Let $k > 1$ be a fixed integer. Prove that there are infinite positive integers $n$ such that $$ lcm \, (n, n + 1, n + 2, ... , n + k) > lcm \, (n + 1, n + 2, n + 3,... , n + k + 1).$$

Suppose that $a$ and $b$ are non-negative integers satisfying $a + b + ab + a^b = 42$. Find the sum of all possible values of $a + b$. Let $T = TNYWR$. Suppose that a sequence $\{a_n\}$ is defined via $a_1 = 11, a_2 = T$, and $a_n = a_{n-1} + 2a_{n-2}$ for $n \ge 3$. Find $a_{19} + a_{20}$.

The sequence $a_0$, $a_1$, $a_2,$ $\ldots$ is defined as follows: \[a_0=2, \qquad a_{k+1}=2a_k^2-1 \quad\text{for }k \geq 0.\] Prove that if an odd prime $p$ divides $a_n$, then $2^{n+3}$ divides $p^2-1$. [hide="comment"] Hi guys , Here is a nice problem: Let be given a sequence $a_n$ such that $a_0=2$ and $a_{n+1}=2a_n^2-1$ . Show that if $p$ is an odd prime such that $p|a_n$ then we have $p^2\equiv 1\pmod{2^{n+3}}$ Here are some futher question proposed by me :Prove or disprove that : 1) $gcd(n,a_n)=1$ 2) for every odd prime number $p$ we have $a_m\equiv \pm 1\pmod{p}$ where $m=\frac{p^2-1}{2^k}$ where $k=1$ or $2$ Thanks kiu si u [i]Edited by Orl.[/i] [/hide]

If a square is partitioned into $ n$ convex polygons, determine the maximum possible number of edges in the obtained figure. (You may wish to use the following theorem of Euler: If a polygon is partitioned into $ n$ polygons with $ v$ vertices and $ e$ edges in the resulting figure, then $ v\minus{}e\plus{}n\equal{}1$.)

Let $ABC$ be a triangle with $\angle A=60^\circ$; $AD$, $BE$, and $CF$ be its bisectors; $P, Q$ be the projections of $A$ to $EF$ and $BC$ respectively; and $R$ be the second common point of the circle $DEF$ with $AD$. Prove that $P, Q, R$ are collinear.

Let a, b be positive integers such that $5 \nmid a, b$ and $5^5 \mid a^5+b^5$. What is the minimum possible value of $a + b$?

[b]p1.[/b] Jack made $3$ quarts of fruit drink from orange and apple juice. His drink contains $45\%$ of orange juice. Nick prefers more orange juice in the drink. How much orange juice should he add to the drink to obtain a drink composed of $60\%$ of orange juice? [b]p2.[/b] A square is tiled by smaller squares as shown in the figure. Find the area of the black square in the middle if the perimeter of the big square $ABCD$ is $40$ cm. [img]https://cdn.artofproblemsolving.com/attachments/8/c/d54925cba07f63ec8578048f46e1e730cb8df3.png[/img] [b]p3.[/b] For one particular number $a > 0$ the function f satisfies the equality $f(x + a) =\frac{1 + f(x)}{1 - f(x)}$ for all $x$. Show that $f$ is a periodic function. (A function $f$ is periodic with the period $T$ if $f(x + T) = f(x)$ for any $x$.) [b]p4.[/b] If $a, b, c, x, y, z$ are numbers so that $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}= 1$ and $\frac{a}{x}+\frac{b}{y}+\frac{c}{z}= 0$. Show that $\frac{x^2}{a^2} +\frac{y^2}{b^2} +\frac{z^2}{c^2} = 1$ [b]p5.[/b] Is it possible that a four-digit number $AABB$ is a perfect square? (Same letters denote the same digits). [b]p6.[/b] A finite number of arcs of a circle are painted black (see figure). The total length of these arcs is less than $\frac15$ of the circumference. Show that it is possible to inscribe a square in the circle so that all vertices of the square are in the unpainted portion of the circle. [img]https://cdn.artofproblemsolving.com/attachments/2/c/bdfa61917a47f3de5dd3684627792a9ebf05d5.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider a checkered $3m\times 3m$ square, where $m$ is an integer greater than $1.$ A frog sits on the lower left corner cell $S$ and wants to get to the upper right corner cell $F.$ The frog can hop from any cell to either the next cell to the right or the next cell upwards. Some cells can be [i]sticky[/i], and the frog gets trapped once it hops on such a cell. A set $X$ of cells is called [i]blocking[/i] if the frog cannot reach $F$ from $S$ when all the cells of $X$ are sticky. A blocking set is [i] minimal[/i] if it does not contain a smaller blocking set.[list=a][*]Prove that there exists a minimal blocking set containing at least $3m^2-3m$ cells. [*]Prove that every minimal blocking set containing at most $3m^2$ cells.

Let $ABC$ be a triangle with circumcenter $O$ and circumcircle $\omega$. Let $D$ be the foot of the altitude from $A$ to $\overline{BC}$. Let $P$ and $Q$ be points on the circumcircles of triangles $AOB$ and $AOC$, respectively, such that $A$, $P$, and $Q$ are collinear. Prove that if the circumcircle of triangle $OPQ$ is tangent to $\omega$ at $T$, then $\angle BTD=\angle CAP$. [i]Tiger Zhang[/i]

Compute the number of ways to color the vertices of a regular heptagon red, green, or blue (with rotations and reflections distinct) such that no isosceles triangle whose vertices are vertices of the heptagon has all three vertices the same color.

