Let $n$ be an odd integer greater than $1.$ Let $A$ be an $n\times n$ symmetric matrix such that each row and column consists of some permutation of the integers $1,2, \ldots, n.$ Show that each of the integers $1,2, \ldots, n$ must appear in the main diagonal of $A$.

For some real number $r$, the polynomial $8x^3-4x^2-42x+45$ is divisible by $(x-r)^2$. Which of the following numbers is closest to $r$? $\text{(A)} \ 1.22 \qquad \text{(B)} \ 1.32 \qquad \text{(C)} \ 1.42 \qquad \text{(D)} \ 1.52 \qquad \text{(E)} \ 1.62$

Albert rolls a fair six-sided die thirteen times. For each time he rolls a number that is strictly greater than the previous number he rolled, he gains a point, where his first roll does not gain him a point. Find the expected number of points that Albert receives. [i]Proposed by Nathan Ramesh

Let $ABC$ be an isosceles triangle with $AB=AC=20$ . Let $P$ be a point inside the triangle $ABC$ such that the sum of the distances of $P$ to $AB$ and $AC$ is $1$ . Describe the locus of all such points inside triangle $ABC$.

[color=darkred]On a table there are $k\ge 2$ piles having $n_1,n_2,\ldots,n_k$ pencils respectively. A [i]move[/i] consists in choosing two piles having $a$ and $b$ pencils respectively, $a\ge b$ and transferring $b$ pencils from the first pile to the second one. Find the necessary and sufficient condition for $n_1,n_2,\ldots,n_k$ , such that there exists a succession of moves through which all pencils are transferred to the same pile.[/color]

Find all functions $f: \mathbb{Z}^+\to \mathbb{R}$, which satisfies $f(n+1)\geq f(n)$ for all $n\geq 1$ and $f(mn)=f(m)f(n)$ for all $(m,n)=1$.

A dart board is a regular octagon divided into regions as shown. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is probability that the dart lands within the center square? [asy] unitsize(10mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; pair A=(0,1), B=(1,0), C=(1+sqrt(2),0), D=(2+sqrt(2),1), E=(2+sqrt(2),1+sqrt(2)), F=(1+sqrt(2),2+sqrt(2)), G=(1,2+sqrt(2)), H=(0,1+sqrt(2)); draw(A--B--C--D--E--F--G--H--cycle); draw(A--D); draw(B--G); draw(C--F); draw(E--H); [/asy] $ \textbf{(A)}\ \frac{\sqrt{2} - 1}{2} \qquad\textbf{(B)}\ \frac{1}{4} \qquad\textbf{(C)}\ \frac{2 - \sqrt{2}}{2} \qquad\textbf{(D)}\ \frac{\sqrt{2}}{4} \qquad\textbf{(E)}\ 2 - \sqrt{2}$

Let $C_1$ and $C_2$ be two concentric circles in the plane with radii $R$ and $3R$ respectively. Show that the orthocenter of any triangle inscribed in circle $C_1$ lies in the interior of circle $C_2$. Conversely, show that every point in the interior of $C_2$ is the orthocenter of some triangle inscribed in $C_1$.

Triangle $\Delta GRB$ is dissected into $25$ small triangles as shown. All vertices of these triangles are painted in three colors so that the following conditions are satisfied: Vertex $G$ is painted in green, vertex $R$ in red, and $B$ in blue; Each vertex on side $GR$ is either green or red, each vertex on $RB$ is either red or blue, and each vertex on $GB$ is either green or blue. The vertices inside the big triangle are arbitrarily colored. Prove that, regardless of the way of coloring, at least one of the $25$ small triangles has vertices of three different colors.

As shown in the figure, in the acute angle $\vartriangle ABC$, $AB > AC$, $M$ is the midpoint of the minor arc $BC$ of the circumcircle $\Omega$ of $\vartriangle ABC$. $K$ is the intersection point of the bisector of the exterior angle $\angle BAC$ and the extension line of $BC$. From point $A$ draw a line perpendicular on $BC$ and take a point $D$ (different from $A$) on that line , such that $DM = AM$. Let the circumscribed circle of $\vartriangle ADK$ intersect the circle $\Omega$ at point $A$ and at another point $T$. Prove that $AT$ bisects line segment $BC$. [img]https://cdn.artofproblemsolving.com/attachments/1/3/6fde30405101620828d63ae31b8c0ffcec972f.png[/img]

For any positive integer $n $ let $ d(n) $ denote the number of positive divisors of $ n $. Do there exist positive integers $ a $ and $b $, such that $ d(a)=d(b)$ and $ d(a^2 ) = d(b^2 ) $, but $ d(a^3 ) \ne d(b^3 ) $ ?

