Let $n\ge 2$ be a positive integer. Fill up a $n\times n$ table with the numbers $1,2,...,n^2$ exactly once each. Two cells are termed adjacent if they have a common edge. It is known that for any two adjacent cells, the numbers they contain differ by at most $n$. Show that there exist a $2\times 2$ square of adjacent cells such that the diagonally opposite pairs sum to the same number.

What is the units digit of $7^{2009}$?

For n real numbers $a_{1},\, a_{2},\, \ldots\, , a_{n},$ let $d$ denote the difference between the greatest and smallest of them and $S = \sum_{i<j}\left |a_i-a_j \right|.$ Prove that \[(n-1)d\le S\le\frac{n^{2}}{4}d\] and find when each equality holds.

Construct a set $A\subset[0,1]\times[0,1]$ such that $A$ is dense in $[0,1]\times[0,1]$ and every vertical and every horizontal line intersects $A$ in at most one point.

In a square $ABCD$, let $P$ and $Q$ be points on the sides $BC$ and $CD$ respectively, different from its endpoints, such that $BP=CQ$. Consider points $X$ and $Y$ such that $X\neq Y$, in the segments $AP$ and $AQ$ respectively. Show that, for every $X$ and $Y$ chosen, there exists a triangle whose sides have lengths $BX$, $XY$ and $DY$.

Let $f(x)$ be a polynomial. Prove that if $\int_0^1 f(x)g_n(x)\ dx=0\ (n=0,\ 1,\ 2,\ \cdots)$, then all coefficients of $f(x)$ are 0 for each case as follows. (1) $g_n(x)=(1+x)^n$ (2) $g_n(x)=\sin n\pi x$ (3) $g_n(x)=e^{nx}$

Jerry writes down all binary strings of length $10$ without any two consecutive $1$s. How many $1$s does Jerry write?

In a plane, two circles $ C_1 $ and $ C_2 $ and a line $ m $ are given. Find a point on the line $ m $ from which one can draw tangents to the circles $ C_1 $ and $ C_2 $ with equal inclination to the line $ m $.

Given positive integer $n$. Prove that for any integers $a_1,a_2,\cdots,a_n,$ at least $\lceil \tfrac{n(n-6)}{19} \rceil$ numbers from the set $\{ 1,2, \cdots, \tfrac{n(n-1)}{2} \}$ cannot be represented as $a_i-a_j (1 \le i, j \le n)$.

The diagram shows a semicircle with diameter $20$ and the circle with greatest diameter that fits inside the semicircle. The area of the shaded region is $N\pi$, where $N$ is a positive integer. Find $N$.

Let $n \ge 3$ be an integer. Find the minimal value of the real number $k_n$ such that for all positive numbers $x_1, x_2, ..., x_n$ with product $1$, we have $$\frac{1}{\sqrt{1 + k_nx_1}}+\frac{1}{\sqrt{1 + k_nx_2}}+ ... + \frac{1}{\sqrt{1 + k_nx_n}} \le n - 1.$$

Let $P$ and $Q$ be points on $AC$ and $AB$, respectively, of triangle $\triangle ABC$ such that $PB=PC$ and $PQ\perp AB$. Suppose $\frac{AQ}{QB}=\frac{AP}{PB}.$ Find $\angle CBA$, in degrees. [i]Proposed by Nathan Ramesh

If an acute-angled triangle $ABC$ is given, construct an equilateral triangle $A'B'C'$ in space such that lines $AA',BB', CC'$ pass through a given point.

If $f(x) = ax^4-bx^2+x+5$ and $f(-3) = 2$, then $f(3) =$ $\textbf{(A)}\ -5 \qquad \textbf{(B)}\ -2 \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 8$

Let $ABCDE$ be a convex pentagon such that $AB=BC=CD$, $\angle{EAB}=\angle{BCD}$, and $\angle{EDC}=\angle{CBA}$. Prove that the perpendicular line from $E$ to $BC$ and the line segments $AC$ and $BD$ are concurrent.

The number $2020$ has three different prime factors. What is their sum?

Find the real number $k$ such that $a$, $b$, $c$, and $d$ are real numbers that satisfy the system of equations \begin{align*} abcd &= 2007,\\ a &= \sqrt{55 + \sqrt{k+a}},\\ b &= \sqrt{55 - \sqrt{k+b}},\\ c &= \sqrt{55 + \sqrt{k-c}},\\ d &= \sqrt{55 - \sqrt{k-d}}. \end{align*}

Two different $3$ digit numbers are picked and then for every of them is calculated sum of all $5$ numbers which are getting when digits of picked number change place (etc. if one of the number is $707$, the sum is $2401=770+77+77+770+707$). Do the given results must be different?