Given a finite sequence with terms in the set $A=\{0,1,…,121\}$ , it is allowed to replace each term by a number from the set $A$ so that like terms are replaced by like numbers, and different terms by different numbers. (Terms may remain without replacement.) The objective is to obtain, from a given sequence, through several such changes, a new sequence with sum divisible by $121$ . Show that it is possible to achieve the objective for every initial sequence. [hide=original wording]Dada una secuencia finita con términos en el conjunto A={0,1,…,121} , está permitido reemplazar cada término por un número del conjunto A de modo que términos iguales se reemplacen por números iguales, y términos distintos por números distintos. (Pueden quedar términos sin reemplazar.) El objetivo es obtener, a partir de una sucesión dada, mediante varios de tales cambios, una nueva sucesión con suma divisible por 121 . Demostrar que es posible lograr el objetivo para toda sucesión inicial.[/hide]

$n$ persons sit around a round table. The number of persons having the same gender than the person at the right of them is the same as the number of those it isn't true for. Show that $4|n$.

On the sides $AB$ and $AD$ of the square $ABCD$, the points $N$ and $P$ are selected, respectively, so that $PN = NC$, the point $Q$ Is a point on the segment $AN$ for which $\angle NCB = \angle QPN$. Prove that $\angle BCQ = \tfrac {1} {2} \angle PQA$.

Compute the sum of the two smallest positive integers $b$ with the following property: there are at least ten integers $0 \le n < b$ such that $n^2$ and $n$ end in the same digit in base $b$.

Find all values of $a$ such that $x^6 - 6x^5 + 12x^4 + ax^3 + 12x^2 - 6x +1$ is nonnegative for all real $x$.

Let $p = 2027$ be the smallest prime greater than $2018$, and let $P(X) = X^{2031}+X^{2030}+X^{2029}-X^5-10X^4-10X^3+2018X^2$. Let $\mathrm{GF}(p)$ be the integers modulo $p$, and let $\mathrm{GF}(p)(X)$ be the set of rational functions with coefficients in $\mathrm{GF}(p)$ (so that all coefficients are taken modulo $p$). That is, $\mathrm{GF}(p)(X)$ is the set of fractions $\frac{P(X)}{Q(X)}$ of polynomials with coefficients in $\mathrm{GF}(p)$, where $Q(X)$ is not the zero polynomial. Let $D\colon \mathrm{GF}(p)(X)\to \mathrm{GF}(p)(X)$ be a function satisfying \[ D\left(\frac fg\right) = \frac{D(f)\cdot g - f\cdot D(g)}{g^2} \]for any $f,g\in \mathrm{GF}(p)(X)$ with $g\neq 0$, and such that for any nonconstant polynomial $f$, $D(f)$ is a polynomial with degree less than that of $f$. If the number of possible values of $D(P(X))$ can be written as $a^b$, where $a$, $b$ are positive integers with $a$ minimized, compute $ab$. [i]Proposed by Brandon Wang[/i]

$f$ and $g$ are real-valued functions defined on the real line. For all $x$ and $y, f(x+y)+f(x-y)=2f(x)g(y)$. $f$ is not identically zero and $|f(x)|\le1$ for all $x$. Prove that $|g(x)|\le1$ for all $x$.

Let $\vartriangle ABC$ be a triangle with $AB > AC$, its incircle is tangent to $BC$ at $D$. Let $DE$ be a diameter of the incircle, and let $F$ be the intersection between line $AE$ and side $BC$. Find the ratio between the areas of $\vartriangle DEF$ and $\vartriangle ABC$ in terms of the three side lengths of$\vartriangle ABC$.

Triangle $ABC$ has $\angle A = 60^o$, $\angle B = 45$, and $AC = 6$. Let $D$ be on $AB$ such that $AD = 3$. There is exactly one point $E$ on $BC$ such that $\overline{DE}$ divides $ABC$ into two cyclic polygons. Compute $DE^2$.

The sum $\sum_{x=-5}^5\sum_{y=-5}^5\frac{2^x3^y}{(1+2^x)(1+3^y)}$ can be expressed as a fraction $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

In rectangle $ABCD$ with area $1$, point $M$ is selected on $\overline{AB}$ and points $X$, $Y$ are selected on $\overline{CD}$ such that $AX < AY$ . Suppose that $AM = BM$. Given that the area of triangle $MXY$ is $\frac{1}{2014}$ , compute the area of trapezoid $AXY B$.

Given $3n$ cards, each of them will be written with a number from the following sequence: $$2, 3, ..., n, n + 1, n + 3, n + 4, ..., 2n + 1, 2n + 2, 2n + 4, ..., 3n + 3$$ with each number used exactly once. Then every card is arranged from left to right in random order. Determine the probability such that for every $i$ with $1\le i \le 3n$, the number written on the $i$-th card, counted from the left, is greater than or equal to $i$.

Let $n$ be the least positive integer for which $149^n - 2^n$ is divisible by $3^3 \cdot 5^5 \cdot 7^7$. Find the number of positive divisors of $n$.

Let $k\ge2$ be an integer. Find the smallest integer $n \ge k+1$ with the property that there exists a set of $n$ distinct real numbers such that each of its elements can be written as a sum of $k$ other distinct elements of the set.

Show that there is a set of $2002$ distinct positive integers such that the sum of one or more elements of the set is never a square, cube, or higher power.