The sequence $(a_n)$ is defined by $a_1 =\sqrt2$ and $a_{n+1} =\sqrt{2-\sqrt{4-a_n^2}}$. Let $b_n =2^{n+1}a_n$. Prove that $b_n \le 7$ and $b_n < b_{n+1}$ for all $n$.

Let $f: [0,1]\rightarrow\mathbb{R}$ be a continuous function satisfying $xf(y)+yf(x)\le 1$ for every $x,y\in[0,1]$. (a) Show that $\int^1_0 f(x)dx \le \frac{\pi}4$. (b) Find such a funtion for which equality occurs.

Find all positive integers $n$ such that \[3^n+4^n+\cdots+(n+2)^n=(n+3)^n.\]

Let $a,b,c$ three positive numbers less than $ 1$. Prove that cannot occur simultaneously these three inequalities: $$a (1- b)>\frac14$$ $$b (1-c)>\frac14 $$ $$c (1-a)>\frac14$$

Let $ A$, $ M$, and $ C$ be nonnegative integers such that $ A \plus{} M \plus{} C \equal{} 12$. What is the maximum value of $ A \cdot M \cdot C \plus{} A\cdot M \plus{} M \cdot C \plus{} C\cdot A$? $ \textbf{(A)}\ 62 \qquad \textbf{(B)}\ 72 \qquad \textbf{(C)}\ 92 \qquad \textbf{(D)}\ 102 \qquad \textbf{(E)}\ 112$

$a,b,c$ are the length of three sides of a triangle. Let $A= \frac{a^2 +bc}{b+c}+\frac{b^2 +ca}{c+a}+\frac{c^2 +ab}{a+b}$, $B=\frac{1}{\sqrt{(a+b-c)(b+c-a)}}+\frac{1}{\sqrt{(b+c-a)(c+a-b)}}$$+\frac{1}{\sqrt{(c+a-b)(a+b-c)}}$. Prove that $AB \ge 9$.

Let $a,b,c$ be distinct positive real numbers, and let $k$ be a positive integer greater than $3$. Show that \[\left\lvert\frac{a^{k+1}(b-c)+b^{k+1}(c-a)+c^{k+1}(a-b)}{a^k(b-c)+b^k(c-a)+c^k(a-b)}\right\rvert\ge \frac{k+1}{3(k-1)}(a+b+c)\] and \[\left\lvert\frac{a^{k+2}(b-c)+b^{k+2}(c-a)+c^{k+2}(a-b)}{a^k(b-c)+b^k(c-a)+c^k(a-b)}\right\rvert\ge \frac{(k+1)(k+2)}{3k(k-1)}(a^2+b^2+c^2).\] [i]Calvin Deng.[/i]

The product of $10$ integers is $1024$. What is the greatest possible sum of these $10$ integers?

Let $a, b, c, d$ be positive reals strictly smaller than $1$, such that $a+b+c+d=2$. Prove that $$\sqrt{(1-a)(1-b)(1-c)(1-d)} \leq \frac{ac+bd}{2}. $$

Let $u_1, u_2, \dots, u_{2019}$ be real numbers satisfying \[u_{1}+u_{2}+\cdots+u_{2019}=0 \quad \text { and } \quad u_{1}^{2}+u_{2}^{2}+\cdots+u_{2019}^{2}=1.\] Let $a=\min \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$ and $b=\max \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$. Prove that \[ a b \leqslant-\frac{1}{2019}. \]

[asy] size(200); dotfactor=3; pair A=(0,0),B=(1,0),C=(2,0),D=(3,0),X=(1.2,0.7); draw(A--D); dot(A);dot(B);dot(C);dot(D); draw(arc((0.4,0.4),0.4,180,110),arrow = Arrow(TeXHead)); draw(arc((2.6,0.4),0.4,0,70),arrow = Arrow(TeXHead)); draw(B--X,dotted); draw(C--X,dotted); label("$A$",A,SW); label("$B$",B,S); label("$C$",C,S); label("$D$",D,S); label("x",X,fontsize(5pt)); //Credit to TheMaskedMagician for the diagram [/asy] Points $A , B, C$, and $D$ are distinct and lie, in the given order, on a straight line. Line segments $AB, AC$, and $AD$ have lengths $x, y$, and $z$ , respectively. If line segments $AB$ and $CD$ may be rotated about points $B$ and $C$, respectively, so that points $A$ and $D$ coincide, to form a triangle with positive area, then which of the following three inequalities must be satisfied? $\textbf{I. }x<\frac{z}{2}\qquad\textbf{II. }y<x+\frac{z}{2}\qquad\textbf{III. }y<\frac{z}{2}$ $\textbf{(A) }\textbf{I. }\text{only}\qquad\textbf{(B) }\textbf{II. }\text{only}\qquad$ $\textbf{(C) }\textbf{I. }\text{and }\textbf{II. }\text{only}\qquad\textbf{(D) }\textbf{II. }\text{and }\textbf{III. }\text{only}\qquad\textbf{(E) }\textbf{I. },\textbf{II. },\text{and }\textbf{III. }$

