For a positive integer $m$, prove that the number of pairs of positive integers $(x,y)$ which satisfies the following two conditions is even or $0$. (i): $x^2-3y^2+2=16m$ (ii): $2y \le x-1$

Find all triplets of nonnegative integers $(x, y, z)$ that satisfy: $x^2-3y^2=y^2-3z^2=22$.

Introduce a standard scalar product in $\mathbb{R}^4.$ Let $V$ be a partial vector space in $\mathbb{R}^4$ produced by $\left( \begin{array}{c} 1 \\ -1 \\ -1 \\ 1 \end{array} \right),\left( \begin{array}{c} 1 \\-1 \\ 1 \\ -1 \end{array} \right).$ Find a pair of base of orthogonal complement $W$ for $V$ in $\mathbb{R}^4.$

The integers $1$ through $1000$ are located on the circumference of a circle in natural order. Starting with $1$, every fifteenth number (i.e.,$1, 16, 31, \cdots$ ) is marked. The marking is continued until an already marked number is reached. How many of the numbers will be left unmarked?

Let $K$ be a point in the interior of an acute triangle $ABC$ and $ARBPCQ$ be a convex hexagon whose vertices lie on the circumcircle $\Gamma$ of the triangle $ABC.$ Let $A_1$ be the second point where the circle passing through $K$ and tangent to $\Gamma$ at $A$ intersects the line $AP.$ The points $B_1$ and $C_1$ are defined similarly. Prove that \[ \min\left\{\frac{PA_1}{AA_1}, \: \frac{QB_1}{BB_1}, \: \frac{RC_1}{CC_1}\right\} \leq 1.\]

The lateral edges of a right square pyramid are of length $a$. Let $ABCD$ be the base of the pyramid, $E$ its top vertex and $F$ the midpoint of $CE$. Assuming that $BDF$ is an equilateral triangle, compute the volume of the pyramid.

Let $I$ be the incenter and $O$ be the circumcenter of triangle $ABC,$ where $\angle A < \angle B < \angle C.$ Points $P$ and $Q$ are such that $AIOP$ and $BIOQ$ are isosceles trapezoids ($AI \parallel OP,$ $BI \parallel OQ$). Prove that $CP = CQ.$ [i]Proposed by Volodymyr Brayman and Matthew Kurskyi[/i]

Consider the equation \[(3x^3+xy^2)(x^2y+3y^3)=(x-y)^7\] (a) Prove that there are infinitely many pairs $(x,y)$ of positive integers satisfying the equation. (b) Describe all pairs $(x,y)$ of positive integers satisfying the equation.

Determine all $f:\mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ such that $f(m)\geq m$ and $f(m+n) \mid f(m)+f(n)$ for all $m,n\in \mathbb{Z}^+$

From an $n\times n$ square divided into $n^2$ unit squares, one corner unit square is cut off. Find all positive integers $n$ for which it is possible to tile the remaining part of the square with $L$-trominos. [img]https://cdn.artofproblemsolving.com/attachments/0/4/d13e6e7016d943b867f44375a2205b10ccf552.png[/img]

Find an integer $n$, where $100 \leq n \leq 1997$, such that \[ \frac{2^n+2}{n} \] is also an integer.

Given a triangle $ABC$, let $ P$ be the point which minimizes the sum of squares of distances from the sides of the triangle. Let $D, E, F$ the projections of $ P$ on the sides of the triangle $ABC$. Show that $P$ is the barycenter of $DEF$. (Jack D’Aurizio)

the positive divisors $d_1,d_2,\cdots,d_k$ of a positive integer $n$ are ordered \[1=d_1<d_2<\cdots<d_k=n\] Suppose $d_7^2+d_{15}^2=d_{16}^2$. Find all possible values of $d_{17}$.

Let $x, y$ and $z$ be positive real numbers such that $x + y + z = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$ . Prove that $xy + yz + zx \ge 3$.

Let $ABCD$ be a tetrahedron such that \[AB^2+CD^2=AC^2+BD^2=AD^2+BC^2.\] Show that at least one of its faces is an acute triangle.

Real numbers $x$ and $y$ satisfy $x + y = 4$ and $x \cdot y = -2$. What is the value of \[x + \frac{x^3}{y^2} + \frac{y^3}{x^2} + y?\] $\textbf{(A)}\ 360\qquad\textbf{(B)}\ 400\qquad\textbf{(C)}\ 420\qquad\textbf{(D)}\ 440\qquad\textbf{(E)}\ 480$

Define $f_{0}(x)=e^x$ and $f_{n+1}(x)=x f_{n}'(x)$. Show that $\sum_{n=0}^{\infty} \frac{f_{n}(1)}{n!}=e^e$.

