Find all positive integers $n$ such that $n^4 - n^3 + 3n^2 + 5$ is a perfect square.

Five mathematicians find a bag of $100$ gold coins in a room. They agree to split up the coins according to the following plan: • The oldest person in the room proposes a division of the coins among those present. (No coin may be split.) Then all present, including the proposer, vote on the proposal. • If at least $50\%$ of those present vote in favor of the proposal, the coins are distributed accordingly and everyone goes home. (In particular, a proposal wins on a tie vote.) • If fewer than $50\%$ of those present vote in favor of the proposal, the proposer must leave the room, receiving no coins. Then the process is repeated: the oldest person remaining proposes a division, and so on. • There is no communication or discussion of any kind allowed, other than what is needed for the proposer to state his or her proposal, and the voters to cast their vote. Assume that each person is equally intelligent and each behaves optimally to maximize his or her share. How much will each person get?

Find all functions $ f:\mathbb{N} \rightarrow \mathbb{ N }_0 $ satisfy the following conditions: i) $ f(ab)=f(a)+f(b)-f(\gcd(a,b)), \forall a,b \in \mathbb{N} $ ii) For all primes $ p $ and natural numbers $ a $, $ f(a)\ge f(ap) \Rightarrow f(a)+f(p) \ge f(a)f(p)+1 $

An (unordered) partition $P$ of a positive integer $n$ is an $n$-tuple of nonnegative integers $P=(x_1,x_2,\cdots,x_n)$ such that $\sum_{k=1}^n kx_k=n$. For positive integer $m\leq n$, and a partition $Q=(y_1,y_2,\cdots,y_m)$ of $m$, $Q$ is called compatible to $P$ if $y_i\leq x_i$ for $i=1,2,\cdots,m$. Let $S(n)$ be the number of partitions $P$ of $n$ such that for each odd $m<n$, $m$ has exactly one partition compatible to $P$ and for each even $m<n$, $m$ has exactly two partitions compatible to $P$. Find $S(2010)$.

Find all positive integers $ k$ with the following property: There exists an integer $ a$ so that $ (a\plus{}k)^{3}\minus{}a^{3}$ is a multiple of $ 2007$.

The vertices of $\triangle ABC$ are $A = (0,0)$, $B = (0,420)$, and $C = (560,0)$. The six faces of a die are labeled with two $A$'s, two $B$'s, and two $C$'s. Point $P_1 = (k,m)$ is chosen in the interior of $\triangle ABC$, and points $P_2$, $P_3$, $P_4, \dots$ are generated by rolling the die repeatedly and applying the rule: If the die shows label $L$, where $L \in \{A, B, C\}$, and $P_n$ is the most recently obtained point, then $P_{n + 1}$ is the midpoint of $\overline{P_n L}$. Given that $P_7 = (14,92)$, what is $k + m$?

Let $m,n$ and $a_1,a_2,\dots,a_m$ be arbitrary positive integers. Ali and Mohammad Play the following game. At each step, Ali chooses $b_1,b_2,\dots,b_m \in \mathbb{N}$ and then Mohammad chosses a positive integers $s$ and obtains a new sequence $\{c_i=a_i+b_{i+s}\}_{i=1}^m$, where $$b_{m+1}=b_1,\ b_{m+2}=b_2, \dots,\ b_{m+s}=b_s$$ The goal of Ali is to make all the numbers divisible by $n$ in a finite number of steps. FInd all positive integers $m$ and $n$ such that Ali has a winning strategy, no matter how the initial values $a_1, a_2,\dots,a_m$ are. [hide=clarification] after we create the $c_i$ s, this sequence becomes the sequence that we continue playing on, as in it is our 'new' $a_i$[/hide] Proposed by Shayan Gholami

A regular pentagon $A_1A_2A_3A_4A_5$ with side length $s$ is given. At each point $A_i$, a sphere $K_i$ of radius $\frac{s}{2}$ is constructed. There are two spheres $K_1$ and $K_2$ each of radius $\frac{s}{2}$ touching all the five spheres $K_i.$ Decide whether $K_1$ and $K_2$ intersect each other, touch each other, or have no common points.

Prove that $$\left( \frac{1}{\sqrt1+\sqrt2}+\frac{1}{\sqrt2+\sqrt3}+\frac{1}{\sqrt3+\sqrt4}+...+\frac{1}{\sqrt{2015}+\sqrt{2016}}\right)^2(2017+24\sqrt{14})=2015^2$$

Let $(u_1, \ldots, u_n)$ be an ordered $n$tuple. For each $k, 1 \leq k \leq n$, define $v_k=\sqrt[k]{u_1u_2 \cdots u_k}$. Prove that \[\sum_{k=1}^n v_k \leq e \cdot \sum_{k=1}^n u_k.\] ($e$ is the base of the natural logarithm).

Do there exist polynomials $p(x), q(x), r(x)$ whose coefficients are positive integers such that $p(x) = (x^{2}-3x+3) q(x)$ and $q(x) = (\frac{x^{2}}{20}-\frac{x}{15}+\frac{1}{12}) r(x)$?

For any positive integer $n$, let $f(n)$ be the maximum number of groups formed by a total of $n$ people such that the following holds: every group consists of an even number of members, and every two groups share an odd number of members. Compute $\textstyle\sum_{n=1}^{2022}f(n)\text{ mod }1000$.

