Prove that there are no integers $x$ and $y$ for which $$x^2 +3xy-2y^2 =122.$$

Given the polynomial $a_0x^n+a_1x^{n-1}+\cdots+a_{n-1}x+a_n$, where $n$ is a positive integer or zero, and $a_0$ is a positive integer. The remaining $a$'s are integers or zero. Set $h=n+a_0+|a_1|+|a_2|+\cdots+|a_n|$. [See example 25 for the meaning of $|x|$.] The number of polynomials with $h=3$ is: $ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 9 $

Danyang is doing math. He starts to draw an isosceles triangle, but only manages to draws an angle of $70^{\circ}$ before he has to leave for recess. Find the sum of all possible values for the smallest angle in Danyang's triangle.

The numbers $1$ to $n^2$ are written in an n×n squared paper in the usual ordering. Any sequence of right and downwards steps from a square to an adjacent one (by side) starting at square $1$ and ending at square $n^2$ is called a path. Denote by $L(C)$ the sum of the numbers through which path $C$ goes. (a) For a fixed $n$, let $M$ and $m$ be the largest and smallest $L(C)$ possible. Prove that $M-m$ is a perfect cube. (b) Prove that for no $n$ can one find a path $C$ with $L(C ) = 1996$.

For even $n$, prove that $\sum_{i=1}^{n}{\left((-1)^{i+1}\cdot\frac{1}{i}\right)}=2\sum_{i=1}^{n/2}{\frac{1}{n+2i}}$.

For a positive real number $r$, let $G(r)$ be the minimum value of $|r - \sqrt{m^2+2n^2}|$ for all integers $m$ and $n$. Prove or disprove the assertion that $\lim_{r\to \infty}G(r)$ exists and equals $0.$

In the tetrahedron $ABCD$ the radius of its inscribed sphere is $r$ and the radiuses of the exinscribed spheres (each tangent with a face of the tetrahedron and with the planes of the other faces) are $r_A, r_B, r_C, r_D.$ Prove the inequality $$\frac{1}{\sqrt{r_A^2-r_Ar_B+r_B^2}}+\frac{1}{\sqrt{r_B^2-r_Br_C+r_C^2}}+\frac{1}{\sqrt{r_C^2-r_Cr_D+r_D^2}}+\frac{1}{\sqrt{r_D^2-r_Dr_A+r_A^2}}\leq\frac{2}{r}.$$

Let $ABC$ be an isosceles triangle, and point $D$ in its interior such that $$D \hat{B} C=30^\circ, D \hat{B}A=50^\circ, D \hat{C}B=55^\circ$$ (a) Prove that $\hat B=\hat C=80^\circ$. (b) Find the measure of the angle $D \hat{A} C$.

Find the remainder when $$20^{20}+21^{21}-21^{20}-20^{21}$$ is divided by $100$.

In a triangle $ABC$, $\angle A$ is $60^\circ$. On sides $AB$ and $AC$ we make two equilateral triangles (outside the triangle $ABC$) $ABK$ and $ACL$. $CK$ and $AB$ intersect at $S$ , $AC$ and $BL$ intersect at $R$ , $BL$ and $CK$ intersect at $T$. Prove the radical centre of circumcircle of triangles $BSK, CLR$ and $BTC$ is on the median of vertex $A$ in triangle $ABC$. [i]Proposed by Ali Zamani[/i]

Find the largest positive integer $n$ such that for each prime $p$ with $2<p<n$ the difference $n-p$ is also prime.

Find the smallest natural number nonzero n so that it exists in real numbers $x_1, x_2,..., x_n$ which simultaneously check the conditions: 1) $x_i \in [1/2 , 2]$ , $i = 1, 2,... , n$ 2) $x_1+x_2+...+x_n \ge \frac{7n}{6}$ 3) $\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}\ge \frac{4n}{3}$

Three numbers $N,n,r$ are such that the digits of $N,n,r$ taken together are formed by $1,2,3,4,5,6,7,8,9$ without repetition. If $N = n^2 - r$, find all possible combinations of $N,n,r$.

Let $n \geq 2$ be a natural number, and let $\left( a_{1};\;a_{2};\;...;\;a_{n}\right)$ be a permutation of $\left(1;\;2;\;...;\;n\right)$. For any integer $k$ with $1 \leq k \leq n$, we place $a_k$ raisins on the position $k$ of the real number axis. [The real number axis is the $x$-axis of a Cartesian coordinate system.] Now, we place three children A, B, C on the positions $x_A$, $x_B$, $x_C$, each of the numbers $x_A$, $x_B$, $x_C$ being an element of $\left\{1;\;2;\;...;\;n\right\}$. [It is not forbidden to place different children on the same place!] For any $k$, the $a_k$ raisins placed on the position $k$ are equally handed out to those children whose positions are next to $k$. [So, if there is only one child lying next to $k$, then he gets the raisin. If there are two children lying next to $k$ (either both on the same position or symmetric with respect to $k$), then each of them gets one half of the raisin. Etc..] After all raisins are distributed, a child is unhappy if he could have received more raisins than he actually has received if he had moved to another place (while the other children would rest on their places). For which $n$ does there exist a configuration $\left( a_{1};\;a_{2};\;...;\;a_{n}\right)$ and numbers $x_A$, $x_B$, $x_C$, such that all three children are happy?

