Given a rebus: $$AB + BC + CA = XY + YZ + ZX = KL + LM + MK $$ where different letters correspond to different numbers, and same letters correspond to the same numbers. Determine the value of $ AXK + BYL + CZM $. [i]Proposed by Giorgi Arabidze[/i]

What is \[\frac{2+4+6}{1+3+5} - \frac{1+3+5}{2+4+6}?\] $ \textbf{(A)}\ -1 \qquad \textbf{(B)}\ \frac{5}{36} \qquad \textbf{(C)}\ \frac{7}{12} \qquad \textbf{(D)}\ \frac{147}{60} \qquad \textbf{(E)}\ \frac{43}{3} $

Let $ U$ be a real normed space such that, for an finite-dimensional, real normed space $ X,U$ contains a subspace isometrically isomorphic to $ X$. Prove that every (not necessarily closed) subspace $ V$ of $ U$ of finite codimension has the same property. (We call $ V$ of finite codimension if there exists a finite-dimensional subspace $ N$ of $ U$ such that $ V\plus{}N\equal{}U$.) [i]A. Bosznay[/i]

Prove that there is no $m$ such that ($1978^m - 1$) is divisible by ($1000^m - 1$).

The two angles of the squares are adjacent, and the extension of the diagonals of one square intersect the diagonal of another square at point $O$ (see figure). Prove that $O$ is the midpoint of $AB$. [img]https://cdn.artofproblemsolving.com/attachments/7/8/8daaaa55c38e15c4a8ac7492c38707f05475cc.png[/img]

$\textbf{N3:}$ For any positive integer $n$, define $rad(n)$ to be the product of all prime divisors of $n$ (without multiplicities), and in particular $rad(1)=1$. Consider an infinite sequence of positive integers $\{a_n\}_{n=1}^{\infty}$ satisfying that \begin{align*} a_{n+1} = a_n + rad(a_n), \: \forall n \in \mathbb{N} \end{align*} Show that there exist positive integers $t,s$ such that $a_t$ is the product of the $s$ smallest primes. [i]Proposed by ltf0501[/i]

The complex number $z$ satisfies $z + |z| = 2 + 8i$. What is $|z|^{2}$? Note: if $z = a + bi$, then $|z| = \sqrt{a^{2} + b^{2}}$. $ \textbf{(A)}\ 68\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 169\qquad\textbf{(D)}\ 208\qquad\textbf{(E)}\ 289 $

Three lights are placed horizontally on a line on the ceiling. All the lights are initially off. Every second, Neil picks one of the three lights uniformly at random to switch: if it is off, he switches it on; if it is on, he switches it off. When a light is switched, any lights directly to the left or right of that light also get turned on (if they were off) or off (if they were on). The expected number of lights that are on after Neil has flipped switches three times can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

Find all $a,b,c \in \mathbb{N}$ such that \[a^2b|a^3+b^3+c^3,\qquad b^2c|a^3+b^3+c^3, \qquad c^2a|a^3+b^3+c^3.\] [PS: The original problem was this: Find all $a,b,c \in \mathbb{N}$ such that \[a^2b|a^3+b^3+c^3,\qquad b^2c|a^3+b^3+c^3, \qquad \color{red}{c^2b}|a^3+b^3+c^3.\] But I think the author meant $c^2a|a^3+b^3+c^3$, just because of symmetry]

The adjacent sides of the decagon shown meet at right angles. What is its perimeter? [asy]defaultpen(linewidth(.8pt)); dotfactor=4; dot(origin);dot((12,0));dot((12,1));dot((9,1));dot((9,7));dot((7,7));dot((7,10));dot((3,10));dot((3,8));dot((0,8)); draw(origin--(12,0)--(12,1)--(9,1)--(9,7)--(7,7)--(7,10)--(3,10)--(3,8)--(0,8)--cycle); label("$8$",midpoint(origin--(0,8)),W); label("$2$",midpoint((3,8)--(3,10)),W); label("$12$",midpoint(origin--(12,0)),S);[/asy]$ \textbf{(A)}\ 22\qquad \textbf{(B)}\ 32\qquad \textbf{(C)}\ 34\qquad \textbf{(D)}\ 44\qquad \textbf{(E)}\ 50$

Let $ABCD$ be a quadrilateral and let $l$ be a line. Let $l$ intersect the lines $AB,CD,BC,DA,AC,BD$ at points $X,X',Y,Y',Z,Z'$ respectively. Given that these six points on $l$ are in the order $X,Y,Z,X',Y',Z'$, show that the circles with diameter $XX',YY',ZZ'$ are coaxal.

