Evaluate $\displaystyle \int_1^\infty \left(\frac{\ln x}{x}\right)^{2011} dx$.

How many three-digit positive integers contain both even and odd digits?

In terms of \(a\), \(b\), and a prime \(p\), find an expression which gives the number of \(x \in \{0, 1, \ldots, p-1\}\) such that the remainder of \(ax\) upon division by \(p\) is less than the remainder of \(bx\) upon division by \(p\).

Let $p$ be a prime number, and $X$ be the set of cubes modulo $p$, including $0$. Denote by $C_2(k)$ the number of ordered pairs $(x, y) \in X \times X$ such that $x + y \equiv k \pmod p$. Likewise, denote by $C_3(k)$ the number of ordered pairs $(x, y, z) \in X \times X \times X$ such that $x + y + z \equiv k \pmod p$. Prove that there are integers $a, b$ such that for all $k$ not in $X$, we have \[ C_3(k) = a\cdot C_2(k) + b. \] [i]Proposed by Murilo Corato, Brazil.[/i]

Is it possible to assign every integral point $(x,y)$ of the plane a positive integer $a_{x,y}$ such that for every two integers $i$ and $j$ the following equality holds $$a_{i,j}=\gcd(a_{i-1,j},a_{i+1,j})+\gcd(a_{i,j-1},a_{i,j+1})$$ [i]M. Shutro[/i]

Let $n$ be a positive integer, and denote by $\mathbb{Z}_n$ the ring of integers modulo $n$. Suppose that there exists a function $f:\mathbb{Z}_n\to\mathbb{Z}_n$ satisfying the following three properties: (i) $f(x)\neq x$, (ii) $f(f(x))=x$, (iii) $f(f(f(x+1)+1)+1)=x$ for all $x\in\mathbb{Z}_n$. Prove that $n\equiv 2 \pmod4$. (Proposed by Ander Lamaison Vidarte, Berlin Mathematical School, Germany)

Neel and Roshan are going to the Newark Liberty International Airport to catch separate flights. Neel plans to arrive at some random time between 5:30 am and 6:30 am, while Roshan plans to arrive at some random time between 5:40 am and 6:40 am. The two want to meet, however briefly, before going through airport security. As such, they agree that each will wait for $n$ minutes once he arrives at the airport before going through security. What is the smallest $n$ they can select such that they meet with at least 50% probability? The answer will be of the form $a + b\sqrt{c}$ for integers $a$, $b$, and $c$, where $c$ has no perfect square factor other than $1$. Report $a + b + c.$

Determine the elements of the sets $A = \{x \in N | x \ne 4a + 7b, a, b \in N\}$, $B = \{x \in N | x\ne 3a + 11b, a, b \in N\}$.

[b]Q[/b] Let $n\geq 2$ be a fixed positive integer and let $d_1,d_2,...,d_m$ be all positive divisors of $n$. Prove that: $$\frac{d_1+d_2+...+d_m}{m}\geq \sqrt{n+\frac{1}{4}}$$Also find the value of $n$ for which the equality holds. [i]Proposed by dangerousliri [/i]

Let $P(x)$ be a polynomial of degree $n\geq 2$ with rational coefficients such that $P(x) $ has $ n$ pairwise different reel roots forming an arithmetic progression .Prove that among the roots of $P(x) $ there are two that are also the roots of some polynomial of degree $2$ with rational coefficients .

How many $0$s are there at the end of the decimal representation of $2000!$? $ \textbf{(A)}\ 222 \qquad\textbf{(B)}\ 499 \qquad\textbf{(C)}\ 625 \qquad\textbf{(D)}\ 999 \qquad\textbf{(E)}\ \text{None of the preceding} $

Barbara, Edward, Abhinav, and Alex took turns writing this test. Working alone, they could finish it in $10$, $9$, $11$, and $12$ days, respectively. If only one person works on the test per day, and nobody works on it unless everyone else has spent at least as many days working on it, how many days (an integer) did it take to write this test?

Carson and Emily attend different schools. Emily’s school has four times as many students as Carson’s school. The total number of students in both schools combined is $10105$. How many students go to Carson’s school?

Let $ABCD$ be a square and let $P$ be a point on side $AB$. The point $Q$ lies outside the square such that $\angle ABQ = \angle ADP$ and $\angle AQB = 90^{\circ}$. The point $R$ lies on the side $BC$ such that $\angle BAR = \angle ADQ$. Prove that the lines $AR, CQ$ and $DP$ pass through a common point.

