Let $N$ be the product of all odd primes less than $2^4$. What remainder does $N$ leave when divided by $2^4$? $\text{(A) }5\qquad\text{(B) }7\qquad\text{(C) }9\qquad\text{(D) }11\qquad\text{(E) }13$

How many $8$-digit numbers are $*2*0*1*7$, where four unknown numbers are replaced by stars, which are divisible by $7$?

Find the roots $ r_i \in \mathbb{R}$ of the polynomial \[ p(x) \equal{} x^n \plus{} n \cdot x^{n\minus{}1} \plus{} a_2 \cdot x^{n\minus{}2} \plus{} \ldots \plus{} a_n\] satisfying \[ \sum^{16}_{k\equal{}1} r^{16}_k \equal{} n.\]

Let $ N\equal{}100^2\plus{}99^2\minus{}98^2\minus{}97^2\plus{}96^2\plus{}\cdots\plus{}4^2\plus{}3^2\minus{}2^2\minus{}1^2$, where the additions and subtractions alternate in pairs. Find the remainder when $ N$ is divided by $ 1000$.

[b]2.[/b] Show that threre exist a compact set $K \subset \mathbb{R}$ and a set $A \subset \mathbb{R}$ of type $F_{\sigma}$ such that the set $\{ x\in \mathbb{R} : K+x \subset A\}$ is not Borel-measurable (here $K+x = \{y+x : y \in K\}$). ([b]M.16[/b]) [M. Laczkovich]

Is it possible to draw $10$ bus routes with stops such that for any $8$ routes there is a stop that does not belong to any of the routes, but any $9$ routes pass through all the stops?

Let $H = \{ \lfloor i\sqrt{2}\rfloor : i \in \mathbb Z_{>0}\} = \{1,2,4,5,7,\dots \}$ and let $n$ be a positive integer. Prove that there exists a constant $C$ such that, if $A\subseteq \{1,2,\dots, n\}$ satisfies $|A| \ge C\sqrt{n}$, then there exist $a,b\in A$ such that $a-b\in H$. (Here $\mathbb Z_{>0}$ is the set of positive integers, and $\lfloor z\rfloor$ denotes the greatest integer less than or equal to $z$.)

In a city's bus route system, any two routes share exactly one stop, and every route includes at least four stops. Prove that the stops can be classified into two groups such that each route includes stops from each group.

Triangle $ABC$. Let $B_1$ and $C_1$ be such points, that $AB= BB_1, AC=CC_1$ and $B_1, C_1$ lie on the circumscribed circle $\Gamma$ of $\vartriangle ABC$. Perpendiculars drawn from from points $B_1$ and $C_1$ on the lines $AB$ and $AC$ intersect $\Gamma$ at points $B_2$ and $C_2$ respectively, these points lie on smaller arcs $AB$ and $AC$ of circle $\Gamma$ respectively, Prove that $BB_2 \parallel CC_2$.

Let $F:\mathbb R^{\mathbb R}\to\mathbb R^{\mathbb R}$ be a function (from the set of real-valued functions to itself) such that $$F(F(f)\circ g+g)=f\circ F(g)+F(F(F(g)))$$ for all $f,g:\mathbb R\to\mathbb R$. Prove that there exists a function $\sigma:\mathbb R\to\mathbb R$ such that $$F(f)=\sigma\circ f\circ\sigma$$ for all $f:\mathbb R\to\mathbb R$.

Consider a quadrilateral $ABCD$ and points $K, L, M, N$ on sides $AB, BC, CD$ and $AD$, respectively, such that $KB = BL = a, MD = DN = b$ and $KL \nparallel MN$. Find the set of all the intersection points of $KL$ with $MN$ as $a$ and $b$ vary.

The Central American Olympiad is an annual competition. The ninth Olympiad is held in 2007. Find all the positive integers $n$ such that $n$ divides the number of the year in which the $n$-th Olympiad takes place.

Find all the integers $n$ for which $\frac{8n-25}{n+5}$ is cube of a rational number.

A ball with mass $m$ projected horizontally off the end of a table with an initial kinetic energy $K$. At a time $t$ after it leaves the end of the table it has kinetic energy $3K$. What is $t$? Neglect air resistance. (A) $(3/g)\sqrt{K/m}$ (B) $(2/g)\sqrt{K/m}$ (C) $(1/g)\sqrt{8K/m}$ (D) $(K/g)\sqrt{6/m}$ (E) $(2K/g)\sqrt{1/m}$

Find all sequences $(a_n)$ of positive integers satisfying the equality $a_n=a_{a_{n-1}}+a_{a_{n+1}}$ a) for all $n\ge 2$ b) for all $n \ge 3$ (I. Gorodnin)

$x_1, x_2, ... , x_8$ is a permutation of $1, 2, ..., 8$. A move is to take $x_3$ or $x_8$ and place it at the start to from a new sequence. Show that by a sequence of moves we can always arrive at $1, 2, ..., 8$.

Determine all values of $n$ such that it is possible to divide a triangle in $n$ smaller triangles such that there are not three collinear vertices and such that each vertex belongs to the same number of segments.

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} $

Point $ I_1$ is the reflection of incentre $ I$ of triangle $ ABC$ across the side $ BC$. The circumcircle of $ BCI_1$ intersects the line $ II_1$ again at point $ P$. It is known that $ P$ lies outside the incircle of the triangle $ ABC$. Two tangents drawn from $ P$ to the latter circle touch it at points $ X$ and $ Y$. Prove that the line $ XY$ contains a medial line of the triangle $ ABC$. [i]Author: L. Emelyanov[/i]

The picture shows a triangle divided into $25$ smaller triangles, numbered $1$ to $25$. Is it possible to place the same numbers in the square cells 5$\times 5$ so that any two numbers written in adjacent triangles were are also written in adjacent cells of the square? (The cells of a square are considered adjacent if they have a common side.) [img]https://cdn.artofproblemsolving.com/attachments/4/3/758fe5531ab3e576ef4712c095b393f8dff397.png[/img]

All arrangements of letters $VNNWHTAAIE$ are listed in lexicographic (dictionary) order. If $AAEHINNTVW$ is the first entry, what entry number is $VANNAWHITE$?

Given an integer $c$, the sequence $a_0, a_1, a_2, ...$ is generated using the recurrence relation $a_0 = c$ and $a_i = a^i_{i-1} + 2021a_{i-1}$ for all $i \ge 1$. Given that $a_0 = c$, let $f(c)$ be the smallest positive integer $n$ such that $a_n - 1$ is a multiple of $47$. Compute $$\sum^{46}_{k=1} f(k).$$

There are $n$ line segments on the plane, no three intersecting at a point, and each pair intersecting once in their respective interiors. Tony and his $2n - 1$ friends each stand at a distinct endpoint of a line segment. Tony wishes to send Christmas presents to each of his friends as follows: First, he chooses an endpoint of each segment as a “sink”. Then he places the present at the endpoint of the segment he is at. The present moves as follows : $\bullet$ If it is on a line segment, it moves towards the sink. $\bullet$ When it reaches an intersection of two segments, it changes the line segment it travels on and starts moving towards the new sink. If the present reaches an endpoint, the friend on that endpoint can receive their present. Prove that Tony can send presents to exactly $n$ of his $2n - 1$ friends.

Let us call the [i]median [/i] of a system of $2n$ points of a plane a straight line passing through exactly two of them, on both sides of which there are points of this system equally. What is the smallest number of [i]medians [/i] that a system of $2n$ points, no three of which lie on the same line?

Could the set $\{1,2,3,...,3000\}$ contain a subset of $2000$ elements such that none of them is twice the size of another?