster. If, on the other hand, the speed is reduced by $10$ kilometers per hour, the trip takes $1$ hour further. How long is the distance traveled?

Let $a, b,c$ be positive real numbers. Prove that $ \sqrt{a^3b+a^3c}+\sqrt{b^3c+b^3a}+\sqrt{c^3a+c^3b}\ge \frac43 (ab+bc+ca)$

For each non-negative integer $n$, $F_n(x)$ is a polynomial in $x$ of degree $n$. Prove that if the identity \[F_n(2x)=\sum_{r=0}^{n} (-1)^{n-r} \binom nr 2^r F_r(x)\] holds for each n, then \[F_n(tx)=\sum_{r=0}^{n} \binom nr t^r (1-t)^{n-r} F_r(x)\]

Let $f, g:\mathbb{N}\to\mathbb{N}$ be functions that satisfy the following equation: \[f(f(n))+g(f(n)) = n,\ \forall\ n\in\mathbb{N}\ .\] Prove that $g$ is the zero function on $\mathbb{N}$.

It is known that $\frac{7}{13} + \sin \phi = \cos \phi$ for some real $\phi$. What is sin $2\phi$?

Let $ S$ be a finite set of points in the plane such that no three of them are on a line. For each convex polygon $ P$ whose vertices are in $ S$, let $ a(P)$ be the number of vertices of $ P$, and let $ b(P)$ be the number of points of $ S$ which are outside $ P$. A line segment, a point, and the empty set are considered as convex polygons of $ 2$, $ 1$, and $ 0$ vertices respectively. Prove that for every real number $ x$ \[\sum_{P}{x^{a(P)}(1 \minus{} x)^{b(P)}} \equal{} 1,\] where the sum is taken over all convex polygons with vertices in $ S$. [i]Alternative formulation[/i]: Let $ M$ be a finite point set in the plane and no three points are collinear. A subset $ A$ of $ M$ will be called round if its elements is the set of vertices of a convex $ A \minus{}$gon $ V(A).$ For each round subset let $ r(A)$ be the number of points from $ M$ which are exterior from the convex $ A \minus{}$gon $ V(A).$ Subsets with $ 0,1$ and 2 elements are always round, its corresponding polygons are the empty set, a point or a segment, respectively (for which all other points that are not vertices of the polygon are exterior). For each round subset $ A$ of $ M$ construct the polynomial \[ P_A(x) \equal{} x^{|A|}(1 \minus{} x)^{r(A)}. \] Show that the sum of polynomials for all round subsets is exactly the polynomial $ P(x) \equal{} 1.$ [i]Proposed by Federico Ardila, Colombia[/i]

Find all functions $f: \mathbb{R}\to [0;+\infty)$ such that: \[f(x^2+y^2)=f(x^2-y^2)+f(2xy)\] for all real numbers $x$ and $y$. [i]Laurentiu Panaitopol[/i]

Julia has $289$ coins stored in boxes: All the boxes contain the same number of coins (which is greater than $1$) and in each box there are coins from the same country, The coins from Bolivia are more than $6\%$ of the total, those from Chile are more than $12\%$ of the total, those of Mexico are more than $24\% $of the total and those of Peru more than $36\%$ of the total. Can Julia have any coins from Uruguay?

Find all functions $f:\mathbb R\rightarrow \mathbb R$ such that for all real numbers $x,y$: $f(x)f(y)+f(xy)\leq x+y$.

Call a quadratic [i]invasive[/i] if it has $2$ distinct real roots. Let $P$ be a quadratic polynomial with real coefficients. Prove that $P(x)$ is invasive [b]if and only if[/b] there exists a real number $c \neq 0$ such that $P(x) + P(x - c)$ is invasive.

If $x,y, z$ are arbitrary positive numbers with $xyz = 1$, prove the inequality $$x^2+y^2+z^2 + xy+yz + zx \ge 2(\sqrt{x} +\sqrt{y}+ \sqrt{z})$$.

Find all real pairs $(a,b)$ that solve the system of equation \begin{align*} a^2+b^2 &= 25, \\ 3(a+b)-ab &= 15. \end{align*} [i](German MO 2016 - Problem 1)[/i]