Prove that if $a$ is an integer and $d$ is a positive divisor of the number $a^4+a^3+2a^2-4a+3$, then $d$ is a fourth power modulo $13$.

On a circle $4n$ points are chosen ($n \ge 1$). The points are alternately colored yellow and blue. The yellow points are divided into $n$ pairs and the points in each pair are connected with a yellow line segment. In the same manner the blue points are divided into $n$ pairs and the points in each pair are connected with a blue segment. Assume that no three of the segments pass through a single point. Show that there are at least $n$ intersection points of blue and yellow segments.

Imagine a square scheme consisting of $n\times n$ fields with edge length $1$, where $n$ is an arbitrary positive integer. What is the maximum possible length of a route you can follow along the edges of the fields from point $A$ in the lower left corner to point $B$ in the upper right corner if you must never return to one point where you have been before? (The figure shows for $n = 5$ an example of a permitted route and an example of a not permitted route). [img]https://cdn.artofproblemsolving.com/attachments/6/e/92931d87f11b9fb3120b8dccc2c37c35a04456.png[/img]

Determine all triples of positive integers $a, b, c$ that satisfy a) $[a, b] + [a, c] + [b, c] = [a, b, c]$. b) $[a, b] + [a, c] + [b, c] = [a, b, c] + (a, b, c)$. Remark: Here $[x, y$] denotes the least common multiple of positive integers $x$ and $y$, and $(x, y)$ denotes their greatest common divisor.

In the multiplication magic square below, $l, m, n, p, q, r, s, t, u$ are positive integers. The product of any three numbers in any row, column or diagonal is equal to a constant $k$, where $k$ is a number between $11, 000$ and $12, 500$. Find the value of $k$. \begin{tabular}{|l|l|l|} \hline $l$ & $m$ & $n$ \\ \hline $p$ & $q$ & $r$ \\ \hline $s$ & $t$ & $u$ \\ \hline \end{tabular}

On straight line $\ell$ there are points $A$, $B$, $C$ and $D$, following in the indicated order: $AB = a$, $BC = b$, $CD = c$. Segments $AD$ and $BC$ serve as chords of two circles, and the sum of the angular values of the arcs of these circles located on one side of $\ell$ is equal to $360^o$. A third circle passes through $A$ and $B$, intersecting the first two at points $K$ and $M$. The straight line $KM$ intersects $\ell$ at point $E$. Find $AE$.

Let $A, B, C, D, E$ be five distinct points on a circle $\Gamma$ in the clockwise order and let the extensions of $CD$ and $AE$ meet at a point $Y$ outside $\Gamma$. Suppose $X$ is a point on the extension of $AC$ such that $XB$ is tangent to $\Gamma$ at $B$. Prove that $XY = XB$ if and only if $XY$ is parallel $DE$.

Let $I$ be the incenter of the $ABC$ triangle. The circumference that passes through $I$ and has center in $A$ intersects the circumscribed circumference of the $ABC$ triangle at points $M$ and $N$. Prove that the line $MN$ is tangent to the inscribed circle of the $ABC$ triangle.

The vertices of the quadrilateral $ABCD$ lie on a single circle. The diagonals of this rectangle divide the angles of the rectangle at vertices $A$ and $B$ and divides the angles at vertices $C$ and $D$ in a $1: 2$ ratio. Find angles of the quadrilateral $ABCD$.

Natalie has a copy of the unit interval $[0,1]$ that is colored white. She also has a black marker, and she colors the interval in the following manner: at each step, she selects a value $x\in [0,1]$ uniformly at random, and (a) If $x\leq\tfrac12$ she colors the interval $[x,x+\tfrac12]$ with her marker. (b) If $x>\tfrac12$ she colors the intervals $[x,1]$ and $[0,x-\tfrac12]$ with her marker. What is the expected value of the number of steps Natalie will need to color the entire interval black?

It is given a tetrahedron $ABCD$ for which two points of opposite edges are mutually perpendicular. Prove that: (a) the four altitudes of $ABCD$ intersects at a common point $H$; (b) $AH+BH+CH+DH<p+2R$, where $p$ is the sum of the lengths of all edges of $ABCD$ and $R$ is the radii of the sphere circumscribed around $ABCD$. [i]H. Lesov[/i]