Let $ a, b \in \mathbb{Z}$ which are not perfect squares. Prove that if \[ x^2 \minus{} ay^2 \minus{} bz^2 \plus{} abw^2 \equal{} 0\] has a nontrivial solution in integers, then so does \[ x^2 \minus{} ay^2 \minus{} bz^2 \equal{} 0.\]

For a positive integer $n$, let $S(n), P(n)$ denote the sum and the product of all the digits of $n$ respectively. 1) Find all values of n such that $n = P(n)$: 2) Determine all values of n such that $n = S(n) + P(n)$.

How many four-digit integers $abcd$, with $a \neq 0$, have the property that the three two-digit integers $ab<bc<cd$ form an increasing arithmetic sequence? One such number is $4692$, where $a=4$, $b=6$, $c=9$, and $d=2$. $\textbf{(A)}\ 9\qquad\textbf{(B)}\ 15\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 17\qquad\textbf{(E)}\ 20$

Let $m \in {\mathbb R}$ and $$x^2+(m-4)x+(m^2-3m+3)=0$$ equations roots are $x_1$ and $x_2$ and $x_1^2+x_2^2=6$. Find all $m$ values.

For any natural number $m \ge 1$ and any real number $x \ge 0$ we define expression $$E (x, m) = \frac{(1^4 + x) (3^4 + x) (5^4 + x) ... [(2m -1)^ 4 + x]}{(2^4 + x) (4^4 + x) (6^4 + x) ... [(2m )^ 4 + x]}.$$ It is known that $E\left(\frac{1}{4},m\right)=\frac{1}{1013}.$ . Determine the value of $m$

Given a positive integer $k$, call $n$ [i]good[/i] if among $$\binom{n}{0},\binom{n}{1},\binom{n}{2},...,\binom{n}{n}$$ at least $0.99n$ of them are divisible by $k$. Show that exists some positive integer $N$ such that among $1,2,...,N$, there are at least $0.99N$ good numbers.

The repeating decimal $0.328181818181...$ can equivalently be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

For how many integers $1\leq n\leq 9999$ is there a solution to the congruence \[\phi(n)\equiv 2\,\,\,\pmod{12},\] where $\phi(n)$ is the Euler phi-function?

Given a nonisosceles, nonright triangle ABC, let O denote the center of its circumscribed circle, and let $A_1$, $B_1$, and $C_1$ be the midpoints of sides BC, CA, and AB, respectively. Point $A_2$ is located on the ray $OA_1$ so that $OAA_1$ is similar to $OA_2A$. Points $B_2$ and $C_2$ on rays $OB_1$ and $OC_1$, respectively, are defined similarly. Prove that lines $AA_2$, $BB_2$, and $CC_2$ are concurrent, i.e. these three lines intersect at a point.

There are $2000$ cities in the country, some pairs of cities are connected by roads. It is known that no more than $N$ different non-self-intersecting cyclic routes of odd length. Prove that the country can be divided into $2N +2$ republics so that no two cities from the same republic are connected by a road.

A parallelogram $ABCD$ is given. On the diagonal BD, a point $P$ is selected such that $AP = BD$ is satisfied. Point $Q$ is the midpoint of segment $CP$. Prove that $\angle BQD = 90^o$. [img]https://cdn.artofproblemsolving.com/attachments/2/0/4bc69ec0330e2afa6b560c56da5dd783b16efb.png[/img] .

For $ a,b,c\geq 0$ with $ a\plus{}b\plus{}c\equal{}1$, prove that $ \sqrt{a\plus{}\frac{(b\minus{}c)^2}{4}}\plus{}\sqrt{b}\plus{}\sqrt{c}\leq \sqrt{3}$

Determine all ordered pairs $(m, n)$ of positive integers such that \[\frac{n^{3}+1}{mn-1}\] is an integer.

Let $a$, $b$, $c$ be positive real numbers for which \[ \frac{5}{a} = b+c, \quad \frac{10}{b} = c+a, \quad \text{and} \quad \frac{13}{c} = a+b. \] If $a+b+c = \frac mn$ for relatively prime positive integers $m$ and $n$, compute $m+n$. [i]Proposed by Evan Chen[/i]

Kevin is in first grade, so his teacher asks him to calculate $20+1\cdot 6+k$, where $k$ is a real number revealed to Kevin. However, since Kevin is rude to his Aunt Sally, he instead calculates $(20+1)\cdot (6+k)$. Surprisingly, Kevin gets the correct answer! Assuming Kevin did his computations correctly, what was his answer? [i]Proposed by James Lin[/i]

There are $n$ coins in a row. Two players take turns picking a coin and flipping it. The location of the heads and tails should not repeat. Loses the one who can not make a move. Which of player can always win, no matter how his opponent plays?

Let $ p \geq 2$ be a prime number. Eduardo and Fernando play the following game making moves alternately: in each move, the current player chooses an index $i$ in the set $\{0,1,2,\ldots, p-1 \}$ that was not chosen before by either of the two players and then chooses an element $a_i$ from the set $\{0,1,2,3,4,5,6,7,8,9\}$. Eduardo has the first move. The game ends after all the indices have been chosen .Then the following number is computed: $$M=a_0+a_110+a_210^2+\cdots+a_{p-1}10^{p-1}= \sum_{i=0}^{p-1}a_i.10^i$$. The goal of Eduardo is to make $M$ divisible by $p$, and the goal of Fernando is to prevent this. Prove that Eduardo has a winning strategy. [i]Proposed by Amine Natik, Morocco[/i]

All edges of a tetrahedron $ABCD$ are equal. The tetrahedron $ABCD$ is inscribed in a sphere. $CC'$ and $DD'$ are diameters. Find the angle between the planes $ABC$' and $ACD'$. (A Zaslavskiy)

A box contains five cards, numbered 1, 2, 3, 4, and 5. Three cards are selected randomly without replacement from the box. What is the probability that 4 is the largest value selected? $\textbf{(A) }\frac{1}{10}\qquad\textbf{(B) }\frac{1}{5}\qquad\textbf{(C) }\frac{3}{10}\qquad\textbf{(D) }\frac{2}{5}\qquad\textbf{(E) }\frac{1}{2}$

Let $R{}$ and $r{}$ be the radii of the circumscribed and inscribed circles of the acute-angled triangle $ABC{}$ respectively. The point $M{}$ is the midpoint of its largest side $BC.$ The tangents to its circumscribed circle at $B{}$ and $C{}$ intersect at $X{}$. Prove that \[\frac{r}{R}\geqslant\frac{AM}{AX}.\]

Find all functions $ f $ defined on real numbers and taking values in the set of real numbers such that $ f(x+y)+f(y+z)+f(z+x) \geq f(x+2y+3z) $ for all real numbers $ x,y,z $. [hide]There is an infinity of such functions. Every function with the property that $ 3 \inf f \geq \sup f $ is a good one. I wonder if there is a way to find all the solutions. It seems very strange.[/hide]

The number of integral points on the circle with center $(199,0)$, radius of $199$ is________.

Construct a trapezoid given the midpoints of the legs, the point of intersection of the diagonals and the foot of the perpendicular, drawn from this point on the larger base.

Write the number $1000$ as the sum of at least two consecutive positive integers. How many (different) ways are there to write it?

Let $a,b,c$ be the sidelengths of a triangle such that $a^2+b^2 > 5c^2$ holds. Prove that $c$ is the shortest side of the triangle.