In a quadrilateral $ABCD$, it is given that $AB$ is parallel to $CD$ and the diagonals $AC$ and $BD$ are perpendicular to each other. Show that (a) $AD \cdot BC \geq AB \cdot CD$ (b) $AD + BC \geq AB + CD.$

Let a [i]lattice tetrahedron[/i] denote a tetrahedron whose vertices have integer coordinates. Given a lattice tetrahedron, a [i]move[/i] consists of picking some vertex and moving it parallel to one of the three edges of the face opposite the vertex so that it lands on a different point with integer coordinates. Prove that any two lattice tetrahedra with the same volume can be transformed into each other by a series of moves

[b]p1.[/b] What is $(2 + 4 + ... + 20) - (1 + 3 + ...+ 19)$? [b]p2.[/b] Two ants start on opposite vertices of a dodecagon ($12$-gon). Each second, they randomly move to an adjacent vertex. What is the probability they meet after four moves? [b]p3.[/b] How many distinct $8$-letter strings can be made using $8$ of the $9$ letters from the words $FORK$ and $KNIFE$ (e.g., $FORKNIFE$)? [b]p4.[/b] Let $ABC$ be an equilateral triangle with side length $8$ and let $D$ be a point on segment $BC$ such that $BD = 2$. Given that $E$ is the midpoint of $AD$, what is the value of $CE^2 - BE^2$? [b][color=#f00](mistyped p4)[/color][/b] Let $ABC$ be an equilateral triangle with side length $8$ and let $D$ be a point on segment $BC$ such that $BD = 2$. Given that $E$ is the midpoint of $AD$, what is the value of $CE^2 + BE^2$? [b]p5.[/b] You have two fair six-sided dice, one labeled $1$ to $6$, and for the other one, each face is labeled $1$, $2$, $3$, or $4$ (not necessarily all numbers are used). Let $p$ be the probability that when the two dice are rolled, the number on the special die is smaller than the number on the normal die. Given that $p = 1/2$, how many distinct combinations of $1$, $2$, $3$, $4$ can appear on the special die? The arrangement of the numbers on the die does not matter. [b]p6.[/b] Let $\omega_1$ and $\omega_2$ be two circles with centers $A$ and $B$ and radii $3$ and $13$, respectively. Suppose $AB = 10$ and that $C$ is the midpoint of $AB$. Let $\ell$ be a line that passes through $C$ and is tangent to $\omega_1$ at $P$. Given that $\ell$ intersects $\omega_2$ at $X$ and $Y$ such that $XP < Y P$, what is $XP$? [b]p7.[/b] Let $f(x)$ be a cubic polynomial. Given that $f(1) = 13$, $f(4) = 19$, $f(7) = 7$, and $f(10) = 13$, find $f(13)$. [b]p8.[/b] For all integers $0 \le n \le 202$ not divisible by seven, define $f(n) = \{\sqrt{7n}\}$. For what value $n$ does $f(n)$ take its minimum value? (Note: $\{x\} = x - \lfloor x \rfloor$, where $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.) [b]p9.[/b] Let $ABC$ be a triangle with $AB = 14$ and $AC = 25$. Let the incenter of $ABC$ be $I$. Let line $AI$ intersect the circumcircle of $BIC$ at $D$ (different from $I$). Given that line $DC$ is tangent to the circumcircle of $ABC$, find the area of triangle $BCD$. [b]p10.[/b] Evaluate the infinite sum $$\frac{4^2 + 3}{1 \cdot 3 \cdot 5 \cdot 7} +\frac{6^2 + 3}{3 \cdot 5 \cdot 7 \cdot 9}+\frac{8^2 + 3}{5 \cdot 7 \cdot 9 \cdot 11}+ ...