Determine all real $a$, such that the solutions to the system of equations $\begin{cases} \frac{3x-5}{3}+\frac{3x+5}{4}\geq \frac{x}{7}-\frac{1}{15}\\ (2x-a)^3+(2x+a)(1-4x^2)+16x^2a-6x^2a+a^3\leq 2a^2+a \end{cases}$ form an interval with length $\frac{32}{225}$.

Given the sequence $(a_n) $ satisfies $1=a_1< a_2 < a_3< \cdots<a_n $ and there exist real number $m$ such that $$\displaystyle\sum_{i=1}^{n-1} \sqrt[3]{\frac{a_{i+1}-a_i}{(2+a_i)^4}}\leq m $$ for any positive integer $ n $ not less than 2 . Find the minimum of $m.$

$\boxed{\text{A3}}$If $a,b$ be positive real numbers, show that:$$ \displaystyle{\sqrt{\dfrac{a^2+ab+b^2}{3}}+\sqrt{ab}\leq a+b}$$

On a highway, a pedestrian and a cyclist were going in the same direction, while a cart and a car were coming from the opposite direction. All were travelling at different constant speeds. The cyclist caught up with the pedestrian at $10$ o'clock. After a time interval, she met the cart, and after another time interval equal to the first, she met the car. After a third time interval, the car met the pedestrian, and after another time interval equal to the third, the car caught up with the cart. If the pedestrian met the car at $11$ o'clock, when did he meet the cart?

Given a polynomial $P$ and a positive integer $k$, we denote the $k$-fold composition of $P$ by $P^{\circ k}$. A polynomial $P$ with real coefficients is called [b]perfect[/b] if for each integer $n$ there is a positive integer $k$ so that $P^{\circ k}(n)$ is an integer. Is it true that for each perfect polynomial $P$, there exists a positive $m$ so that for each integer $n$ there is $0<k\leq m$ for which $P^{\circ k}(n)$ is an integer?

The width of a long winding river is not greater than $1$ km. This means by definition that from any point of each bank of the river one can reach the other bank swimming $1$ km or less. Is it true that a boat can move along the river so that its distances from both banks are never greater than (a) $0.7$ km? (b) $0.8$ km? (Grigory Kondakov, Moscow) You may assume that the banks consist of segments and arcs of circles.

The product of $10$ natural numbers is equal to $10^{10}$. What is the largest possible sum of these numbers?

If $ y\equal{}x\plus{}\frac{1}{x}$, then $ x^4\plus{}x^3\minus{}4x^2\plus{}x\plus{}1\equal{}0$ becomes: $ \textbf{(A)}\ x^2(y^2\plus{}y\minus{}2)\equal{}0 \qquad\textbf{(B)}\ x^2(y^2\plus{}y\minus{}3)\equal{}0\\ \textbf{(C)}\ x^2(y^2\plus{}y\minus{}4)\equal{}0 \qquad\textbf{(D)}\ x^2(y^2\plus{}y\minus{}6)\equal{}0\\ \textbf{(E)}\ \text{none of these}$

Find all real values of $x$ for which $$\frac{1}{\sqrt{x} + \sqrt{x - 2}} +\frac{1}{\sqrt{x+2} + \sqrt{x }} =\frac14.$$

Let $ A$ and $ B$ be the endpoints of a semicircular arc of radius $ 2$. The arc is divided into seven congruent arcs by six equally spaced points $ C_1,C_2,\ldots,C_6$. All chords of the form $ \overline{AC_i}$ or $ \overline{BC_i}$ are drawn. Let $ n$ be the product of the lengths of these twelve chords. Find the remainder when $ n$ is divided by $ 1000$.

