For any positive real numbers $a, b, c, d$, prove the following inequality: $$(a^2+b^2)(b^2+c^2)(c^2+d^2)(d^2+a^2) \geq 64abcd|(a-b)(b-c)(c-d)(d-a)|$$ [i]Proposed by Anton Trygub[/i]

Let $(a_1, a_2,..., a_n)$ and $(b_1, b_2, ..., b_n)$ be two sequences of positive real numbers. Let $\pi$ be a permutation of the set $\{1, 2, ..., n\}$, for which the sum $a_{\pi(1)}(b_{\pi(1)}+b_{\pi(2)}+...+b_{\pi(n)})+a_{\pi(2)}(b_{\pi(3)}+b_{\pi(3)}+...+b_{\pi(n)})+...+a_{\pi(n)}b_{\pi(n)}$ is minimal. Proce for this permutation $\pi$, that $$ \frac{a_{\pi(1)}}{b_{\pi(1)}}\le \frac{a_{\pi(2})}{b_{\pi(2)}}\le ... \le \frac{a_{\pi(n)}}{b_{\pi(n)}}$$ Application: In an idealized role-playing game you fight against $n$ opponents at the same time. In order to minimize the damage you suffer yourself, you should first take care of your opponent for the ratio of the time it takes to defeat him (if you only focus on him), and the damage it does per second is minimal; next, one should fight the opponent with the second smallest such ratio, and so on.

Consider the sequence $a(n)$ defined by the following conditions:$$a(1) = 1\,\,\,\, a(n + 1) = a(n) + [\sqrt{a(n)}] \,\,\, , \,\,\,\, n = 1,2,3,...$$ How many perfect squares no greater in value than $1000 000$ will be found among the first terms of the sequence? ( (Note: $[x]$ means the integer part of $x$, that is the greatest integer not greater than $x$.) (A Andjans)

Find the number of positive integer solutions to $(x_1 + x_2 + x_3)(y_1 + y_2 + y_3 + y_4) = 91$

Two players take turns entering a symbol in an empty cell of a $1 \times n$ chessboard, where $n$ is an integer greater than $1$. Aaron always enters the symbol $X$ and Betty always enters the symbol $O$. Two identical symbols may not occupy adjacent cells. A player without a move loses the game. If Aaron goes first, which player has a winning strategy?

Write two ones, then a $2$ between them, then a $3$ between the numbers whose sum is $3$, then a $4$ between the numbers whose sum is $4$, as shown below: $$(1, 1), (1, 2, 1),(1, 3, 2, 3, 1), (1, 4, 3, 2, 3, 4, 1)$$ and so on. Prove that the number of times $n$ appears, ($n\ge 2$), is equal to the number of positive integers less than $n$ and relative prime with $n$..

Let $ABC$ be a triangle and let $\omega$ be its incircle. Denote by $D_1$ and $E_1$ the points where $\omega$ is tangent to sides $BC$ and $AC$, respectively. Denote by $D_2$ and $E_2$ the points on sides $BC$ and $AC$, respectively, such that $CD_2=BD_1$ and $CE_2=AE_1$, and denote by $P$ the point of intersection of segments $AD_2$ and $BE_2$. Circle $\omega$ intersects segment $AD_2$ at two points, the closer of which to the vertex $A$ is denoted by $Q$. Prove that $AQ=D_2P$.

Let $ ABCDE$ be a convex pentagon such that \[ \angle BAC \equal{} \angle CAD \equal{} \angle DAE\qquad \text{and}\qquad \angle ABC \equal{} \angle ACD \equal{} \angle ADE. \]The diagonals $BD$ and $CE$ meet at $P$. Prove that the line $AP$ bisects the side $CD$. [i]Proposed by Zuming Feng, USA[/i]

For any positive integer $d$, prove there are infinitely many positive integers $n$ such that $d(n!)-1$ is a composite number.

Let f be a function that satisfies the following conditions: $(i)$ If $x > y$ and $f(y) - y \geq v \geq f(x) - x$, then $f(z) = v + z$, for some number $z$ between $x$ and $y$. $(ii)$ The equation $f(x) = 0$ has at least one solution, and among the solutions of this equation, there is one that is not smaller than all the other solutions; $(iii)$ $f(0) = 1$. $(iv)$ $f(1987) \leq 1988$. $(v)$ $f(x)f(y) = f(xf(y) + yf(x) - xy)$. Find $f(1987)$. [i]Proposed by Australia.[/i]

Rectangles $ABCD$ and $EFGH$ are drawn such that $D,E,C,F$ are collinear. Also, $A,D,H,G$ all lie on a circle. If $BC=16,$ $AB=107,$ $FG=17,$ and $EF=184,$ what is the length of $CE$? [asy] unitsize(1 cm); pair A, B, C, D, E, F, G, H; A = (0,0); B = (5,0); C = (5,1.5); D = (0,1.5); E = (1,1.5); F = (8,1.5); G = (8,3.5); H = (1,3.5); draw(A--B--C--D--cycle); draw(E--F--G--H--cycle); dot("A", A, SW); dot("B", B, SE); dot("C", C, SE); dot("D", D, NW); dot("E", E, NW); dot("F", F, SE); dot("G", G, NE); dot("H", H, NW); [/asy]