Evaluate $\prod_{k=1}^{254}\log_{k+1}(k + 2)^{u_k}$, where $u_k = \begin{cases}- k & \text{if} \,\, k \,\, \text{is odd}\\ \frac{1}{k-1} & \text{if} \,\, k \,\, \text{is even} \end{cases}$

Let $ f:[0,1]\longrightarrow\mathbb{R} $ be a continuous and nondecreasing function. [b]a)[/b] Show that the sequence $ \left( \frac{1}{2^n}\sum_{i=1}^{2^n} f\left(\frac{i}{2^n}\right) \right)_{n\ge 1} $ is nonincreasing. [b]b)[/b] Prove that, if there exists some natural index at which the sequence above is equal to $ \int_0^1 f(x)dx, $ then $ f $ is constant.

There are $n$ parallel lines on a plane, and there is a set $S$ of distinct points. Each point in $S$ lies on one of the $n$ lines and is colored either red or blue. Determine the minimum value of $n$ such that if $S$ satisfies the following condition, it is guaranteed that there are infinitely many red points and infinitely many blue points. [list] [*] Each line contains at least one red point and at least one blue point from $S$. [*] Consider a triangle formed by three elements of $S$ located on three distinct lines. If two of the vertices of the triangle are red, there must exist a blue point, not one of the vertices, either inside or on the boundary of the triangle. Similarly, if two of the vertices are blue, there must exist a red point, not one of the vertices, either inside or on the boundary of the triangle. [/list]

On a line $\ell$ there exists $3$ points $A, B$, and $C$ where $B$ is located between $A$ and $C$. Let $\Gamma_1, \Gamma_2, \Gamma_3$ be circles with $AC, AB$, and $BC$ as diameter respectively; $BD$ is a segment, perpendicular to $\ell$ with $D$ on $\Gamma_1$. Circles $\Gamma_4, \Gamma_5, \Gamma_6$ and $\Gamma_7$ satisfies the following conditions: $\bullet$ $\Gamma_4$ touches $\Gamma_1, \Gamma_2$, and$ BD$. $\bullet$ $\Gamma_5$ touches $\Gamma_1, \Gamma_3$, and $BD$. $\bullet$ $\Gamma_6$ touches $\Gamma_1$ internally, and touches $\Gamma_2$ and $\Gamma_3$ externally. $\bullet$ $\Gamma_7$ passes through $B$ and the tangent points of $\Gamma_2$ with $\Gamma_6$, and $\Gamma_3$ with $\Gamma_6$. Show that the circles $\Gamma_4, \Gamma_5$, and $\Gamma_7$ are congruent.

In isosceles triangle $ABC$ with base side $AB$, on side $BC$ it is given point $M$. Let $O$ be a circumcenter and $S$ incenter of triangle $ABC$. Prove that $$ SM \mid \mid AC \Leftrightarrow OM \perp BS$$

The polynomial $ax^2 + bx + c$ crosses the $x$-axis at $x = 10$ and $x = -6$ and crosses the $y$-axis at $y = 10$. Compute $a + b + c$.

Let $n \geq 2$ be a positive integer, and suppose buildings of height $1, 2, \ldots, n$ are built in a row on a street. Two distinct buildings are said to be $\emph{roof-friendly}$ if every building between the two is shorter than both buildings in the pair. For example, if the buildings are arranged $5, 3, 6, 2, 1, 4,$ there are $8$ roof-friendly pairs: $(5, 3), (5, 6), (3, 6), (6, 2), (6, 4), (2, 1),$ $(2, 4), (1, 4).$ Find, with proof, the minimum and maximum possible number of roof-friendly pairs of buildings, in terms of $n.$

How many three-digit numbers satisfy the property that the middle digit is the average of the first and the last digits? $ \textbf{(A)}\ 41\qquad \textbf{(B)}\ 42\qquad \textbf{(C)}\ 43\qquad \textbf{(D)}\ 44\qquad \textbf{(E)}\ 45$

Consider parallelogram $ABCD$ and let $E$ be the midpoint of $BC$. In segment $DE$ a point $F$ is chosen such that $AF$ is perpendicular to $DE$. Prove that $\angle CDE=\angle EFB$.

Determine all two-digit positive integers $n =\overline{ab}$ (in the decimal system) with the property that for all integers $x$ the difference $x^a - x^b$ is divisible by $n$.

In triangle $ABC$, we have that $AB=AC$, $BC=16$, and that the area of $\triangle ABC$ is $120$. Compute the length of $AB$.

Let $ABC$ be an acute triangle. Let $D$ be a point on the line parallel to $AC$ that passes through $B$, such that $\angle BDC = 2\angle BAC$ as well as such that $ABDC$ is a convex quadrilateral. Show that $BD + DC = AC$.

Find all solutions of the differential equation $zz" - 2z'z' = 0$ which pass through the point $x=1, z=1.$

Problem 3 (MAR CP 1992) : From the digits $1,2,...,9$, we write all the numbers formed by these nine digits (the nine digits are all distinct), and we order them in increasing order as follows : $123456789$, $123456798$, ..., $987654321$. What is the $100000th$ number ?