Determine the largest $k\ge0$ such that the inequality \[\left(\sum_{j=1}^n x_j\right)^2\left(\sum_{j=1}^n x_jx_{j+1}\right)\ge k\sum_{j=1}^n x_j^2x_{j+1}^2\] holds for every $n\ge2$ and any $n$-tuple $x_1,\ldots,x_n$ of non-negative numbers (given that $x_{n+1}=x_1$)

Let $a_0$, $a_1$, $a_2$, ... be an infinite sequence of real numbers satisfying the equation $a_n=\left|a_{n+1}-a_{n+2}\right|$ for all $n\geq 0$, where $a_0$ and $a_1$ are two different positive reals. Can this sequence $a_0$, $a_1$, $a_2$, ... be bounded? [i]Proposed by Mihai Bălună, Romania[/i]

Let $\mathbb{R}^+$ the set of positive real numbers. Determine all the functions $f, g: \mathbb{R}^+ \rightarrow \mathbb{R}^+$ such that, for all positive real numbers $x, y$ we have that $$f(x + g(y)) = f(x + y) + g(y) \text{ and } g(x + f(y)) = g(x + y) + f(y)$$

Bartek patiently performs operations on fractions. In each move, he adds its inverse to the current result, obtaining a new result. Bartek starts with the number $1$: after the first move, he receives the result 2, after the second move, the result is $\frac{5}{2}$, after the third move $\frac{29}{10}$, etc. After $300$ moves, Bartek receives the result $x$. Determine the largest integer not greater than $x$.

[b]p1.[/b] Compute $$\left(\frac{1}{10}\right)^{\frac12}\left(\frac{1}{10^2}\right)^{\frac{1}{2^4}}\left(\frac{1}{10^3}\right)^{\frac{1}{2^3}} ...$$ [b]p2.[/b] Consider the sequence $1, 2, 2, 3, 3, 3, 4, 4, 4, 4,...,$ where the positive integer $m$ appears $m$ times. Let $d(n)$ denote the $n$th element of this sequence starting with $n = 1$. Find a closed-form formula for $d(n)$. [b]p3.[/b] Let $0 < \theta < \frac{\pi}{2}$, prove that $$ \left( \frac{\sin^2 \theta}{2}+\frac{2}{\cos^2 \theta} \right)^{\frac14}+ \left( \frac{\cos^2 \theta}{2}+\frac{2}{\sin^2 \theta} \right)^{\frac14} \ge (68)^{\frac14} $$ and determine the value of \theta when the inequality holds as equality. [b]p4.[/b] In $\vartriangle ABC$, parallel lines to $AB$ and $AC$ are drawn from a point $Q$ lying on side $BC$. If $a$ is used to represent the ratio of the area of parallelogram $ADQE$ to the area of the triangle $\vartriangle ABC$, (i) find the maximum value of $a$. (ii) find the ratio $\frac{BQ}{QC}$ when $a =\frac{24}{49}.$ [img]https://cdn.artofproblemsolving.com/attachments/5/8/eaa58df0d55e6e648855425e581a6ba0ad3ea6.png[/img] [b]p5.[/b] Prove the following inequality $$\frac{1}{2009} < \frac{1}{2} \cdot \frac{3}{4} \cdot \frac{5}{6} \cdot \frac{7}{8}...\frac{2007}{2008}<\frac{1}{40}$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].Thanks to gauss202 for sending the problems.

Let $n,s,t$ be three positive integers, and let $A_1,\ldots, A_s, B_1,\ldots, B_t$ be non-necessarily distinct subsets of $\{1,2,\ldots,n\}$. For any subset $S$ of $\{1,\ldots,n\}$, define $f(S)$ to be the number of $i\in\{1,\ldots,s\}$ with $S\subseteq A_i$ and $g(S)$ to be the number of $j\in\{1,\ldots,t\}$ with $S\subseteq B_j$. Assume that for any $1\leq x<y\leq n$, we have $f(\{x,y\})=g(\{x,y\})$. Show that if $t<n$, then there exists some $1\leq x\leq n$ so that $f(\{x\})\geq g(\{x\})$. [i]Proposed by usjl[/i]

Let $a_1,a_2,...$ be an infinite sequence of integers that satisfies $a_{n+2}=a_{n+1}+a_n$ for all $n \ge 1$. There exists a positive integer $k$ such that $a_k=a_{k+2018}$. Prove that there exists a term of the sequence which is equal to zero.