[b]p1.[/b] What is the smallest possible value for the product of two real numbers that differ by ten? [b]p2.[/b] Determine the number of positive integers $n$ with $1 \le n \le 400$ that satisfy the following: $\bullet$ $n$ is a square number. $\bullet$ $n$ is one more than a multiple of $5$. $\bullet$ $n$ is even. [b]p3.[/b] How many positive integers less than $2019$ are either a perfect cube or a perfect square but not both? [b]p4.[/b] Felicia draws the heart-shaped figure $GOAT$ that is made of two semicircles of equal area and an equilateral triangle, as shown below. If $GO = 2$, what is the area of the figure? [img]https://cdn.artofproblemsolving.com/attachments/3/c/388daa657351100f408ab3f1185f9ab32fcca5.png[/img] [b]p5.[/b] For distinct digits $A, B$, and $ C$: $$\begin{tabular}{cccc} & A & A \\ & B & B \\ + & C & C \\ \hline A & B & C \\ \end{tabular}$$ Compute $A \cdot B \cdot C$. [b]p6 [/b] What is the difference between the largest and smallest value for $lcm(a,b,c)$, where $a,b$, and $c$ are distinct positive integers between $1$ and $10$, inclusive? [b]p7.[/b] Let $A$ and $B$ be points on the circumference of a circle with center $O$ such that $\angle AOB = 100^o$. If $X$ is the midpoint of minor arc $AB$ and $Y$ is on the circumference of the circle such that $XY\perp AO$, find the measure of $\angle OBY$ . [b]p8. [/b]When Ben works at twice his normal rate and Sammy works at his normal rate, they can finish a project together in $6$ hours. When Ben works at his normal rate and Sammy works as three times his normal rate, they can finish the same project together in $4$ hours. How many hours does it take Ben and Sammy to finish that project if they each work together at their normal rates? [b][b]p9.[/b][/b] How many positive integer divisors $n$ of $20000$ are there such that when $20000$ is divided by $n$, the quotient is divisible by a square number greater than $ 1$? [b]p10.[/b] What’s the maximum number of Friday the $13$th’s that can occur in a year? [b]p11.[/b] Let circle $\omega$ pass through points $B$ and $C$ of triangle $ABC$. Suppose $\omega$ intersects segment $AB$ at a point $D \ne B$ and intersects segment $AC$ at a point $E \ne C$. If $AD = DC = 12$, $DB = 3$, and $EC = 8$, determine the length of $EB$. [b]p12.[/b] Let $a,b$ be integers that satisfy the equation $2a^2 - b^2 + ab = 18$. Find the ordered pair $(a,b)$. [b]p13.[/b] Let $a,b,c$ be nonzero complex numbers such that $a -\frac{1}{b}= 8, b -\frac{1}{c}= 10, c -\frac{1}{a}= 12.$ Find $abc -\frac{1}{abc}$ . [b]p14.[/b] Let $\vartriangle ABC$ be an equilateral triangle of side length $1$. Let $\omega_0$ be the incircle of $\vartriangle ABC$, and for $n > 0$, define the infinite progression of circles $\omega_n$ as follows: $\bullet$ $\omega_n$ is tangent to $AB$ and $AC$ and externally tangent to $\omega_{n-1}$. $\bullet$ The area of $\omega_n$ is strictly less than the area of $\omega_{n-1}$. Determine the total area enclosed by all $\omega_i$ for $i \ge 0$. [b]p15.[/b] Determine the remainder when $13^{2020} +11^{2020}$ is divided by $144$. [b]p16.[/b] Let $x$ be a solution to $x +\frac{1}{x}= 1$. Compute $x^{2019} +\frac{1}{x^{2019}}$ . [b]p17. [/b]The positive integers are colored black and white such that if $n$ is one color, then $2n$ is the other color. If all of the odd numbers are colored black, then how many numbers between $100$ and $200$ inclusive are colored white? [b]p18.[/b] What is the expected number of rolls it will take to get all six values of a six-sided die face-up at least once? [b]p19.[/b] Let $\vartriangle ABC$ have side lengths $AB = 19$, $BC = 2019$, and $AC = 2020$. Let $D,E$ be the feet of the angle bisectors drawn from $A$ and $B$, and let $X,Y$ to be the feet of the altitudes from $C$ to $AD$ and $C$ to $BE$, respectively. Determine the length of $XY$ . [b]p20.[/b] Suppose I have $5$ unit cubes of cheese that I want to divide evenly amongst $3$ hungry mice. I can cut the cheese into smaller blocks, but cannot combine blocks into a bigger block. Over all possible choices of cuts in the cheese, what’s the largest possible volume of the smallest block of cheese? PS. You had better use hide for answers.

