Let $a, b, c$ be reals such that $$a^2(c^2-2b-1)+b^2(a^2-2c-1)+c^2(b^2-2a-1)=0.$$ Show that $$3(a^2+b^2+c^2)+4(a+b+c)+3 \geq 6abc.$$

For any given integers $m,n$ such that $2\leq m<n$ and $(m,n)=1$. Determine the smallest positive integer $k$ satisfying the following condition: for any $m$-element subset $I$ of $\{1,2,\cdots,n\}$ if $\sum_{i\in I}i> k$, then there exists a sequence of $n$ real numbers $a_1\leq a_2 \leq \cdots \leq a_n$ such that $$\frac1m\sum_{i\in I} a_i>\frac1n\sum_{i=1}^na_i$$

[b]p1.[/b] Suppose $x \ne 1$ or $10$ and logarithms are computed to the base $10$. Define $y= 10^{\frac{1}{1-\log x}}$ and $z = ^{\frac{1}{1-\log y}}$ . Prove that $x= 10^{\frac{1}{1-\log z}}$ [b]p2.[/b] If $n$ is an odd number and $x_1, x_2, x_3,..., x_n$ is an arbitrary arrangement of the integers $1, 2,3,..., n$, prove that the product $$(x_1 -1)(x_2-2)(x_3- 3)... (x_n-n)$$ is an even number (possibly negative or zero). [b]p3.[/b] Prove that $\frac{1 \cdot 3 \cdot 5 \cdot \cdot \cdot (2n-1)}{2 \cdot 4 \cdot 6 \cdot \cdot \cdot(2n} < \sqrt{\frac{1}{2n + 1}}$ for all integers $n = 1,2,3,...$ [b]p4.[/b] Prove that if three angles of a convex polygon are each $60^o$, then the polygon must be an equilateral triangle. [b]p5.[/b] Find all solutions, real and complex, of $$4 \left(x^2+\frac{1}{x^2} \right)-4 \left( x+\frac{1}{x} \right)-7=0$$ [b]p6.[/b] A man is $\frac38$ of the way across a narrow railroad bridge when he hears a train approaching at $60$ miles per hour. No matter which way he runs he can [u]just [/u] escape being hit by the train. How fast can he run? Prove your assertion. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all functions $f:\mathbb Z\to \mathbb Z$ such that for all surjective functions $g:\mathbb Z\to \mathbb Z$, $f+g$ is also surjective. (A function $g$ is surjective over $\mathbb Z$ if for all integers $y$, there exists an integer $x$ such that $g(x)=y$.) [i]Proposed by Sean Li[/i]

Let $P(x), Q(x), $ and $R(x)$ be three monic quadratic polynomials with only real roots, satisfying $$P(Q(x))=(x-1)(x-3)(x-5)(x-7)$$$$Q(R(x))=(x-2)(x-4)(x-6)(x-8)$$ for all real numbers $x.$ What is $P(0)+Q(0)+R(0)?$ [i]Proposed by Kyle Lee[/i]

Find all functions satisfying the equality $$(x-1)f \left(\dfrac{x+1}{x-1}\right)- f(x) = x$$ for all $x \ne 1$.

Let $a$, $b$, $c$, $d$ and $e$ be distinct positive real numbers such that $a^2+b^2+c^2+d^2+e^2=ab+ac+ad+ae+bc+bd+be+cd+ce+de$ $a)$ Prove that among these $5$ numbers there exists triplet such that they cannot be sides of a triangle $b)$ Prove that, for $a)$, there exists at least $6$ different triplets

What is the largest positive integer $n$ such that $$\frac{a^2}{\frac{b}{29} + \frac{c}{31}}+\frac{b^2}{\frac{c}{29} + \frac{a}{31}}+\frac{c^2}{\frac{a}{29} + \frac{b}{31}} \ge n(a+b+c)$$holds for all positive real numbers $a,b,c$.

The angle formed by the rays $y=x$ and $y=2x$ ($x \ge 0$) cuts off two arcs from a given parabola $y=x^2+px+q$. Prove that the projection of one arc onto the $x$-axis is shorter by $1$ than that of the second arc.

Let $P_1(x, y)$ and $P_2(x, y)$ be two relatively prime polynomials with complex coefficients. Let $Q(x, y)$ and $R(x, y)$ be polynomials with complex coefficients and each of degree not exceeding $d$. Prove that there exist two integers $A_1, A_2$ not simultaneously zero with $|A_i| \leq d + 1 \ (i = 1, 2)$ and such that the polynomial $A_1P_1(x, y) + A_2P_2(x, y)$ is coprime to $Q(x, y)$ and $R(x, y).$

Positive integers $x,y,z$ satisfy $x^3+xy+x^2+xz+y+z=301$. Compute $y+z-x$. [i]2015 CCA Math Bonanza Individual Round #12[/i]

Given two odd natural numbers $ a,b$ prove that for each $ n\in\mathbb{N}$ there exists $ m\in\mathbb{N}$ such that either $ a^mb^2-1$ or $ b^ma^2-1$ is multiple of $ 2^n.$

Determine all tuples of positive integers $(x, y, z, t)$ such that: $$ xyz = t!$$ $$ (x+1)(y+1)(z+1) = (t+1)!$$ holds simultaneously.

Let $a_1, a_2, \cdots , a_n$ be real numbers, with $n\geq 3,$ such that $a_1 + a_2 +\cdots +a_n = 0$ and $$ 2a_k\leq a_{k-1} + a_{k+1} \ \ \ \text{for} \ \ \ k = 2, 3, \cdots , n-1.$$ Find the least number $\lambda(n),$ such that for all $k\in \{ 1, 2, \cdots, n\} $ it is satisfied that $|a_k|\leq \lambda (n)\cdot \max \{|a_1|, |a_n|\} .$

Let $P(x)$ be a polynomial of degree $n$ such that $P(x)=3^k$ for $0\le k \le n$. Find $P(n+1)$

Find all functions $f: \mathbb R \to \mathbb R$ such that for all reals $x$ and $y$, \[f(f(x)-y^{2})=f(x)^{2}-2f(x)y^{2}+f(f(y)).\]

The function $f$ is defined on the set of integers and satisfies \[ f(n)=\begin{cases} n-3 & \text{if } n\ge 1000 \\ f(f(n+5)) & \text{if } n<1000\end{cases} \] Find $f(84)$.

Find all functions $f : R \to R$ that satisfy $f (xy + f(xy)) = 2x f(y)$ for all $x, y \in R$

Without using a calculator, find out which number is greater: $$|\sqrt[3]{5}-\sqrt3|-\sqrt3| \,\,\,\, \text{or} \,\,\,\, 0.01$$

[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.

Find $ \lim_{n\to\infty}(\sum_{i=0}^{n}\frac{1}{n+i})$

Find all functions $f:R \rightarrow R$, which satisfy the equality for any $x,y \in R$: $f(xf(y)+y)+f(xy+x)=f(x+y)+2xy$,

Find all functions $f \colon \mathbb{R_{+}}\to \mathbb{R_{+}}$ satisfying : \[f ( f (x)-x) = 2x\] for all $x > 0$.

When all vertex angles of a convex polygon are equal, call it equiangular. Prove that $ p > 2$ is a prime number, if and only if the lengths of all sides of equiangular $ p$ polygon are rational numbers, it is a regular $ p$ polygon.

Find all functions $f:\mathbb Z\to\mathbb Z$ such that $$f(mf(n+1))=f(m+1)f(n)+f(f(n))+1$$for all integer pairs $(m,n)$.