In the following list of numbers, the integer $n$ appears $n$ times in the list for $1 \leq n \leq 200.$ $$1, 2, 2, 3, 3, 3, 4, 4, 4, 4, …, 200, 200, …, 200$$ What is the median of the numbers in this list? $\textbf{(A)}\ 100.5 \qquad\textbf{(B)}\ 134 \qquad\textbf{(C)}\ 142 \qquad\textbf{(D)}\ 150.5 \qquad\textbf{(E)}\ 167$

Let $\mathbb{N} = \{1,2, \ldots\}.$ Does there exists a function $f: \mathbb{N} \mapsto \mathbb{N}$ such that $\forall n \in \mathbb{N},$ $f^{1989}(n) = 2 \cdot n$ ?

Let $N$ be the sum of all binomial coefficients $\binom{a}{b}$ such that $a$ and $b$ are nonnegative integers and $a+b$ is an even integer less than 100. Find the remainder when $N$ is divided by 144. (Note: $\binom{a}{b} = 0$ if $a<b$, and $\binom{0}{0} = 1$.)

Consider the following sequence $$(a_n)_{n=1}^{\infty}=(1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,1,\dots)$$ Find all pairs $(\alpha, \beta)$ of positive real numbers such that $\lim_{n\to \infty}\frac{\displaystyle\sum_{k=1}^n a_k}{n^{\alpha}}=\beta$. (Proposed by Tomas Barta, Charles University, Prague)

The eight points $A, B,. . ., G$ and $H$ lie on five circles as shown. Each of these letters are represented by one of the eight numbers $1, 2,. . ., 7$ and $ 8$ replaced so that the following conditions are met: (i) Each of the eight numbers is used exactly once. (ii) The sum of the numbers on each of the five circles is the same. How many ways are there to get the letters substituted through the numbers in this way? (Walther Janous) [img]https://cdn.artofproblemsolving.com/attachments/5/e/511cdd2fc31e8067f400369c4fe9cf964ef54c.png[/img]

Find the number of triples $(a, b, c)$ of positive integers such that $a+ab+abc = 11.$

Given is an acute-angled triangle $ABC$. On sides $BC, CA, AB$, find points $A', B', C'$ such that the longest side of triangle $A'B'C'$ is minimal.

Around a circular table an even number of persons have a discussion. After a break they sit again around the circular table in a different order. Prove that there are at least two people such that the number of participants sitting between them before and after a break is the same.

The number of positive integer pairs $(a,b)$ that have $a$ dividing $b$ and $b$ dividing $2013^{2014}$ can be written as $2013n+k$, where $n$ and $k$ are integers and $0\leq k<2013$. What is $k$? Recall $2013=3\cdot 11\cdot 61$.

One commercially available ten-button lock may be opened by depressing -- in any order -- the correct five buttons. The sample shown below has $\{1, 2, 3, 6, 9\}$ as its combination. Suppose that these locks are redesigned so that sets of as many as nine buttons or as few as one button could serve as combinations. How many additional combinations would this allow? [asy] path p=origin--(18,0)--(18,5)--(0,5)--cycle; draw(p^^shift(0,8)*p^^shift(22,0)*p^^shift(22,16)*p^^shift(22,24)*p); filldraw(shift(0,16)*p^^shift(22,8)*p^^shift(22,32)*p^^shift(0,32)*p^^shift(0,24)*p, black, black); draw((-1,-1)--(41,-1)--(41,38)--(-1,38)--cycle, linewidth(2)); int i; for(i=1; i<6; i=i+1) { label(string(6-i), (-3,8*i-5.5), W); label(string(11-i), (43,8*i-5.5), E); }[/asy]

A positive integer $M$ has been represented as a product of primes. Each of these primes is increased by 1 . The product $N$ of the new multipliers is divisible by $M$ . Prove that if we represent $N$ as a product of primes and increase each of them by 1 then the product of the new multipliers will be divisible by $N$ . Alexandr Gribalko

The eight points $A, B,. . ., G$ and $H$ lie on five circles as shown. Each of these letters are represented by one of the eight numbers $1, 2,. . ., 7$ and $ 8$ replaced so that the following conditions are met: (i) Each of the eight numbers is used exactly once. (ii) The sum of the numbers on each of the five circles is the same. How many ways are there to get the letters substituted through the numbers in this way? (Walther Janous) [img]https://cdn.artofproblemsolving.com/attachments/5/e/511cdd2fc31e8067f400369c4fe9cf964ef54c.png[/img]

Let $\mathcal{C}$ be the circumference inscribed in the triangle $ABC,$ which is tangent to sides $BC, AC, AB$ at the points $A' , B' , C' ,$ respectively. The distinct points $K$ and $L$ are taken on $\mathcal{C}$ such that $$\angle AKB'+\angle BKA' =\angle ALB'+\angle BLA'=180^{\circ}.$$ Prove that the points $A', B', C'$ are equidistant from the line $KL.$

A point moves on the sides of the triangle $ABC$, defined by the vertices $A(-1.8, 0)$, $B(3.2, 0)$, $C(0, 2.4)$ . Determine the positions of said point, in which the sum of their distance to the three vertices is absolute maximum or minimum. [img]https://cdn.artofproblemsolving.com/attachments/2/5/9e5bb48cbeefaa5f4c069532bf5605b9c1f5ea.png[/img]

Call a lattice point [i]visible[/i] if the line segment connecting the point and the origin does not pass through another lattice point. Given a positive integer $k$, denote by $S_k$ the set of all visible lattice points $(x, y)$ such that $x^2 + y^2 = k^2$. Let $D$ denote the set of all positive divisors of $2021 \cdot 2025$. Compute the sum \[ \sum_{d \in D} |S_d| \] Here, a lattice point is a point $(x, y)$ on the plane where both $x$ and $y$ are integers, and $|A|$ denotes the number of elements of the set $A$.

In quadrilateral $ABCD$, $\angle{BAD}\cong\angle{ADC}$ and $\angle{ABD}\cong\angle{BCD}$, $AB=8$, $BD=10$, and $BC=6$. The length $CD$ may be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

The straight line $ \overline{AB}$ is divided at $ C$ so that $ AC\equal{}3CB$. Circles are described on $ \overline{AC}$ and $ \overline{CB}$ as diameters and a common tangent meets $ AB$ produced at $ D$. Then $ BD$ equals: $ \textbf{(A)}\ \text{diameter of the smaller circle} \\ \textbf{(B)}\ \text{radius of the smaller circle} \\ \textbf{(C)}\ \text{radius of the larger circle} \\ \textbf{(D)}\ CB\sqrt{3}\\ \textbf{(E)}\ \text{the difference of the two radii}$