On regular hexagon $GOBEAR$ with side length $2$, bears are initially placed at $G, B, A$, forming an equilateral triangle. At time $t = 0$, all of them move clockwise along the sides of the hexagon at the same pace, stopping once they have each traveled $1$ unit. What is the total area swept out by the triangle formed by the three bears during their journey?

Find six distinct positive integers such that the product of any two of them is divisible by their sum. (D. Fomin, Leningrad)

Let $p$ be a prime number. $T(x)$ is a polynomial with integer coefficients and degree from the set $\{0,1,...,p-1\}$ and such that $T(n) \equiv T(m) (mod p)$ for some integers m and n implies that $ m \equiv n (mod p)$. Determine the maximum possible value of degree of $T(x)$

A positive integer $\ell \ge 2$ is called [i]sweet[/i] if there exists a positive integer $n \ge 10$ such that when the leftmost nonzero decimal digit of $n$ is deleted, the resulting number $m$ satisfies $n = m\ell.$ Let $S$ denote the set of all sweet numbers $\ell.$ If the sum $\sum_{\ell \in S} \tfrac{1}{\ell-1}$ can be written as $\tfrac{A}{B}$ for relatively prime positive integers $A,B,$ find $A+B.$

Evaluate $ \int_0^{\frac{\pi}{4}} \frac{\tan x}{1\plus{}\sin x}\ dx$.

$(NET 4)$ A boy has a set of trains and pieces of railroad track. Each piece is a quarter of circle, and by concatenating these pieces, the boy obtained a closed railway. The railway does not intersect itself. In passing through this railway, the train sometimes goes in the clockwise direction, and sometimes in the opposite direction. Prove that the train passes an even number of times through the pieces in the clockwise direction and an even number of times in the counterclockwise direction. Also, prove that the number of pieces is divisible by $4.$

