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?

The nonnegative real numbers $a$ and $b$ satisfy $a + b = 1$. Prove that: $$\frac{1}{2} \leq \frac{a^3+b^3}{a^2+b^2} \leq 1$$ When do we have equality in the right inequality and when in the left inequality? [i]Proposed by Walther Janous [/i]

Let $A$, $B$ be opposite vertices of a unit square with circumcircle $\Gamma$. Let $C$ be a variable point on $\Gamma$. If $C\not\in\{A, B\}$, then let $\omega$ be the incircle of triangle $ABC$, and let $I$ be the center of $\omega$. Let $C_1$ be the point at which $\omega$ meets $\overline{AB}$, and let $D$ be the reflection of $C_1$ over line $CI$. If $C \in\{A, B\}$, let $D = C$. As $C$ varies on $\Gamma$, $D$ traces out a curve $\mathfrak C$ enclosing a region of area $\mathcal A$. Compute $\lfloor 10^4 \mathcal A\rfloor$. [i]Proposed by Brandon Wang[/i]

Two flag poles of height $h$ and $k$ are situated $2a$ units apart on a level surface. Find the set of all points on the surface which are so situated that the angles of elevation of the tops of the poles are equal.

In $1748$, the great Russian mathematician Leonhard Euler published one of his most important works, Introduction to the Analysis of Infinites. In this work, in particular, Euler finds the values of two infinite sums $1 +\frac14 +\frac19+ \frac{1}{16}+...$ and $1 +\frac19+ \frac{1}{16}+...$ (the terms in the first sum are the inverses of the squares of the natural numbers, and in the second are the inverses of the squares of the odd numbers of the natural series). The value of the first sum, as Euler proved, equals $\frac{\pi^2}{6}$. Given this result, find the value of the second sum.

By inserting parentheses, it is possible to give the expression \[ 2\times3\plus{}4\times5 \]several values. How many different values can be obtained? $ \textbf{(A)}\ 2\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 4\qquad \textbf{(D)}\ 5\qquad \textbf{(E)}\ 6$

Let define $P_{n}(x)=x^{n-1}+x^{n-2}+x^{n-3}+ \dots +x+1$ for every positive integer $n$. Prove that for every positive integer $a$ one can find a positive integer $n$ and polynomials $R(x)$ and $Q(x)$ with integer coefficients such that \[P_{n}(x)= [1+ax+x^{2}R(x)] Q(x).\]

Let the incircle of $\triangle ABC$ touch sides $BC$, $CA$, and $AB$ at points $D$, $E$, and $F$, respectively. Let $D'$ be the diametrically opposite point of $D$ with respect to the incircle. Let lines $AD'$ and $AD$ intersect the incircle again at $X$ and $Y$, respectively. Prove that the lines $DX$, $D'Y$, and $EF$ are concurrent, i.e., the lines intersect at the same point. [i](Kritesh Dhakal, Nepal)[/i]

Just wondering: what exactly is Power of a Point?

Trapezoid $ABCD$ is an isosceles trapezoid with $AD=BC$. Point $P$ is the intersection of the diagonals $AC$ and $BD$. If the area of $\triangle ABP$ is $50$ and the area of $\triangle CDP$ is $72$, what is the area of the entire trapezoid? [i]Proposed by David Altizio

In a sequence of natural numbers $ a_1 $, $ a_2 $, $ \dots $, $ a_ {1999} $, $ a_n-a_ {n-1} -a_ {n-2} $ is divisible by $ 100 (3 \leq n \leq 1999) $. It is known that $ a_1 = 19$ and $ a_2 = 99$. Find the remainder of $ a_1 ^ 2 + a_2 ^ 2 + \dots + a_ {1999} ^ 2 $ by $8$.

let $x_i,y_i 1 \le i \le n$ be positive numbers such that : $\displaystyle \sum_{i=1}^n x_i \ge \sum_{i=1}^n x_iy_i$ Prove : $\displaystyle \sum_{i=1}^n x_i \le \sum _{i=1}^n \frac{x_i}{y_i}$

A five-digit integer is said to be [i]balanced [/i]i f the sum of any three of its digits is divisible by any of the other two. How many [i]balanced [/i] numbers are there?

A square grid is composed of $ n^2\equiv 1\pmod 4 $ unit cells that contained each a locust that jumped the same amount of cells in the direccion of columns or lines, without leaving the grid. Prove that, as a result of this, at least two locusts landed on the same cell. [i]Marius Cavachi[/i]

A circle passes through vertex $B$ of the triangle $ABC$, intersects its sides $ AB $and $BC$ at points $K$ and $L$, respectively, and touches the side $ AC$ at its midpoint $M$. The point $N$ on the arc $BL$ (which does not contain $K$) is such that $\angle LKN = \angle ACB$. Find $\angle BAC $ given that the triangle $CKN$ is equilateral.

For each real number $x$, $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$. For example $\lfloor 2.8 \rfloor = 2$. Let $r \ge 0$ be a real number such that for all integers $m, n, m|n$ implies $\lfloor mr \rfloor| \lfloor nr \rfloor$. Prove that $r$ is an integer.

We have a set of $343$ closed jars, each containing blue, yellow and red marbles with the number of marbles from each color being at least $1$ and at most $7$. No two jars have exactly the same contents. Initially all jars are with the caps up. To flip a jar will mean to change its position from cap-up to cap-down or vice versa. It is allowed to choose a triple of positive integers $(b; y; r) \in \{1; 2; ...; 7\}^3$ and flip all the jars whose number of blue, yellow and red marbles differ by not more than $1$ from $b, y, r$, respectively. After $n$ moves all the jars turned out to be with the caps down. Find the number of all possible values of $n$, if $n \le 2021$.

The sequence $(a_n)$ is such that $a_{n+1} = (a_n)^n + n + 1$ for all positive integers $n$, where $a_1$ is some positive integer. Let $k$ be the greatest power of $3$ by which $a_{101}$ is divisible. Find all possible values of $k$. [i]Proposed by Kyrylo Holodnov[/i]

Show that $\frac{(x - y)^7 + (y - z)^7 + (z - x)^7 - (x - y)(y - z)(z - x) ((x - y)^4 + (y - z)^4 + (z - x)^4)} {(x - y)^5 + (y - z)^5 + (z - x)^5} \ge 3$ holds for all triples of distinct integers $x, y, z$. When does equality hold?

In table of dimensions $2n \times 2n$ there are positive integers not greater than $10$, such that numbers lying in unit squares with common vertex are coprime. Prove that there exist at least one number which occurs in table at least $\frac{2n^2}{3}$ times

Let $ABC$ be a scalene triangle. Let $l_A$ be the tangent to the nine-point circle at the foot of the perpendicular from $A$ to $BC$, and let $l_A'$ be the tangent to the nine-point circle from the mid-point of $BC$. The lines $l_A$ and $l_A'$ intersect at $A'$ . Define $B'$ and $C'$ similarly. Show that the lines $AA' , BB'$ and $CC'$ are concurrent.