Prove that for all n=3,4,5.... there excist odd x,y such $2^n=x^2 + 7y^2$ .

A convex quadrilateral is inscribed in a circle of radius $1$. Prove that the difference between its perimeter and the sum of the lengths of its diagonals is greater than zero and less than $2.$

Which of the following integers cannot be written as the sum of four consecutive odd integers? $\textbf{(A)}\text{ 16}\qquad\textbf{(B)}\text{ 40}\qquad\textbf{(C)}\text{ 72}\qquad\textbf{(D)}\text{ 100}\qquad\textbf{(E)}\text{ 200}$

Find the minimum value of \[\frac{x^3+1}{(y-1)(z+1)}+\frac{y^3+1}{(z-1)(x+1)}+\frac{z^3+1}{(x-1)(y+1)}\] where $x,y,z>1$ are reals.

Determine all functions $f : N_0 \to R$ satisfying $f (x+y)+ f (x-y)= f (3x)$ for all $x,y$.

Let $ABC$ be a triangle with $BC=7,AB=5$, and $AC=8$. Let $M,N$ be the midpoints of sides $AC,AB$ respectively, and let $O$ be the circumcenter of $ABC$. Let $BO, CO$ meet $AC, AB$ at $P$ and $Q$, respectively. If $MN$ meets $PQ$ at $R$ and $OR$ meets $BC$ at $S$, then the value of $OS^2$ can be written in the form $\frac{m}{n}$ where $m,n$ are relatively prime positive integers. Find $100m+n$. [i]Proposed by Vincent Huang[/i]

In the fourth annual Swirled Series, the Oakland Alphas are playing the San Francisco Gammas. The first game is played in San Francisco and succeeding games alternate in location. San Francisco has a $50\%$ chance of winning their home games, while Oakland has a probability of $60\%$ of winning at home. Normally, the series will stretch on forever until one team gets a three game lead, in which case they are declared the winners. However, after each game in San Francisco there is a $50\%$ chance of an earthquake, which will cause the series to end with the team that has won more games declared the winner. What is the probability that the Gammas will win?

A coast artillery gun can fire at every angle of elevation between $0^{\circ}$ and $90^{\circ}$ in a fixed vertical plane. If air resistance is neglected and the muzzle velocity is constant ($=v_0 $), determine the set $H$ of points in the plane and above the horizontal which can be hit.

On a circle, Alina draws $2019$ chords, the endpoints of which are all different. A point is considered [i]marked[/i] if it is either $\text{(i)}$ one of the $4038$ endpoints of a chord; or $\text{(ii)}$ an intersection point of at least two chords. Alina labels each marked point. Of the $4038$ points meeting criterion $\text{(i)}$, Alina labels $2019$ points with a $0$ and the other $2019$ points with a $1$. She labels each point meeting criterion $\text{(ii)}$ with an arbitrary integer (not necessarily positive). Along each chord, Alina considers the segments connecting two consecutive marked points. (A chord with $k$ marked points has $k-1$ such segments.) She labels each such segment in yellow with the sum of the labels of its two endpoints and in blue with the absolute value of their difference. Alina finds that the $N + 1$ yellow labels take each value $0, 1, . . . , N$ exactly once. Show that at least one blue label is a multiple of $3$. (A chord is a line segment joining two different points on a circle.)

Three prime numbers $p,q,r$ and a positive integer $n$ are given such that the numbers \[ \frac{p+n}{qr}, \frac{q+n}{rp}, \frac{r+n}{pq} \] are integers. Prove that $p=q=r $. [i]Nazar Agakhanov[/i]

$5$ monkeys, $5$ snakes, and $5$ tigers are standing in line at the local grocery store, with animals of the same species being indistinguishable. A monkey stands at the front of the line and a tiger stands at the end of the line. Unfortunately, monkeys and tigers are sworn enemies, so monkeys and tigers cannot stand in adjacent places in line. Compute the number of possible arrangements of the line.

A tortoise walks $60$ meters per hour and a lizard walks at $240$ meters per hour. There is a rectangle $ABCD$ where $AB =60$ and $AD =120$. Both start from the vertex $A$ and in the same direction ($A \to B \to D \to A$), crossing the edge of the rectangle. The lizard has the habit of advancing two consecutive sides of the rectangle, turning to go back one, turning to go forward two, turning to go back one and so on. How many times and in what places do the tortoise and the lizard meet when the tortoise completes its third turn?