[b]p1.[/b] What is $6\times 7 + 4 \times 7 + 6\times 3 + 4\times 3$? [b]p2.[/b] How many integers $n$ have exactly $\sqrt{n}$ factors? [b]p3.[/b] A triangle has distinct angles $3x+10$, $2x+20$, and $x+30$. What is the value of $x$? [b]p4.[/b] If $4$ people of the Math Club are randomly chosen to be captains, and Henry is one of the $30$ people eligible to be chosen, what is the probability that he is not chosen to be captain? [b]p5.[/b] $a, b, c, d$ is an arithmetic sequence with difference $x$ such that $a, c, d$ is a geometric sequence. If $b$ is $12$, what is $x$? (Note: the difference of an aritmetic sequence can be positive or negative, but not $0$) [b]p6.[/b] What is the smallest positive integer that contains only $0$s and $5$s that is a multiple of $24$. [b]p7.[/b] If $ABC$ is a triangle with side lengths $13$, $14$, and $15$, what is the area of the triangle made by connecting the points at the midpoints of its sides? [b]p8.[/b] How many ways are there to order the numbers $1,2,3,4,5,6,7,8$ such that $1$ and $8$ are not adjacent? [b]p9.[/b] Find all ordered triples of nonnegative integers $(x, y, z)$ such that $x + y + z = xyz$. [b]p10.[/b] Noah inscribes equilateral triangle $ABC$ with area $\sqrt3$ in a cricle. If $BR$ is a diameter of the circle, then what is the arc length of Noah's $ARC$? [b]p11.[/b] Today, $4/12/14$, is a palindromic date, because the number without slashes $41214$ is a palindrome. What is the last palindromic date before the year $3000$? [b]p12.[/b] Every other vertex of a regular hexagon is connected to form an equilateral triangle. What is the ratio of the area of the triangle to that of the hexagon? [b]p13.[/b] How many ways are there to pick four cards from a deck, none of which are the same suit or number as another, if order is not important? [b]p14.[/b] Find all functions $f$ from $R \to R$ such that $f(x + y) + f(x - y) = x^2 + y^2$. [b]p15.[/b] What are the last four digits of $1(1!) + 2(2!) + 3(3!) + ... + 2013(2013!)$/ [b]p16.[/b] In how many distinct ways can a regular octagon be divided up into $6$ non-overlapping triangles? [b]p17.[/b] Find the sum of the solutions to the equation $\frac{1}{x-3} + \frac{1}{x-5} + \frac{1}{x-7} + \frac{1}{x-9} = 2014$ . [b]p18.[/b] How many integers $n$ have the property that $(n+1)(n+2)(n+3)(n+4)$ is a perfect square of an integer? [b]p19.[/b] A quadrilateral is inscribed in a unit circle, and another one is circumscribed. What is the minimum possible area in between the two quadrilaterals? [b]p20.[/b] In blindfolded solitary tic-tac-toe, a player starts with a blank $3$-by-$3$ tic-tac-toe board. On each turn, he randomly places an "$X$" in one of the open spaces on the board. The game ends when the player gets $3$ $X$s in a row, in a column, or in a diagonal as per normal tic-tac-toe rules. (Note that only $X$s are used, not $O$s). What fraction of games will run the maximum $7$ amount of moves? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ n(\ge2) $ be a positive integer. Find the minimum $ m $, so that there exists $x_{ij}(1\le i ,j\le n)$ satisfying: (1)For every $1\le i ,j\le n, x_{ij}=max\{x_{i1},x_{i2},...,x_{ij}\} $ or $ x_{ij}=max\{x_{1j},x_{2j},...,x_{ij}\}.$ (2)For every $1\le i \le n$, there are at most $m$ indices $k$ with $x_{ik}=max\{x_{i1},x_{i2},...,x_{ik}\}.$ (3)For every $1\le j \le n$, there are at most $m$ indices $k$ with $x_{kj}=max\{x_{1j},x_{2j},...,x_{kj}\}.$

Consider a polynomial $P(x) = ax^2 + bx + c$ with $a > 0$ that has two real roots $x_1, x_2$. Prove that the absolute values of both roots are less than or equal to $1$ if and only if $a + b + c \ge 0, a -b + c \ge 0$, and $a - c \ge 0$.

Determine all pairs $(x, y)$ of positive integers such that \[\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.\] [i]Proposed by Titu Andreescu, USA[/i]

A quadrilateral $ABCD$ is such that $\angle A= \angle C=60^o$ and $\angle B=100^o$. Let $O_1$ and $O_2$ be the centers of the incircles of triangles $ABD$ and $CBD$ respectively. Find the angle between the lines $AO_2$ and $CO_1$.

A particle moves on a circle with center $O$, starting from rest at a point $P$ and coming to rest again at a point $Q$, without coming to rest at any intermediate point. Prove that the acceleration vector of the particle does not vanish at any point between $P$ and $ Q$ and that, at some point $R$ between $P$ and $Q$, the acceleration vector points in along the radius $RO.$

Let $ABCD$ be a convex quadrilateral with $\angle BAC = \angle CAD$, $\angle ABC =\angle ACD$, $(AD \cap (BC =\{E\}$, $(AB \cap (DC = \{F\}$. Prove that: a) $AB\cdot DE = BC \cdot CE$ b) $AC^2 < \frac12 (AD \cdot AF + AB \cdot AE).$