Find all functions $f:\mathbb{R}\longrightarrow\mathbb{R}$ such that $f(x+y)+xy=f(x)f(y)$ for all reals $x, y$

In the plane, six circles are given so that none of the circles contain one the center of the other. Show that there is no point that lies in all the circles.

How many possible areas are there in a convex hexagon with all of its angles being equal and its sides having lengths $1, 2, 3, 4, 5$ and $6,$ in any order?

[b]p1.[/b] Let $f(x) = \frac{3x^3+7x^2-12x+2}{x^2+2x-3}$ . Find all integers $n$ such that $f(n)$ is an integer. [b]p2.[/b] How many ways are there to arrange $10$ trees in a line where every tree is either a yew or an oak and no two oak trees are adjacent? [b]p3.[/b] $20$ students sit in a circle in a math class. The teacher randomly selects three students to give a presentation. What is the probability that none of these three students sit next to each other? [b]p4.[/b] Let $f_0(x) = x + |x - 10| - |x + 10|$, and for $n \ge 1$, let $f_n(x) = |f_{n-1}(x)| - 1$. For how many values of $x$ is $f_{10}(x) = 0$? [b]p5.[/b] $2$ red balls, $2$ blue balls, and $6$ yellow balls are in a jar. Zion picks $4$ balls from the jar at random. What is the probability that Zion picks at least $1$ red ball and$ 1$ blue ball? [b]p6.[/b] Let $\vartriangle ABC$ be a right-angled triangle with $\angle ABC = 90^o$ and $AB = 4$. Let $D$ on $AB$ such that $AD = 3DB$ and $\sin \angle ACD = \frac35$ . What is the length of $BC$? [b]p7.[/b] Find the value of of $$\dfrac{1}{1 +\dfrac{1}{2+ \dfrac{1}{1+ \dfrac{1}{2+ \dfrac{1}{1+ ...}}}}}$$ [b]p8.[/b] Consider all possible quadrilaterals $ABCD$ that have the following properties; $ABCD$ has integer side lengths with $AB\parallel CD$, the distance between $\overline{AB}$ and $\overline{CD}$ is $20$, and $AB = 18$. What is the maximum area among all these quadrilaterals, minus the minimum area? [b]p9.[/b] How many perfect cubes exist in the set $\{1^{2018},2^{2017}, 3^{2016},.., 2017^2, 2018^1\}$? [b]p10.[/b] Let $n$ be the number of ways you can fill a $2018\times 2018$ array with the digits $1$ through $9$ such that for every $11\times 3$ rectangle (not necessarily for every $3 \times 11$ rectangle), the sum of the $33$ integers in the rectangle is divisible by $9$. Compute $\log_3 n$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Suppose that $ f$ is a real-valued function for which \[ f(xy)+f(y-x)\geq f(y+x)\] for all real numbers $ x$ and $ y$. a) Give a non-constant polynomial that satisfies the condition. b) Prove that $ f(x)\geq 0$ for all real $ x$.

Let $M$ be a positive real number. Determine the least positive real number $k$ with the following property: for each integer $n>M$, the interval $(n,kn]$ contains a power of $2$.

Find all functions $f : R\rightarrow R$ such that $f ( f (x)+y) = f (x^2 -y)+4 f (x)y$ for all $x,y \in R$ .

All edges of a polyhedron are painted with red or yellow. For an angle of a facet, if the edges determining it are of different colors, then the angle is called [i]excentric[/i]. The[i] excentricity [/i]of a vertex $A$, namely $S_A$, is defined as the number of excentric angles it has. Prove that there exist two vertices $B$ and $C$ such that $S_B + S_C \leq 4$.

The number $734{,}851{,}474{,}594{,}578{,}436{,}096$ is equal to $n^6$ for some positive integer $n$. What is the value of $n$?

In the triangle $ABC$, the angle bisector $AK$ is drawn. The center of the circle inscribed in the triangle $AKC$ coincides with the center of the circle, circumscribed around the triangle $ABC$. Determine the angles of triangle $ABC$.

A circle is inscribed in a triangle and a square is circumscribed around this circle so that no side of the square is parallel to any side of the triangle. Prove that less than half of the square’s perimeter lies outside the triangle.