[b]Q.[/b] Prove that: $$\sum_{\text{cyc}}\tan (\tan A) - 2 \sum_{\text{cyc}} \tan \left(\cot \frac{A}{2}\right) \geqslant -3 \tan (\sqrt 3)$$where $A, B$ and $C$ are the angles of an acute-angled $\triangle ABC$. [i]Proposed by SA2018[/i]

The diagonals of a pentagon $ABCDE$ determine another pentagon $MNPQR$. If $MNPQR$ and $ABCDE$ are similar, must $ABCDE$ be regular?

Let $k$ be a fixed positive integer. The sequence $\{a_{n}\}_{n\ge1}$ is defined by \[a_{1}=k+1, a_{n+1}=a_{n}^{2}-ka_{n}+k.\] Show that if $m \neq n$, then the numbers $a_{m}$ and $a_{n}$ are relatively prime.

The digits from $1$ to $9$ are placed in the figure below with one digit in each square. The sum of three numbers placed in the same horizontal or vertical line is $13$. Show that the marked place says $4$. [img]https://cdn.artofproblemsolving.com/attachments/a/f/517b644caf59bbc57701662f21d57465855dc1.png[/img]

Determine whether there is a tetrahedron $ABCD$ with the longest edge of length 1 such that all its faces are similar right triangles with right angles at vertices $B,C.$ If so, determine which edge is the longest, which is the shortest and what is its length.

Let $R$ be the rectangle in the Cartesian plane with vertices at $(0,0), (2,0), (2,1),$ and $(0,1)$. $R$ can be divided into two unit squares, as shown; the resulting figure has seven edges. [asy] size(3cm); draw((0,0)--(2,0)--(2,1)--(0,1)--cycle); draw((1,0)--(1,1)); [/asy] How many subsets of these seven edges form a connected figure?

[b]p1.[/b] Alex and Sam have a friend Pat, who is younger than they are. Alex, Sam and Pat all share a birthday. When Pat was born, Alex’s age times Sam’s age was $42$. Now Pat’s age is $33$ and Alex’s age is a prime number. How old is Sam now? Show your work and justify your answer. (All ages are whole numbers.) [b]p2.[/b] Let $ABCD$ be a square with side length $2$. The four sides of $ABCD$ are diameters of four semicircles, each of which lies inside the square. The set of all points which lie on or inside two of these semicircles is a four petaled flower. Find (with proof) the area of this flower. [img]https://cdn.artofproblemsolving.com/attachments/5/5/bc724b9f74c3470434c322020997a533986d33.png[/img] [b]p3.[/b] A prime number is called [i]strongly prime[/i] if every integer obtained by permuting its digits is also prime. For example $113$ is strongly prime, since $113$, $131$, and $311$ are all prime numbers. Prove that there is no strongly prime number such that each of the digits $1, 3, 7$, and $9$ appears at least once in its decimal representation. [b]p4.[/b] Suppose $n$ is a positive integer. Let an be the number of permutations of $1, 2, . . . , n$, where $i$ is not in the $i$-th position, for all $i$ with $1 \le i \le n$. For example $a_3 = 2$, where the two permutations that are counted are $231$, and $312$. Let bn be the number of permutations of $1, 2, . . . , n$, where no $i$ is followed by $i + 1$, for all $i$ with $1 \le i \le n - 1$. For example $b_3 = 3$, where the three permutations that are counted are $132$, $213$, and $321$. For every $n \ge 1$, find (with proof) a simple formula for $\frac{a_{n+1}}{b_n}$. Your formula should not involve summations. Use your formula to evaluate $\frac{a_{2020}}{b_{2019}}$. [b]p5.[/b] Let $n \ge 2$ be an integer and $a_1, a_2, ... , a_n$ be positive real numbers such that $a_1 + a_2 +... + a_n = 1$. Prove that $$\sum^n_{k=1}\frac{a_k}{1 + a_{k+1} - a_{k-1}}\ge 1.$$ (Here $a_0 = a_n$ and $a_{n+1} = a_1$.) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the sum of the coefficients of the polynomial $(63x-61)^4$.