Find all positive integers $n$ such that the equation $y^2 + xy + 3x = n(x^2 + xy + 3y)$ has at least a solution $(x, y)$ in positive integers.

On the plane, given an angle $ xOy$. $ M$ be a mobile point on ray $ Ox$ and $ N$ a mobile point on ray $ Oy$. Let $ d$ be the external angle bisector of angle $ xOy$ and $ I$ be the intersection of $ d$ with the perpendicular bisector of $ MN$. Let $ P$, $ Q$ be two points lie on $ d$ such that $ IP \equal{} IQ \equal{} IM \equal{} IN$, and let $ K$ the intersection of $ MQ$ and $ NP$. $ 1.$ Prove that $ K$ always lie on a fixed line. $ 2.$ Let $ d_1$ line perpendicular to $ IM$ at $ M$ and $ d_2$ line perpendicular to $ IN$ at $ N$. Assume that there exist the intersections $ E$, $ F$ of $ d_1$, $ d_2$ from $ d$. Prove that $ EN$, $ FM$ and $ OK$ are concurrent.

Let $s=a+b+c$, where $a$, $b$, and $c$ are integers that are lengths of the sides of a box. The volume of the box is numerically equal to the sum of the lengths of the twelve edges of the box plus its surface area. Find the sum of the possible values of $s$.

Let $M$ be the set of real numbers of the form $\frac{m+n}{\sqrt{m^2+n^2}}$, where $m$ and $n$ are positive integers. Prove that for every pair $x \in M, y \in M$ with $x < y$, there exists an element $z \in M$ such that $x < z < y.$

The expression \[ \frac{P+Q}{P-Q}-\frac{P-Q}{P+Q} \] where $P=x+y$ and $Q=x-y$, is equivalent to: ${ \textbf{(A)}\ \frac{x^2-y^2}{xy}\qquad\textbf{(B)}\ \frac{x^2-y^2}{2xy}\qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\ \frac{x^2+y^2}{xy} \qquad\textbf{(E)}\ \frac{x^2+y^2}{2xy} } $

Solve equation $(3a-bc)(3b-ac)(3c-ab)=1000$ in integers. (V.Brayman)

Suppose that $a, b, c$ are real numbers such that for all positive numbers $x_1,x_2,\dots,x_n$ we have $(\frac{1}{n}\sum_{i=1}^nx_i)^a(\frac{1}{n}\sum_{i=1}^nx_i^2)^b(\frac{1}{n}\sum_{i=1}^nx_i^3)^c\ge 1$ Prove that vector $(a, b, c)$ is a nonnegative linear combination of vectors $(-2,1,0)$ and $(-1,2,-1)$.

Find $c$ if $a$, $b$, and $c$ are positive integers which satisfy $c=(a + bi)^3 - 107i$, where $i^2 = -1$.

For all reals $x,y,z$ prove that $$\sqrt {x^2+\frac {1}{y^2}}+ \sqrt {y^2+\frac {1}{z^2}}+ \sqrt {z^2+\frac {1}{x^2}}\geq 3\sqrt {2}. $$

Let $n$ be a positive integer. A Steiner tree associated with a finite set $S$ of points in the Euclidean $n$-space is a finite collection $T$ of straight-line segments in that space such that any two points in $S$ are joined by a unique path in $T$ , and its length is the sum of the segment lengths. Show that there exists a Steiner tree of length $1+(2^{n-1}-1)\sqrt{3}$ associated with the vertex set of a unit $n$-cube.

How many ways are there to color the edges of a hexagon orange and black if we assume that two hexagons are indistinguishable if one can be rotated into the other? Note that we are saying the colorings OOBBOB and BOBBOO are distinct; we ignore flips.

$5.$Let x be a real number with $0<x<1$ and let $0.c_1c_2c_3...$ be the decimal expansion of x.Denote by $B(x)$ the set of all subsequences of $c_1c_2c_3$ that consist of 6 consecutive digits. For instance , $B(\frac{1}{22})={045454,454545,545454}$ Find the minimum number of elements of $B(x)$ as $x$ varies among all irrational numbers with $0<x<1$

Let $f$ be a function from nonnegative integers to nonnegative integers such that $f(0)=0$ and $$f(m)=f\left(\left\lfloor \frac{m}{2}\right\rfloor\right)+\left\lceil\frac{m}{2}\right\rceil^2$$ for all positive integers $m.$ Compute $$\frac{f(1)}{1\cdot2}+\frac{f(2)}{2\cdot3}+\frac{f(3)}{3\cdot4}+\cdots+\frac{f(31)}{31\cdot32}.$$(Here, $\lfloor z \rfloor$ is the greatest integer less than or equal to $z,$ and $\lceil z \rceil$ is the least positive integer greater than or equal to $z.$)

Let $BB',CC'$ be altitudes of $\triangle ABC$ and assume $AB$ ≠ $AC$.Let $M$ be the midpoint of $BC$ and $H$ be orhocenter of $\triangle ABC$ and $D$ be the intersection of $BC$ and $B'C'$.Show that $DH$ is perpendicular to $AM$.