Given a function $f(x)=a\sin x-\frac{1}{2}\cos 2x+a-\frac{3}{a}+\frac{1}{2}$, where $a\in\mathbb{R}, a\ne 0$. [b](1)[/b] If for any $x\in\mathbb{R}$, inequality $f(x)\le 0$ holds, find all possible value of $a$. [b](2)[/b] If $a\ge 2$, and there exists $x\in\mathbb{R}$, such that $f(x)\le 0$. Find all possible value of $a$.

Let $ S_i$ be the set of all integers $ n$ such that $ 100i\leq n < 100(i \plus{} 1)$. For example, $ S_4$ is the set $ {400,401,402,\ldots,499}$. How many of the sets $ S_0, S_1, S_2, \ldots, S_{999}$ do not contain a perfect square?

Let $n$ be a positive integer. There is a collection of cards that meets the following properties: $\bullet$Each card has a number written in the form $m!$, where $m$ is a positive integer. $\bullet$For every positive integer $t\le n!$, it is possible to choose one or more cards from the collection in such a way $\text{ }$that the sum of the numbers of those cards is $t$. Determine, based on $n$, the smallest number of cards that this collection can have.

In triangle $ABC$, let the circumcenter, incenter, and orthocenter be $O$, $I$, and $H$ respectively. Segments $AO$, $AI$, and $AH$ intersect the circumcircle of triangle $ABC$ at $D$, $E$, and $F$. $CD$ intersects $AE$ at $M$ and $CE$ intersects $AF$ at $N$. Prove that $MN$ is parallel to $BC$.

The numbers 1 through 10 can be arranged along the vertices and sides of a pentagon so that the sum of the three numbers along each side is the same. The diagram below shows an arrangement with sum 16. Find, with proof, the smallest possible value for a sum and give an example of an arrangement with that sum. [asy] int i; pair[] A={dir(18+72*0), dir(18+72*1), dir(18+72*2),dir(18+72*3), dir(18+72*4), dir(18+72*0)}; pair O=origin; int[] v = {7,1,10,4,3}; int[] s = {8, 5, 2, 9, 6}; for(i=0; i<5; i=i+1) { label(string(v[i]), A[i], dir(O--A[i])); label(string(s[i]), A[i]--A[i+1], dir(-90)*dir(A[i]--A[i+1])); } draw(rotate(0)*polygon(5));[/asy]

Let $n\geq 3$ be an integer. Let also $P_1,P_2,...,P_n$ be different two-element-subsets of $M=\{1,2,...,n\}$, such that when for $i,j \in M , i\neq j$ the sets $P_i,P_j$ are not totally disjoint, then there is a $k \in M$ with $P_k = \{ i,j\}$. Prove that every element of $M$ occurse in exactly $2$ of these subsets.

Given a sequence of integers $A_1,A_2,\cdots A_{99}$ such that for every sub-sequence that contains $m$ consecutive elements, there exist not more than $max\{ \frac{m}{3} ,1\}$ odd integers. Let $S=\{ (i,j) \ | i<j \}$ such that $A_i$ is even and $A_j$ is odd. Find $max\{ |S|\}$.

Let $n$ be a natural number, and suppose that the equation $$x_1x_2+x_2x_3+x_3x_4+x_4x_5+\dots +x_{n-1}x_n+x_nx_1=0$$ has a solution with all the $x_i$s equal to $\pm 1$. Prove that $n$ is divisible by $4$.

Define $f(n)$ to be the smallest integer such that for every positive divisor $d \mid n,$ either $n \mid d^d$ or $d^d \mid n^{f(n)}.$ How many positive integers $b < 1000$ which are not squarefree satisfy the equation $f(2023) \cdot f(b) = f(2023b)$?

Prove that $n! \geq n^{\frac{n}{2}}$ for all natural numbers $n$. Also, show that the inequality is strict for $n > 2$.

Around a circular table an even number of persons have a discussion. After a break they sit again around the circular table in a different order. Prove that there are at least two people such that the number of participants sitting between them before and after a break is the same.

Find the smallest positive integer $N$ satisfying the following three properties. $\bullet$ N leaves a remainder of $5$ when divided by $7$. $\bullet$ N leaves a remainder of $6$ when divided by $ 8$. $\bullet$ N leaves a remainder of $7$ when divided by $9$.

Let $N$ be the smallest positive integer divisble by $10^{2023} - 1$ that only has the digits $4$ and $8$ in decimal form (these digits may be repeated). Compute the sum of the digits of $\frac{N}{10^{2023}-1}$ .