Assume real numbers $a_i,b_i\,(i=0,1,\cdots,2n)$ satisfy the following conditions: (1) for $i=0,1,\cdots,2n-1$, we have $a_i+a_{i+1}\geq 0$; (2) for $j=0,1,\cdots,n-1$, we have $a_{2j+1}\leq 0$; (2) for any integer $p,q$, $0\leq p\leq q\leq n$, we have $\sum_{k=2p}^{2q}b_k>0$. Prove that $\sum_{i=0}^{2n}(-1)^i a_i b_i\geq 0$, and determine when the equality holds.

find all positive integer $a\geq 2 $ and $b\geq2$ such that $a$ is even and all the digits of $a^b+1$ are equals.

In an acute triangle $ ABC$, let $ D$ be a point on $ \left[AC\right]$ and $ E$ be a point on $ \left[AB\right]$ such that $ \angle ADB\equal{}\angle AEC\equal{}90{}^\circ$. If perimeter of triangle $ AED$ is 9, circumradius of $ AED$ is $ \frac{9}{5}$ and perimeter of triangle $ ABC$ is 15, then $ \left|BC\right|$ is $\textbf{(A)}\ 5 \qquad\textbf{(B)}\ \frac{24}{5} \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ \frac{48}{5}$

If $p,q,r$ are the roots of the cubic equation $x^3 - 3px^2 + 3q^2 x - r^3 = 0$, then show that $p = q =r$.

Given distinct straight lines $ OA$ and $ OB$. From a point in $ OA$ a perpendicular is drawn to $ OB$; from the foot of this perpendicular a line is drawn perpendicular to $ OA$. From the foot of this second perpendicular a line is drawn perpendicular to $ OB$; and so on indefinitely. The lengths of the first and second perpendiculars are $ a$ and $ b$, respectively. Then the sum of the lengths of the perpendiculars approaches a limit as the number of perpendiculars grows beyond all bounds. This limit is: $ \textbf{(A)}\ \frac {b}{a \minus{} b} \qquad \textbf{(B)}\ \frac {a}{a \minus{} b} \qquad \textbf{(C)}\ \frac {ab}{a \minus{} b} \qquad \textbf{(D)}\ \frac {b^2}{a \minus{} b} \qquad \textbf{(E)}\ \frac {a^2}{a \minus{} b}$

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

Let $p$ be a prime number such that $p\mid (2^{2019}-1) .$ The sequence $a_1,a_2,...,a_n$ satisfies the following conditions: $a_0=2, a_1=1 ,a_{n+1}=a_n+\frac{p^2-1}{4}a_{n-1}$ $(n\geq 1).$ Prove that $p\nmid (a_n+1),$ for any $n\geq 0.$

Find all triples of positive integers $a, b, c$ such that they are pairwise relatively prime and $a^2+3b^2c^2=7^c$.

There are $n$ children in a room. Each child has at least one piece of candy. In Round $1$, Round $2$, etc., additional pieces of candy are distributed among the children according to the following rule: In Round $k$, each child whose number of pieces of candy is relatively prime to $k$ receives an additional piece. Show that after a sufficient number of rounds the children in the room have at most two different numbers of pieces of candy. [i](Proposed by Theresia Eisenkölbl)[/i]

Miguel has a square piece of paper $ABCD$ that he folded along a line $EF$, $E$ on $AB$, and $F$ on $CD$. This fold sent $A$ to point $A'$ on $BC$, distinct from $B$ and $C$. Also, it brought $D$ to point $D'$. $G$ is the intersection of $A'D'$ and $DC$. Prove that the inradius of $GCA'$ is equal to the sum of the inradius of $D'GF$ and $A'BE$.

The sequence of real numbers $a_0,a_1,a_2,\ldots$ is defined recursively by \[a_0=-1,\qquad\sum_{k=0}^n\dfrac{a_{n-k}}{k+1}=0\quad\text{for}\quad n\geq 1.\]Show that $ a_{n} > 0$ for all $ n\geq 1$. [i]Proposed by Mariusz Skalba, Poland[/i]

For any integer $n \ge 2$, let $b_n$ be the least positive integer such that, for any integer $N$, $m$ divides $N$ whenever $m$ divides the digit sum of $N$ written in base $b_n$, for $2 \le m \le n$. Find the integer nearest to $b_{36}/b_{25}$.

The Fibonacci sequence is defined by $F_1 = F_2 = 1$ and $F_{n+2} = F_{n+1}+F_n$ for every integer $n$. A sequence $(a_n)$ of integers is said to be $\textit{phirme}$ if there is a fixed integer $k$ such that $a_n + a_{n+1} = F_{n+k}$ for all $n \geq 1$. Show that if $(a_n)$ is a $\textit{phirme}$ sequence, then there exists an integer $c$ such that $$a_n = F_{n+k-2} + (-1)^nc.$$

Prove that if $a+b+c=0$ then $a^3+b^3+c^3=3abc$