find all pairs of non-negative integer pairs $(m,n)$,satisfies $107^{56}(m^{2}-1)+2m+3=\binom{113^{114}}{n}$

In a plane we choose a cartesian system of coordinates. A point $A(x,y)$ in the plane is called an integer point if and only if both $x$ and $y$ are integers. An integer point $A$ is called invisible if on the segment $(OA)$ there is at least one integer point. Prove that for each positive integer $n$ there exists a square of side $n$ in which all the interior integer points are invisible.

Prove that for any four points in the plane, no three of which are collinear, there exists a circle such that three of the four points are on the circumference and the fourth point is either on the circumference or inside the circle.

Find all triples $(x,y,n)$ of positive integers such that \[ (x+y)(1+xy) = 2^{n} \]

Let $ABCD$ be a trapezoid with bases $ AB$ and $CD$ $(AB>CD)$. Diagonals $AC$ and $BD$ intersect in point $ N$ and lines $AD$ and $BC$ intersect in point $ M$. The circumscribed circles of $ADN$ and $BCN$ intersect in point $ P$, different from point $ N$. Prove that the angles $AMP$ and $BMN$ are equal.

For each integer sequence $a_1, a_2, a_3, \dots, a_n$, a [i]single parity swapping[/i] is to choose $2$ terms in this sequence, say $a_i$ and $a_j$, such that $a_i + a_j$ is odd, then switch their placement, while the other terms stay in place. This creates a new sequence. Find the minimal number of single parity swapping to transform the sequence $1,2,3, \dots, 2025$ to $2025, \dots, 3, 2, 1$, using only single parity swapping.

Let $a\circ b=a+b-ab$. Find all triples $(x,y,z)$ of integers such that \[(x\circ y)\circ z +(y\circ z)\circ x +(z\circ x)\circ y=0\]

How to hang a picture? What a strange question? It's simple. We take a piece of rope, attach its ends to the picture frame on the back side, then drive it into the wall. nail and throw a rope over the nail. The picture is hanging. If you pull out the nail, then, of course, it will fall. But Professor No wonder acted differently. At first, he attached the rope to the painting in the same way, only he took it a little longer. Then he hammered two nails into the wall nearby and threw a rope over these nails in a special way. The painting hangs on these nails, but if you pull out any nail, the painting will fall. Moreover, the professor claims that he can hang a painting on three nails so that the painting hangs on all three, but if any nail is pulled out, the painting will fall. You have two tasks: indicate how you can hang the picture in the right way on a) two nails; b) three nails.

A polynomial $P$ with integer coefficients has at least $13$ distinct integer roots. Prove that if an integer $n$ is not a root of $P$, then $|P(n)| \geq 7 \cdot 6!^2$, and give an example for equality.

If the circumradius of a regular $n$-gon is $1$ and the ratio of its perimeter over its area is $\dfrac{4\sqrt 3}{3}$, what is $n$? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8 $

Let $a,b,k$ be positive integers such that $gcd(a,b)^2+lcm(a,b)^2+a^2b^2=2020^k$ Prove that $k$ is an even number.

In triangle $ABC$, the incircle touches sides $BC,CA$ and $AB$ in $D,E$ and $F$ respectively. Let $X$ be the feet of the altitude of the vertex $D$ on side $EF$ of triangle $DEF$. Prove that $AX,BY$ and $CZ$ are concurrent on the Euler line of the triangle $DEF$.

There are a lot of different real numbers written on the board. It turned out that for each two numbers written, their product was also written. What is the largest possible number of numbers written on the board?

In a parallelogram $ABCD$, points $E$ and $F$ are the midpoints of $AB$ and $BC$, respectively, and $P$ is the intersection of $EC$ and $FD$. Prove that the segments $AP,BP,CP$ and $DP$ divide the parallelogram into four triangles whose areas are in the ratio $1 : 2 : 3 : 4$.