Let $a<b<c<d$ be positive integers which satisfy $ad=bc.$ Prove that $2a+\sqrt{a}+\sqrt{d}<b+c+1.$ [i]Marius Mînea[/i]

Let $a$, $b$, $c$ be positive real numbers such that $\min(ab,bc,ca) \ge 1$. Prove that $$\sqrt[3]{(a^2+1)(b^2+1)(c^2+1)} \le \left(\frac{a+b+c}{3}\right)^2 + 1.$$ [i]Proposed by Tigran Margaryan, Armenia[/i]

Prove that if the distance from a point inside a convex polyhedra with $n$ faces to the vertices of the polyhedra is at most 1, then the sum of the distances from this point to the faces of the polyhedra is smaller than $n-2$. [i]Calin Popescu[/i]

What is the smallest value of $k$ for which the inequality \begin{align*} ad-bc+yz&-xt+(a+c)(y+t)-(b+d)(x+z)\leq \\ &\leq k\left(\sqrt{a^2+b^2}+\sqrt{c^2+d^2}+\sqrt{x^2+y^2}+\sqrt{z^2+t^2}\right)^2 \end{align*} holds for any $8$ real numbers $a,b,c,d,x,y,z,t$? Edit: Fixed a mistake! Thanks @below.

Find all real numbers $a$ such that the inequality $3x^2 + y^2 \ge -ax(x + y)$ holds for all real numbers $x$ and $y$.

Find all real numbers $k$ such that $r^4+kr^3+r^2+4kr+16=0$ is true for exactly one real number $r$.

a) Determine the largest real number $A$ with the following property: For all non-negative real numbers $x,y,z$, one has \[\frac{1+yz}{1+x^2}+\frac{1+zx}{1+y^2}+\frac{1+xy}{1+z^2} \ge A.\] b) For this real number $A$, find all triples $(x,y,z)$ of non-negative real numbers for which equality holds in the above inequality.

Prove that if $a,b,c$ are positive numbers with $abc=1$, then \[\frac{a}{b} +\frac{b}{c} + \frac{c}{a} \ge a + b + c. \]

Let the inequality $$Aa(Bb + Cc) + Bb(Aa + Cc) + Cc(Aa + Bb) > \frac{ABc^2 + BCa^2 + CAb^2}{2}$$ with given $a > 0, b > 0, c > 0$ hold for all $A > 0, B > 0, C > 0$. Is it possible to construct a triangle with sides of lengths $a, b, c$?

If a, b, c are positive real numbers with $ a+b+c=1 $. Find the minimum value of $ \sqrt{a}+\sqrt{b}+\sqrt{c}+\frac{1}{\sqrt{abc}} $

Given natural numbers $m, n$ and $X$ such that $X \ge m$ and $X \ge n$. Prove that one can find two integers $u$ and $v$ such that $|u| + |v| > 0$, $|u| \le \sqrt{X}$, $|v| \le \sqrt{X}$ and $$0 \le mu + nv \le 2 \sqrt{X}.$$

Determine the minimum value of $\dfrac{m^m}{1\cdot 3\cdot 5\cdot \ldots \cdot(2m-1)}$ for positive integers $m$.

Prove that if non-zero complex numbers $\alpha_1,\alpha_2,\alpha_3$ are distinct and noncollinear on the plane, and satisfy $\alpha_1+\alpha_2+\alpha_3=0$, then there holds \[\sum_{i=1}^{3}\left(\frac{|\alpha_{i+1}-\alpha_{i+2}|}{\sqrt{|\alpha_i|}}\left(\frac{1}{\sqrt{|\alpha_{i+1}|}}+\frac{1}{\sqrt{|\alpha_{i+2}|}}-\frac{2}{\sqrt{|\alpha_{i}|}}\right)\right)\leq 0......(*)\] where $\alpha_4=\alpha_1, \alpha_5=\alpha_2$. Verify further the sufficient and necessary condition for the equality holding in $(*)$.

Let $m$ be the smallest integer whose cube root is of the form $n+r$, where $n$ is a positive integer and $r$ is a positive real number less than $1/1000$. Find $n$.