Found problems: 85335
How many rational solutions are there to $5x^2+2y^2=1$?
A round robin tournament is held with $2016$ participants. Each player plays each other player once and exactly one game results in a tie. Let $W$ be the sum of the squares of each team's win total and let $L$ be the sum of the squares of each team's loss total. Find the maximum possible value of $W-L$.
[i]Proposed by Matthew Weiss
Find the sum of all integers $x$ satisfying $1 + 8x \le 358 - 2x \le 6x + 94$.
Let $ABCD$ be a rectangle with center $O$ such that $\angle DAC = 60^o$. Bisector of $\angle DAC$ cuts a $DC$ at $S$, $OS$ and $AD$ intersect at $L$, $BL$ and $AC$ intersect at $M$. Prove that $SM \parallel CL$.
In a certain country, every two cities are connected either by an airline route or by a railroad. Prove that one can always choose a type of transportation in such a way that each city can be reached from any other city with at most two transfers.
Let $p$ be a prime divisor of $x^3 + x^2 - 4x + 1$. Prove that $p$ is a cubic residue modulo $13$.
A cake has the form of an $ n$ x $ n$ square composed of $ n^{2}$ unit squares. Strawberries lie on some of the unit squares so that each row or column contains exactly one strawberry; call this arrangement $\mathcal{A}$.
Let $\mathcal{B}$ be another such arrangement. Suppose that every grid rectangle with one vertex at the top left corner of the cake contains no fewer strawberries of arrangement $\mathcal{B}$ than of arrangement $\mathcal{A}$. Prove that arrangement $\mathcal{B}$ can be obtained from $ \mathcal{A}$ by performing a number of switches, defined as follows:
A switch consists in selecting a grid rectangle with only two strawberries, situated at its top right corner and bottom left corner, and moving these two strawberries to the other two corners of that rectangle.
A frog jumps on the coordinate lattice, starting from the point $(1,1)$, according to the following rules:
(i) From point $(a,b)$ the frog can jump to either $(2a,b)$ or $(a,2b)$;
(ii) If $a>b$, the frog can also jump from $(a,b)$ to $(a-b,b)$, while for $a<b$ it can jump from $(a,b)$ to $(a,b-a)$.
Can the frog get to the point: (a) $(24,40)$; (b) $(40,60)$; (c) $(24,60)$; (d) $(200,4)$?
For every sequence $p_1<p_2<\cdots<p_8$ of eight prime numbers, determine the largest integer $N$ for which the following equation has no solution in positive integers $x_1,\ldots,x_8$:
$$p_1\, p_2\, \cdots\, p_8
\left( \frac{x_1}{p_1}+ \frac{x_2}{p_2}+ ~\cdots~ +\frac{x_8}{p_8} \right)
~~=~~ N $$
[i]Proposed by Gerhard Woeginger, Austria[/i]
Distinct points $A$, $B$ and $C$ lie on a line in this order. Point $D$ lies on the perpendicular bisector of the segment $BC$. Denote by $M$ the midpoint of the segment $BC$. Let $r$ be the radius of the incircle of the triangle $ABD$ and let $R$ be the radius of the circle with center lying outside the triangle $ACD$, tangent to $CD$, $AC$ and $AD$. Prove that $DM=r+R$.
On a plane are given $6$ red, $6$ blue, and $6$ green points, such that no three of the given points lie on a line. Prove that the sum of the areas of the triangles whose vertices are of the same color does not exceed quarter the sum of the areas of all triangles with vertices in the given points.
A marble is placed on each $33$ unit square of a $10*10$ chessboard. After that, the number of marbles in the same row or column with that square is written on each of the remaining empty unit squares. What is the maximum sum of the numbers written on the board?
Plane vectors form a group for addition. Show that this group has a generator system of every set $S$ that contains a Borel subset of positive linear measure of a circular arc.
Find all positive integers $n$ such that equation $$3a^2-b^2=2018^n$$ has a solution in integers $a$ and $b$.