Determine if there exist positive integer pairs $(m,n)$, such that (i) the greatest common divisor of m and $n$ is $1$, and $m \le 2007$, (ii) for any $k=1,2,..., 2007$, $\big[\frac{nk}{m}\big]=\big[\sqrt2 k\big]$ . (Here $[x]$ stands for the greatest integer less than or equal to $x$.)

Given the equation \[ y^4 \plus{} 4y^2x \minus{} 11y^2 \plus{} 4xy \minus{} 8y \plus{} 8x^2 \minus{} 40x \plus{} 52 \equal{} 0,\] find all real solutions.

$ M,N,P,Q,R,S $ are the midpoints of the sides $ AB,BC,CD,DE,EF,FA $ of a convex hexagon $ ABCDEF. $ [b]a)[/b] Show that with the segments $ MQ,NR,PS, $ it can be formed a triangle. [b]b)[/b] Show that a triangle formed with the segments $ MQ,NR,PS $ is right if and only if ether $ MQ\perp NR $ or $ MQ\perp PS $ or $ PS\perp RN. $ [i]Vasile Pop[/i]

For how many positive integer values of $n$ are both $\frac{n}{3}$ and $3n$ three-digit whole numbers? $\textbf{(A)}\ 12\qquad \textbf{(B)}\ 21\qquad \textbf{(C)}\ 27\qquad \textbf{(D)}\ 33\qquad \textbf{(E)}\ 34$

Integers $1, 2, 3, ... ,n$, where $n > 2$, are written on a board. Two numbers $m, k$ such that $1 < m < n, 1 < k < n$ are removed and the average of the remaining numbers is found to be $17$. What is the maximum sum of the two removed numbers?

Let $\Gamma_1$ and $\Gamma_2$ be two circles that tangents each other at point $N$, with $\Gamma_2$ located inside $\Gamma_1$. Let $A, B, C$ be distinct points on $\Gamma_1$ such that $AB$ and $AC$ tangents $\Gamma_2$ at $D$ and $E$, respectively. Line $ND$ cuts $\Gamma_1$ again at $K$, and line $CK$ intersects line $DE$ at $I$. (i) Prove that $CK$ is the angle bisector of $\angle ACB$. (ii) Prove that $IECN$ and $IBDN$ are cyclic quadrilaterals.

Let $a > 1$ and $r > 1$ be real numbers. (a) Prove that if $f : \mathbb{R}^{+}\to\mathbb{ R}^{+}$ is a function satisfying the conditions (i) $f (x)^{2}\leq ax^{r}f (\frac{x}{a})$ for all $x > 0$, (ii) $f (x) < 2^{2000}$ for all $x < \frac{1}{2^{2000}}$, then $f (x) \leq x^{r}a^{1-r}$ for all $x > 0$. (b) Construct a function $f : \mathbb{R}^{+}\to\mathbb{ R}^{+}$ satisfying condition (i) such that for all $x > 0, f (x) > x^{r}a^{1-r}$ .

Prove that for all all non-negative real numbers $a,b,c$ with $a^2+b^2+c^2=1$ \[\sqrt{a+b}+\sqrt{a+c}+\sqrt{b+c} \geq 5abc+2.\]

Assume integer $m \geq 2.$ There are $3m$ people in a meeting, any two of them either shake hands with each other once or not.We call the meeting "$n$-interesting", only if there exists $n(n\leq 3m-1)$ people of them, the time everyone of whom shakes hands with other $3m-1$ people is exactly $1,2,\cdots,n,$ respectively. If in any "$n$-interesting" meeting, there exists $3$ people of them who shake hands with each other, find the minimum value of $n.$

Let be given points $A,B$ on a circle and let $P$ be a variable point on that circle. Let point $M$ be determined by $P$ as the point that is either on segment $PA$ with $AM=MP+PB$ or on segment $PB$ with $AP+MP=PB$. Find the locus of points $M$.

For a positive real number $\alpha$, define \[S(\alpha)=\{ \lfloor n\alpha\rfloor \; \vert \; n=1,2,3,\cdots \}.\] Prove that $\mathbb{N}$ cannot be expressed as the disjoint union of three sets $S(\alpha)$, $S(\beta)$, and $S(\gamma)$.