Given a positive integer $N$. Prove that \[\sum_{m=1}^N \sum_{n=1}^N \frac{1}{mn^2+m^2n+2mn}<\frac{7}{4}.\]Proposed by tan-1

Given integer $n \geq 3$ and $x_1$, $x_2$, …, $x_n$ be $n$ real numbers satisfying $|x_1|+|x_2|+…+|x_n|=1$. Find the minimum of \[|x_1+x_2|+|x_2+x_3|+…+|x_{n-1}+x_n|+|x_n+x_1|.\]Proposed by snap7822

Find all infinite integer sequences $a_1,a_2,\ldots$ satisfying \[a_{n+2}^{a_{n+1}}=a_{n+1}+a_n\]holds for all $n\geq 1$. Define $0^0=1$

find all function $f:\mathbb{R} \to \mathbb{R}$ such that \[f(x^3-xf(y)^2)=xf(x+y)f(x-y)\]holds for all real number $x$, $y$. Proposed by chengbilly

The non-negative numbers $ x_1, x_2, \ldots, x_5$ satisfy $ \sum_{i = 1}^5 \frac {1}{1 + x_i} = 1$. Prove that $ \sum_{i = 1}^5 \frac {x_i}{4 + x_i^2} \leq 1$.

Given positive real $a,b,c$ satisfying \[\frac{1}{\sqrt{a+1}}+\frac{3}{\sqrt{b+3}}+\frac{3}{\sqrt{c+3}}=\frac72\]Prove that $abc\leq 3$. I was asked to propose a inequality for the condition of $abc<3$ inequality since <3 looks like a heart shape, then I construct a equality and with the help of wolfram, I gave the birth of this bad-looking inequality, I’m glad to see any method besides calculus.

Given positive integers $n$, $P_1$, $P_2$, …$P_n$ and two sets \[B=\{ (a_1,a_2,…,a_n)|a_i=0 \vee 1,\ \forall i \in \mathbb{N} \}, S=\{ (x_1,x_2,…,x_n)|1 \leq x_i \leq P_i \wedge x_i \in \mathbb{N} ,\ \forall i \in \mathbb{N} \}\]A function $f:S \to \mathbb{Z}$ is called Real, if and only if for any positive integers $(y_1,y_2,…,y_n)$ and positive integer $a$ which satisfied $ 1 \leq y_i \leq P_i-a$ $\forall i \in \mathbb{N}$, we always have: \begin{align*} \sum_{(a_1,a_2,…,a_n) \in B \wedge 2| \sum_{i=1}^na_i}f(y+a \times a_1,y+a \times a_2,……,y+a \times a_n)&>\\ \sum_{(a_1,a_2,…,a_n) \in B \wedge 2 \nmid \sum_{i=1}^na_i}f(y+a \times a_1,y+a \times a_2,……,y+a \times a_n)&. \end{align*}Find the minimum of $\sum_{i_1=1}^{P_1}\sum_{i_2=1}^{P_2}....\sum_{i_n=1}^{P_n}|f(i_1,i_2,...,i_n)|$, where $f$ is a Real function. Proposed by tob8y

$a$, $b$, $c$ are three distinct real numbers, given $\lambda >0$. Proof that \[\frac{1+ \lambda ^2a^2b^2}{(a-b)^2}+\frac{1+ \lambda ^2b^2c^2}{(b-c)^2}+\frac{1+ \lambda ^2c^2a^2}{(c-a)^2} \geq \frac 32 \lambda.\] RemarkOld problem, can be found here. Double post to have a cleaner thread for collection (as the original one contains a messy quote)

On a $n \times n$ grid, each edge are written with $=$ or $\neq$. We need to filled every cells with color black or white. Find the largest constant $k$, such that for every $n>777771449$ and any layout of $=$ and $\neq$, we can always find a way to colored every cells, such that at least $k \cdot 2n(n-1)$ neighboring cells, there colors conform to the symbols on the edge. (Namely, two cells are filled with the same color if $=$ was written on their edge; two cells are filled with different colors if $\neq$ was written on their edge) Proposed by chengbilly & sn6dh