In a chess tournament there are $n>2$ players. Every two players play against each other exactly once. It is known that exactly $n$ games end as a tie. For any set $S$ of players, including $A$ and $B$, we say that $A$ [i]admires[/i] $B$ [i]in that set [/i]if
i) $A$ does not beat $B$; or
ii) there exists a sequence of players $C_1,C_2,\ldots,C_k$ in $S$, such that $A$ does not beat $C_1$, $C_k$ does not beat $B$, and $C_i$ does not beat $C_{i+1}$ for $1\le i\le k-1$.
A set of four players is said to be [i]harmonic[/i] if each of the four players admires everyone else in the set. Find, in terms of $n$, the largest possible number of harmonic sets.
Find an upper bound for the ratio
$$\frac{x_1x_2+2x_2x_3+x_3x_4}{x_1^2+x_2^2+x_3^2+x_4^2}$$
over all quadruples of real numbers $(x_1,x_2,x_3,x_4)\neq (0,0,0,0)$.
[i]Note.[/i] The smaller the bound, the better the solution.
Let $ p$ be a prime number and let $ 0\leq a_{1}< a_{2}<\cdots < a_{m}< p$ and $ 0\leq b_{1}< b_{2}<\cdots < b_{n}< p$ be arbitrary integers. Let $ k$ be the number of distinct residues modulo $ p$ that $ a_{i}\plus{}b_{j}$ give when $ i$ runs from 1 to $ m$, and $ j$ from 1 to $ n$. Prove that
a) if $ m\plus{}n > p$ then $ k \equal{} p$;
b) if $ m\plus{}n\leq p$ then $ k\geq m\plus{}n\minus{}1$.
Luis' friends decided to play a prank on him in his geometry homework. They erased most of a triangle and, instead, drew an equivalent triangle with the sum of its three side lengths. Help Luis complete his homework by reconstructing the original triangle using only a straightedge and compass. Since Luis' method involves measurements, prove that his method results in a triangle longer than the sum of its three sides.
Find all integers $n$ such that $n^{n-1}-1$ is square-free.
For which non-negative integers $n$ is \[K=5^{2n+3} + 3^{n+3} \cdot 2^n\] prime?
Point $M$ is an arbitrary point in the plane and let points $G$ and $H$ be the intersection points of the tangents from point M and the circle $k$. Let $O$ be the center of the circle $k$ and let $K$ be the orthocenter of the triangle $MGH$. Prove that ${\angle}GMH={\angle}OGK$.
Let $N \geqslant 3$ be an integer. In the country of Sibyl, there are $N^2$ towns arranged as the vertices of an $N \times N$ grid, with each pair of towns corresponding to an adjacent pair of vertices on the grid connected by a road. Several automated drones are given the instruction to traverse a rectangular path starting and ending at the same town, following the roads of the country. It turned out that each road was traversed at least once by some drone. Determine the minimum number of drones that must be operating.
[i]Proposed by Sutanay Bhattacharya and Anant Mudgal[/i]
From a point $O$ lying outside the circle $\omega$, two tangents are drawn touching $\omega$ at points $M$ and $N$. A point $K$ is chosen on the segment $MN$. Let points $P$ and $Q$ be the midpoints of segments $KM$ and $OM$ respectively. The circumcircle of triangle $MPQ$ intersects $\omega$ again at point $L$ ($L \neq M$). Prove that the line $LN$ passes through the centroid of triangle $KMO$.
\[\int_0^{\frac \pi 4} \cot(x)\sqrt{\sin(x)}\,\mathrm dx\]
[i]Proposed by Connor Gordon[/i]
The largest divisor of $2,014,000,000$ is itself. What is its fifth largest divisor?
$\textbf{(A) }125,875,000\qquad\textbf{(B) }201,400,000\qquad\textbf{(C) }251,750,000\qquad\textbf{(D) }402,800,000\qquad\textbf{(E) }503,500,000\qquad$