Let $n$ and $k$ be positive integers. The square in the $i$th row and $j$th column of an $n$-by-$n$ grid contains the number $i+j-k$. For which $n$ and $k$ is it possible to select $n$ squares from the grid, no two in the same row or column, such that the numbers contained in the selected squares are exactly $1,\,2,\,\ldots,\,n$?

A lattice point in the plane is a point both of whose coordinates are integers. Each lattice point has four neighboring points: upper, lower, left, and right. Let $k$ be a circle with radius $r \geq 2$, that does not pass through any lattice point. An interior boundary point is a lattice point lying inside the circle $k$ that has a neighboring point lying outside $k$. Similarly, an exterior boundary point is a lattice point lying outside the circle $k$ that has a neighboring point lying inside $k$. Prove that there are four more exterior boundary points than interior boundary points.

Find all integers $n$, $n \ge 2$, such that the numbers $1!, 2 !,..., (n - 1)!$ give distinct remainders when divided by $n$.

Given a circular arc, find a triangle of the smallest possible area which covers the arc so that the endpoints of the arc lie on the same side of the triangle.

Suppose $r$, $s$, and $t$ are nonzero reals such that the polynomial $x^2 + rx + s$ has $s$ and $t$ as roots, and the polynomial $x^2 + tx + r$ has $5$ as a root. Compute $s$.

Investigate the properties of the tetrahedron $ABCD$ for which there is equality $$\frac{AD}{ \sin \alpha}=\frac{BD}{\sin \beta}=\frac{CD}{ \sin \gamma}$$ where $\alpha, \beta, \gamma$ are the values ​​of the dihedral angles at the edges $AD, BD$ and $CD$, respectively.

Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ that satisfy \[f(f(x)+y)=2x+f(f(y)-x)\quad\text{for all real}\ x,y. \]

Prove the following inequality. \[ \frac{2}{2\plus{}e^{\frac 12}}<\int_0^1 \frac{dx}{1\plus{}xe^{x}}<\frac{2\plus{}e}{2(1\plus{}e)}\]

Suppose that there are $16$ variables $\{a_{i,j}\}_{0\leq i,j\leq 3}$, each of which may be $0$ or $1$. For how many settings of the variables $a_{i,j}$ do there exist positive reals $c_{i,j}$ such that the polynomial \[f(x,y)=\sum_{0\leq i,j\leq 3}a_{i,j}c_{i,j}x^iy^j\] $(x,y\in\mathbb{R})$ is bounded below?

There are two spherical balls of different sizes lying in two corners of a rectangular room, each touching two walls and the floor. If there is a point on each ball which is $5$ inches from each wall which that ball touches and $10$ inches from the floor, then the sum of the diameters of the balls is $\textbf{(A) }20\text{ inches}\qquad\textbf{(B) }30\text{ inches}\qquad\textbf{(C) }40\text{ inches}\qquad$ $\textbf{(D) }60\text{ inches}\qquad \textbf{(E) }\text{not determined by the given information}$

A pair of positive integers $(m,n)$ is called [i]compatible[/i] if $m \ge \tfrac{1}{2} n + 7$ and $n \ge \tfrac{1}{2} m + 7$. A positive integer $k \ge 1$ is called [i]lonely[/i] if $(k,\ell)$ is not compatible for any integer $\ell \ge 1$. Find the sum of all lonely integers. [i]Proposed by Evan Chen[/i]

This problem is generalization of [url=http://www.mathlinks.ro/Forum/viewtopic.php?t=5918]this one[/url]. Suppose $G$ is a graph and $S\subset V(G)$. Suppose we have arbitrarily assign real numbers to each element of $S$. Prove that we can assign numbers to each vertex in $G\backslash S$ that for each $v\in G\backslash S$ number assigned to $v$ is average of its neighbors.

Prove that for each prime number distinct from $2$ and $5$ there exist infinitely many multiples of $p$ of the form $1111...1$.