Given integer $n \geq 3$. There are $n$ dots marked $1$ to $n$ clockwise on a big circle. And between every two neighboring dots, there is a light. At first, every light were dark. A and B are playing a game, A pick up $n$ pairs from $\{ (i,j)|1 \leq i < j \leq n \}$ and for every pairs $(i,j)$. B starts from the point marked $i$ and choose to walk clockwise or counterclockwise to the point marked $j$. And B invert the status of all passing lights (bright $\leftrightarrow$ dark) A hopes the number of dark light can be as much as possible while B hopes the number of bright light can be as much as possible. Suppose A, B are both smart, how many lights are bright in the end? Proposed by BlessingOfHeaven

There are $n$ snails on the plane where the $i$ snail has $a_i$ attack and $d_i$ defense, where $a_i, d_i\in \mathbb{R}$ and each snail has a distinct attack and a distinct defense. We said a 4-tuple of subsets of snails $(S_1, S_2, S_3, S_4)$ is balanced if $|S_1|+|S_3|$ is either $\lceil n/2\rceil$ or $\lfloor n/2\rfloor$ and there exist real numbers $A, D$ such that \begin{align*} S_1=\{i\ |\ a_i\geq A\text{ and } d_i\geq D, 1\leq i\leq n\}\\ S_2=\{i\ |\ a_i<A\text{ and } d_i\geq D, 1\leq i\leq n\}\\ S_3=\{i\ |\ a_i< A\text{ and } d_i< D, 1\leq i\leq n\}\\ S_4=\{i\ |\ a_i\geq A\text{ and } d_i< D, 1\leq i\leq n\} \end{align*}Find the largest integer $k$ such that there is always at least $k$ balanced 4-tuples of subsets. Proposed by redshrimp

The REAL country has $n$ islands, and there are $n-1$ two-way bridges connecting these islands. Any two islands can be reached through a series of bridges. Arctan, the king of the REAL country, found that it is too difficult to manage $n$ islands, so he wants to bomb some islands and their connecting bridges to divide the country into multiple small areas. Arctan wants the number of connected islands in each group is less than $\delta n$ after bombing these islands, and the island he bomb must be a connected area. Besides, Arctan wants the number of islands to be bombed to be as less as possible. Find all real numbers $\delta$ so that for any positive integer $n$ and the layout of the bridge, the method of bombing the islands is the only one. Proposed by chengbilly

Given integer $n\geq 2$, two invisible rabbits (rabbits) discussed their strategy and was sent to two points $A, B$ with distance $n$ units on a plane, respectively. However, they do not know whether they are on the same or different side of the plane (when facing each other, the might view the left/right direction as the same or different). They both can see points $A,B$, and they need to hop to each other's starting point. Each hop would measure $1$ unit in distance, and they would jump and land at the same time for each round. However, if at any time they landed no more than $1$ unit away, they'll both turn into deer. Find the minimum number of round they need to reach their destiny while ensuring they won't turn into deer. Proposed by redshrimp

On an $m\times n$ grid there's a snail in each cell. Each round, the snail army can choose four snail and make them perform the complete Quadrilateral Dance, which is rotating the four snails clockwise by $90$ degrees, moving each one to an adjacent cell. Find all pairs of positive integers $(m,n)$ such that the snails can achieve any permutation by performing a finite number of times of Quadrilateral Dance. Proposed by chengbilly

On a plane there is an invisible rabbit (rabbit) hiding on a lattice point. We want to put $n$ hunters on some lattice points to catch the rabbit. In a turn each hunter can choose to shoot to left/right or top/bottom direction. On the $i$th turn there will be these steps in order 1. The rabbit jumps to an adjacent lattice point on the top, bottom, left, or right. 2. item Each hunter moves to an adjacent lattice point on the top, bottom, left or right (each hunter can move to different direction). Then they shoot a bullet which travels $\frac{334111214}{334111213}i$ units on the directions they chose. If a bullet hits the rabbit then it is caught. Find the smallest number $n$ such that the rabbit would definitely be caught in a finite number of turns. Proposed by tob8y

Given quadrilateral $ABCD$. $AC$ and $BD$ meets at $E$, and $M, N$ are the midpoints of $AC, BD$, respectively. Let the circumcircles of $ABE$ and $CDE$ meets again at $X\neq E$. Prove that $E, M, N, X$ are concyclic. Proposed by chengbilly