$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In $1960$ only $5\%$ of the working adults in Carlin City worked at home. By $1970$ the "at-home" work force increased to $8\%$. In $1980$ there were approximately $15\%$ working at home, and in $1990$ there were $30\%$. The graph that best illustrates this is [asy] unitsize(13); draw((0,4)--(0,0)--(7,0)); draw((0,1)--(.2,1)); draw((0,2)--(.2,2)); draw((0,3)--(.2,3)); draw((2,0)--(2,.2)); draw((4,0)--(4,.2)); draw((6,0)--(6,.2)); for (int a = 1; a < 4; ++a) { for (int b = 1; b < 4; ++b) { draw((2*a,b-.1)--(2*a,b+.1)); draw((2*a-.1,b)--(2*a+.1,b)); } } label("1960",(0,0),S); label("1970",(2,0),S); label("1980",(4,0),S); label("1990",(6,0),S); label("10",(0,1),W); label("20",(0,2),W); label("30",(0,3),W); label("$\%$",(0,4),N); draw((12,4)--(12,0)--(19,0)); draw((12,1)--(12.2,1)); draw((12,2)--(12.2,2)); draw((12,3)--(12.2,3)); draw((14,0)--(14,.2)); draw((16,0)--(16,.2)); draw((18,0)--(18,.2)); for (int a = 1; a < 4; ++a) { for (int b = 1; b < 4; ++b) { draw((2*a+12,b-.1)--(2*a+12,b+.1)); draw((2*a+11.9,b)--(2*a+12.1,b)); } } label("1960",(12,0),S); label("1970",(14,0),S); label("1980",(16,0),S); label("1990",(18,0),S); label("10",(12,1),W); label("20",(12,2),W); label("30",(12,3),W); label("$\%$",(12,4),N); draw((0,12)--(0,8)--(7,8)); draw((0,9)--(.2,9)); draw((0,10)--(.2,10)); draw((0,11)--(.2,11)); draw((2,8)--(2,8.2)); draw((4,8)--(4,8.2)); draw((6,8)--(6,8.2)); for (int a = 1; a < 4; ++a) { for (int b = 1; b < 4; ++b) { draw((2*a,b+7.9)--(2*a,b+8.1)); draw((2*a-.1,b+8)--(2*a+.1,b+8)); } } label("1960",(0,8),S); label("1970",(2,8),S); label("1980",(4,8),S); label("1990",(6,8),S); label("10",(0,9),W); label("20",(0,10),W); label("30",(0,11),W); label("$\%$",(0,12),N); draw((12,12)--(12,8)--(19,8)); draw((12,9)--(12.2,9)); draw((12,10)--(12.2,10)); draw((12,11)--(12.2,11)); draw((14,8)--(14,8.2)); draw((16,8)--(16,8.2)); draw((18,8)--(18,8.2)); for (int a = 1; a < 4; ++a) { for (int b = 1; b < 4; ++b) { draw((2*a+12,b+7.9)--(2*a+12,b+8.1)); draw((2*a+11.9,b+8)--(2*a+12.1,b+8)); } } label("1960",(12,8),S); label("1970",(14,8),S); label("1980",(16,8),S); label("1990",(18,8),S); label("10",(12,9),W); label("20",(12,10),W); label("30",(12,11),W); label("$\%$",(12,12),N); draw((24,12)--(24,8)--(31,8)); draw((24,9)--(24.2,9)); draw((24,10)--(24.2,10)); draw((24,11)--(24.2,11)); draw((26,8)--(26,8.2)); draw((28,8)--(28,8.2)); draw((30,8)--(30,8.2)); for (int a = 1; a < 4; ++a) { for (int b = 1; b < 4; ++b) { draw((2*a+24,b+7.9)--(2*a+24,b+8.1)); draw((2*a+23.9,b+8)--(2*a+24.1,b+8)); } } label("1960",(24,8),S); label("1970",(26,8),S); label("1980",(28,8),S); label("1990",(30,8),S); label("10",(24,9),W); label("20",(24,10),W); label("30",(24,11),W); label("$\%$",(24,12),N); draw((0,9)--(2,9.25)--(4,10)--(6,11)); draw((12,8.5)--(14,9)--(16,10)--(18,10.5)); draw((24,8.5)--(26,8.8)--(28,10.5)--(30,11)); draw((0,0.5)--(2,1)--(4,2.8)--(6,3)); draw((12,0.5)--(14,.8)--(16,1.5)--(18,3)); label("(A)",(-1,12),W); label("(B)",(11,12),W); label("(C)",(23,12),W); label("(D)",(-1,4),W); label("(E)",(11,4),W);[/asy]

A rhombus is formed by two radii and two chords of a circle whose radius is $ 16$ feet. The area of the rhombus in square feet is: $ \textbf{(A)}\ 128 \qquad\textbf{(B)}\ 128\sqrt {3} \qquad\textbf{(C)}\ 256 \qquad\textbf{(D)}\ 512 \qquad\textbf{(E)}\ 512\sqrt {3}$

A positive integer $ m$ is called [i]good[/i] if there is a positive integer $ n$ such that $ m$ is the quotient of $ n$ by the number of positive integer divisors of $ n$ (including $ 1$ and $ n$ itself). Prove that $ 1, 2, \ldots, 17$ are good numbers and that $ 18$ is not a good number.

1. In a triangle $ABC$, $O$ is the circumcenter and $I$ is the incenter. $X$ is the reflection of $I$ to $O$. $A_1$ is foot of the perpendicular from $X$ to $BC$. $B_1$ and $C_1$ are defined similarly. prove that $AA_1$,$BB_1$ and $CC_1$ are concurrent.(12 points)