In an acute-angled triangle $ABC$, the orthocenter is $H$. $I_H$ is the incenter of $\vartriangle BHC$. The bisector of $\angle BAC$ intersects the perpendicular from $I_H$ to the side $BC$ at point $K$. Let $F$ be the foot of the perpendicular from $K$ to $AB$. Prove that $2KF+BC=BH +HC$ A. Voidelevich

Acute triangle $ABC$ has altitudes $AD$, $BE$, and $CF$. Point $D$ is projected onto $AB$ and $AC$ to points $D_c$ and $D_b$ respectively. Likewise, $E$ is projected to $E_a$ on $BC$ and $E_c$ on $AB$, and $F$ is projected to $F_a$ on $BC$ and $F_b$ on $AC$. Lines $D_bD_c$, $E_cE_a$, $F_aF_b$ bound a triangle of area $T_1$, and lines $E_cF_b$, $D_bE_a$, $F_aD_c$ bound a triangle of area $T_2$. What is the smallest possible value of the ratio $T_2/T_1$?

Let $n$ be a positive integer. Two players $A$ and $B$ play a game in which they take turns choosing positive integers $k \le n$. The rules of the game are: (i) A player cannot choose a number that has been chosen by either player on any previous turn. (ii) A player cannot choose a number consecutive to any of those the player has already chosen on any previous turn. (iii) The game is a draw if all numbers have been chosen; otherwise the player who cannot choose a number anymore loses the game. The player $A$ takes the first turn. Determine the outcome of the game, assuming that both players use optimal strategies. [i]Proposed by Finland[/i]

Suppose that you are given the foot of the altitude from vertex $A$ of a scalene triangle $ABC$, the midpoint of the arc with endpoints $B$ and $C$, not containing $A$ of the circumscribed circle of $ABC$, and also a third point $P$. Construct the triangle from these three points if $P$ is the a) orthocenter b) centroid c) incenter of the triangle.

Area of convex quadrilateral $ABCD$ is $1$. Prove that we can find four points on its side (vertex included) or inside, satisfying: area of triangles comprised of any three points of the four points is larger than $\frac{1}{4}$.

Show that if $x, y, z$ are positive integers, then $(xy+1)(yz+1)(zx+1)$ is a perfect square if and only if $xy+1$, $yz+1$, $zx+1$ are all perfect squares.

Given a polynomial with integer coefficients $$P(x) = x^{20} + a_{19}x^{19} +... + a_1x + a_0,$$ having $20$ different real roots. Determine the maximum number of roots such a polynomial $P$ can have in the interval $(99, 100)$.

A town's population increased by $1,200$ people, and then this new population decreased by $11 \%$. The town now had $32$ less people than it did before the $1,200$ increase. What is the original population? $ \textbf{(A)}\ 1,200 \qquad\textbf{(B)}\ 11,200 \qquad\textbf{(C)}\ 9,968 \qquad\textbf{(D)}\ 10,000 \qquad\textbf{(E)}\ \text{none of these} $

For each date of year $1998$, we calculate day$^{month}$ −year and determine the greatest power of $3$ that divides it. For example, for April $21$ we get $21^4 - 1998 =192483 = 3^3 \cdot 7129$, which is divisible by $3^3$ and not by $3^4$ . Find all dates for which this power of $3$ is the greatest.

Let $a_1,a_2,...,a_9$ be a sequence of numbers satisfying $0 < p \le a_i \le q$ for each $i = 1,2,..., 9$. Prove that $\frac{a_1}{a_9}+\frac{a_2}{a_8}+...+\frac{a_9}{a_1} \le 1 + \frac{4(p^2+q^2)}{pq}$

In the $xyz$-plane given points $P,\ Q$ on the planes $z=2,\ z=1$ respectively. Let $R$ be the intersection point of the line $PQ$ and the $xy$-plane. (1) Let $P(0,\ 0,\ 2)$. When the point $Q$ moves on the perimeter of the circle with center $(0,\ 0,\ 1)$ , radius 1 on the plane $z=1$, find the equation of the locus of the point $R$. (2) Take 4 points $A(1,\ 1,\ 1) , B(1,-1,\ 1), C(-1,-1,\ 1)$ and $D(-1,\ 1,\ 1)$ on the plane $z=2$. When the point $P$ moves on the perimeter of the circle with center $(0,\ 0,\ 2)$ , radius 1 on the plane $z=2$ and the point $Q$ moves on the perimeter of the square $ABCD$, draw the domain swept by the point $R$ on the $xy$-plane, then find the area.

For every positive integer $ n$, we define $ n?$ as $ 1?\equal{}1$ and $ n?\equal{}\frac{n}{(n\minus{}1)?}$ for $ n \ge 2$. Prove that $ \sqrt{1992}<1992?<\frac{4}{3} \sqrt{1992}.$