Triangle $ABC$ has circumcenter $O$. $D$ is an arbitrary point on $BC$, and $AD$ intersects $\odot(ABC)$ at $E$. $S$ is a point on $\odot(ABC)$ such that $D, O, E, S$ are colinear. $AS$ intersects $BC$ at $P$. $Q$ is a point on $BC$ such that $D, O, A, Q$ are concylic. Prove that $\odot(ABC)$ is tangent to $\odot (APQ)$. Proposed by chengbilly

Triangle $ABC$ has circumcircle $\Omega$ and incircle $\omega$, where $\omega$ is tangent to $BC, CA, AB$ at $D,E,F$, respectively. $T$ is an arbitrary point on $\omega$. $EF$ meets $BC$ at $K$, $AT$ meets $\Omega$ again at $P$, $PK$ meets $\Omega$ again at $S$. $X$ is a point on $\Omega$ such that $S, D, X$ are colinear. Let $Y$ be the intersection of $AX$ and $EF$, prove that $YT$ is tangent to $\omega$. Proposed by chengbilly

Given triangle $ABC$ with $AB<AC$ and its circumcircle $\Omega$. Let $I$ be the incenter of $ABC$, and the feet from $I$ to $BC$ is $D$. The circle with center $A$ and radius $AI$ intersects $\Omega$ at $E$ and $F$. $P$ is a point on $EF$ such that $DP$ is parallel to $AI$. Prove that $AP$ and $MI$ intersects on $\Omega$ where $M$ is the midpoint of arc $BAC$. RemarkIn the test, the incenter called $O$ and the circumcircle called $Luna$ $Cabrera$ You have to prove $AP \cap MO \in Luna$ $Cabrera$ Proposed by BlessingOfHeaven

Triangle $ABC$ satisfying $AB<AC$ has circumcircle $\Omega$. $E, F$ lies on $AC, AB$, respectively, such that $BCEF$ is cyclic. $T$ lies on $EF$ such that $\odot(TEF)$ is tangent to $BC$ at $T$. $A'$ is the antipode of $A$ on $\Omega$. $TA', TA$ intersects $\Omega$ again at $X, Y$, respectively, and $EF$ intersects $\odot(TXY)$ again at $W$. Prove that $\measuredangle WBA=\measuredangle ACW$ Proposed by BlessingOfHeaven

$ABCD$ is a cyclic quadrilateral and $AC$ intersects $BD$ at $E$. $M, N$ are the midpoints of $AB, CD$, respectively. $\odot(AMN)$ meets $\odot(ABCD)$ again at $P$. $\odot(CMN)$ meets $\odot(ABCD)$ again at $Q$. $\odot(PEQ)$ meets $BD$ again at $T$. Prove that $M,N,T$ are colinear. Proposed by chengbilly

Triangle $ABC$ has circumcenter $O$ and incenter $I$. The incircle is tangent to $AC, AB$ at $E, F$, respectively. $H$ is the orthocenter of $\triangle BIC$. $\odot(AEF)$ and $\odot(ABC)$ intersects again at $S$. $BC, AH$ intersects $OI$ again at $J, K$, respectively. Prove that $H, K, J, S$ are concyclic. Proposed by chengbilly

Proof that for every primes $p$, $q$ \[p^{q^2-q+1}+q^{p^2-p+1}-p-q\]is never a perfect square. Proposed by chengbilly

Find all positive integers $(m,n)$ such that $$11^n+2^n+6=m^3$$

Find all positive integers $n$ such that $$n(2^n-1)$$is a perfect square

Given a set of integers $S$ satisfies that: for any $a,b,c\in S$ ($a,b,c$ can be the same), $ab+c\in S$ Find all pairs of integers $(x,y)$ such that if $x,y\in S$, then $S=\mathbb{Z}$.

Find all positive integers $n$ such that $$2^n+15|3^n+200$$

Find all functions $f:\mathbb{Q}^+\to\mathbb{Q}^+$ such that \[xy(f(x)-f(y))|x-f(f(y))\]holds for all positive rationals $x$, $y$ (we define that $a|b$ if and only if exist $n \in \mathbb{Z}$ such that $b=an$) Proposed by supercarry & windleaf1A

Find all functions $f:\mathbb{N}\to\mathbb{N}$ such that $$|xf(y)-yf(x)|$$is a perfect square for every $x,y \in \mathbb{N}$

Find all integers $(a,b)$ satisfying: there is an integer $k>1$ such that $$a^k+b^k-1\ |\ a^n+b^n-1$$holds for all integer $n\geq k$ (we define that $0|0$)