For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\] over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$ [i]Proposed by Shahjalal Shohag, Bangladesh[/i]

A sequence $(u_{n})$ is defined by \[ u_{0}=2 \quad u_{1}=\frac{5}{2}, u_{n+1}=u_{n}(u_{n-1}^{2}-2)-u_{1} \quad \textnormal{for } n=1,\ldots \] Prove that for any positive integer $n$ we have \[ [u_{n}]=2^{\frac{(2^{n}-(-1)^{n})}{3}} \](where [x] denotes the smallest integer $\leq$ x)$.$

If $k$ is an integer, let $\mathrm{c}(k)$ denote the largest cube that is less than or equal to $k$. Find all positive integers $p$ for which the following sequence is bounded: $a_0 = p$ and $a_{n+1} = 3a_n-2\mathrm{c}(a_n)$ for $n \geqslant 0$.

find all the function $f,g:R\rightarrow R$ such that (1)for every $x,y\in R$ we have $f(xg(y+1))+y=xf(y)+f(x+g(y))$ (2)$f(0)+g(0)=0$

Let the sequence $ a(n), n \equal{} 1,2,3, \ldots$ be generated as follows with $ a(1) \equal{} 0,$ and for $ n > 1:$ \[ a(n) \equal{} a\left( \left \lfloor \frac{n}{2} \right \rfloor \right) \plus{} (\minus{}1)^{\frac{n(n\plus{}1)}{2}}.\] 1.) Determine the maximum and minimum value of $ a(n)$ over $ n \leq 1996$ and find all $ n \leq 1996$ for which these extreme values are attained. 2.) How many terms $ a(n), n \leq 1996,$ are equal to 0?

Given three functions $$P(x) = (x^2-1)^{2023}, Q(x) = (2x+1)^{14}, R(x) = \left(2x+1+\frac 2x \right)^{34}.$$ Initially, we pick a set $S$ containing two of these functions, and we perform some [i]operations[/i] on it. Allowed operations include: - We can take two functions $p,q \in S$ and add one of $p+q, p-q$, or $pq$ to $S$. - We can take a function $p \in S$ and add $p^k$ to $S$ for $k$ is an arbitrary positive integer of our choice. - We can take a function $p \in S$ and choose a real number $t$, and add to $S$ one of the function $p+t, p-t, pt$. Show that no matter how we pick $S$ in the beginning, there is no way we can perform finitely many operations on $S$ that would eventually yield the third function not in $S$.

Let $ n \ge 1$ and $ k \ge 3$ be integers. A circle is divided into $ n$ sectors $ a_1,a_2,\dots,a_n$. We will color the $ n$ sectors with $ k$ different colors such that $ a_i$ and $ a_{i \plus{} 1}$ have different color for each $ i \equal{} 1,2,\dots,n$ where $ a_{n \plus{} 1}\equal{}a_1$. Find the number of ways to do such coloring.

A set $M$ of elements $u, v, w$ is called a semigroup if an operation is defined in it is which uniquely assigns an element $w$ from $M$ to every ordered pair $(u, v)$ of elements from $M$ (you write $u \otimes v = w$) and if this algebraic operation is associative, i.e. if for all elements $u, v,w$ from $M$: $$(u \otimes v) \otimes w = u \otimes (v \otimes w).$$ Now let $c$ be a positive real number and let $M$ be the set of all non-negative real numbers that are smaller than $c$. For each two numbers $u, v$ from $M$ we define: $$u \otimes v = \dfrac{u + v}{1 + \dfrac{uv}{c^2}}$$ Investigate a) whether $M$ is a semigroup; b) whether this semigroup is regular, i.e. whether from $u \otimes v_1 = u\otimes v_2$ always $v_1 = v_2$ and from $v_1 \otimes u = v_2 \otimes u$ also $v_1 = v_2$ follows.