Show that there is no polyhedron whose projection on the plane is a nondegenerate triangle.

Let $n$ be a positive integer. In $n$-dimensional space, consider the $2^n$ points whose coordinates are all $\pm 1$. Imagine placing an $n$-dimensional ball of radius 1 centered at each of these $2^n$ points. Let $B_n$ be the largest $n$-dimensional ball centered at the origin that does not intersect the interior of any of the original $2^n$ balls. What is the smallest value of $n$ such that $B_n$ contains a point with a coordinate greater than 2?

Consider those functions $ f: \mathbb{N} \mapsto \mathbb{N}$ which satisfy the condition \[ f(m \plus{} n) \geq f(m) \plus{} f(f(n)) \minus{} 1 \] for all $ m,n \in \mathbb{N}.$ Find all possible values of $ f(2007).$ [i]Author: Nikolai Nikolov, Bulgaria[/i]

For a set $S$ we denote its cardinality by $|S|$. Let $e_1,e_2,\ldots,e_k$ be non-negative integers. Let $A_k$ (respectively $B_k$) be the set of all $k$-tuples $(f_1,f_2,\ldots,f_k)$ of integers such that $0\leq f_i\leq e_i$ for all $i$ and $\sum_{i=1}^k f_i$ is even (respectively odd). Show that $|A_k|-|B_k|=0 \textrm{ or } 1$.

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that $$\displaystyle{f(f(x)) = \frac{x^2 - x}{2}\cdot f(x) + 2-x,}$$ for all $x \in \mathbb{R}.$ Find all possible values of $f(2).$

In the $8\times 8$ grid shown, fill in $12$ of the grid cells with the numbers $1-12$ so that the following conditions are satisfied: [list] [*]Each cell contains at most one number, and each number from $1-12$ is used exactly once. [*]Two cells that both contain numbers may not touch, even at a point. [*]A clue outside the grid pointing at a row or column gives the sum of all the numbers in that row or column. Rows and columns without clues have an unknown sum.[/list] You do not need to prove that your configuration is the only one possible; you merely need to find a configuration that satisfies the constraints above. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.) [asy] size(150); defaultpen(linewidth(0.8)); path arrow=(-1/8,1/8)--(1/8,0)--(-1/8,-1/8)--cycle; int sumRows[]={3,13,20,0,21,0,18,3}; int sumCols[]={24,1,3,0,20,13,0,11}; for(int i=0;i<=8;i=i+1) draw((i,0)--(i,8)^^(0,i)--(8,i)); for(int j=0;j<=7;j=j+1) { if(sumRows[j]>0) { filldraw(shift(-1/4,j+1/2)*arrow,black); label("$"+(string)sumRows[j]+"$",(-7/8,j+1/2)); } if(sumCols[j]>0) { filldraw(shift(j+1/2,8+3/8)*(rotate(270,origin)*arrow),black); label("$"+(string)sumCols[j]+"$",(j+1/2,9)); } } [/asy]

Prove that there exists $N\in\mathbb{N}$ so that for all integer $n > N$, one may find $2019$ pairwise co-prime positive integers with \[n=a_1+a_2+\cdots+a_{2019}\] and \[2019<a_1<a_2<\cdots<a_{2019}\]

Let the sequence $\{a_n\}$ satisfy $a_{n+1}=a_n^3+103,n=1,2,...$. Prove that at most one integer $n$ such that $a_n$ is a perfect square.

Two equal rectangles of area $10$ are arranged as follows. Find the area of the gray rectangle. [img]https://cdn.artofproblemsolving.com/attachments/7/1/112b07530a2ef42e5b2cf83a2cb9fb11dfc9e6.png[/img]

Let $a,b,c$ be positive integers such that the numbers $k=b^c+a, l=a^b+c, m=c^a+b$ to be prime numbers. Prove that at least two of the numbers $k,l,m$ are equal.

Let $ABC$ be a triangle each of whose angles is greater than $30^{\circ}$. Suppose a circle centered with $P$ cuts segments $BC$ in $T,Q; CA$ in $K,L$ and $AB$ in $M,N$ such that they are on a circle in counterclockwise direction in that order.Suppose further $PQK,PLM,PNT$ are equilateral. Prove that: $a)$ The radius of the circle is $\frac{2abc}{a^2+b^2+c^2+4\sqrt{3}S}$ where $S$ is area. $b) a\cdot AP=b\cdot BP=c\cdot PC.$