Show that \[ a^n + b^n + c^n \geq ab^{n-1} + bc^{n-1} + ca^{n-1} \] for real $a,b,c \geq 0$ and $n$ a positive integer.

Prove that from any set of one hundred different whole numbers one can choose either one number which is divisible by $100$, or several numbers whose sum is divisible by $100$.

[u]Round 1[/u] [b]p1.[/b] An iscoceles triangle has one angle equal to $100$ degrees, what is the degree measure of one of the two remaining angles. [b]p2.[/b] Tanmay picks four cards from a standard deck of $52$ cards at random. What is the probability he gets exactly one Ace, exactly exactly one King, exactly one Queen, exactly one Jack and exactly one Ten? [b]p3.[/b] What is the sum of all the factors of $2014$? [u]Round 2[/u] [b]p4.[/b] Which number under $1000$ has the greatest number of factors? [b]p5.[/b] How many $10$ digit primes have all distinct digits? [b]p6.[/b] In a far o universe called Manhattan, the distance between two points on the plane $P = (x_1, y_1)$ and $Q = (x_2, y_2)$ is defined as $d(P,Q) = |x_1-x_2|+|y_1-y_2|$. Let $S$ be the region of points that are a distance of $\le 7$ away from the origin $(0, 0)$. What is the area of $S$? [u]Round 3[/u] [b]p7.[/b] How many factors does $13! + 14! + 15!$ have? [b]p8.[/b] How many zeroes does $45!$ have consecutively at the very end in its representation in base $45$? [b]p9.[/b] A sequence of circles $\omega_0$, $\omega_1$, $\omega_2$, ... is drawn such that: $\bullet$ $\omega_0$ has a radius of $1$. $\bullet$ $\omega_{i+1}$ has twice the radius of $\omega_i$. $\bullet$ $\omega_i$ is internally tangent to $\omega_{i+1}$. Let $A$ be a point on $\omega_0$ and $B$ be a point on $\omega_{10}$. What is the maximum possible value of $AB$? [u]Round 4[/u] [b]p10.[/b] A $3-4-5$ triangle is constructed. Then a similar triangle is constructed with the shortest side of the first triangle being the new hypotenuse for the second triangle. This happens an infinite amount of times. What is the maximum area of the resulting figure? [b]p11.[/b] If an unfair coin is flipped $4$ times and has a $3/64$ chance of coming heads exactly thrice, what is the probability the coin comes tails on a single flip. [b]p12.[/b] Find all triples of positive integers $(a, b, c)$ that satisfy $2a = 1+bc$, $2b = 1+ac$, and $2c = 1 + ab$. [u]Round 5[/u] [b]p13.[/b] $6$ numbered points on a plane are placed so that they can create a regular hexagon $P_1P_2P_3P_4P_5P_6$ if connected. If a triangle is drawn to include a certain amount of points in it, how many triangles are there that hold a different set of points? (note: the triangle with $P_1$ and $P_2$ is not the same as the one with $P_3$ and $P_4$). [b]p14.[/b] Let $S$ be the set of all numbers of the form $n(2n + 1)(3n + 2)(4n + 3)(5n + 4)$ for $n \ge 1$. What is the largest number that divides every member of $S$? [b]p15. [/b]Jordan tosses a fair coin until he gets heads at least twice. What is the expected number of flips of the coin that he will make? PS. You should use hide for answers. Rounds 6-10 have been posted [url=https://artofproblemsolving.com/community/c3h3156859p28695035]here[/url].. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The centre of a circle is the origin of $N$ vectors whose ends divide the circle in $N$ equal arcs . Some of the vectors are blue and some are red . We calculate the sum of the angles formed between each pair consisting of a red vector and a blue vector (the angle being measured anticlockwise from red to blue) and divide this sum by the total number of such angles . Prove that the "mean angle" thus obtained is $180^o$. (V. P roizvolov)

In a circle of radius $ 5$ units, $ CD$ and $ AB$ are perpendicular diameters. A chord $ CH$ cutting $ AB$ at $ K$ is $ 8$ units long. The diameter $ AB$ is divided into two segments whose dimensions are: $ \textbf{(A)}\ 1.25, 8.75 \qquad\textbf{(B)}\ 2.75,7.25 \qquad\textbf{(C)}\ 2,8 \qquad\textbf{(D)}\ 4,6$ $ \textbf{(E)}\ \text{none of these}$

Let $k$ be a circle with centre $M$ and let $AB$ be a diameter of $k$. Furthermore, let $C$ be a point on $k$ such that $AC = AM$. Let $D$ be the point on the line $AC$ such that $CD = AB$ and $C$ lies between $A$ and $D$. Let $E$ be the second intersection of the circumcircle of $BCD$ with line $AB$ and $F$ be the intersection of the lines $ED$ and $BC$. The line $AF$ cuts the segment $BD$ in $X$. Determine the ratio $BX/XD$.

Let ${c\equiv c\left(O, R\right)}$ be a circle with center ${O}$ and radius ${R}$ and ${A, B}$ be two points on it, not belonging to the same diameter. The bisector of angle${\angle{ABO}}$ intersects the circle ${c}$ at point ${C}$, the circumcircle of the triangle $AOB$ , say ${c_1}$ at point ${K}$ and the circumcircle of the triangle $AOC$ , say ${{c}_{2}}$ at point ${L}$. Prove that point ${K}$ is the circumcircle of the triangle $AOC$ and that point ${L}$ is the incenter of the triangle $AOB$. Evangelos Psychas (Greece)

In a trapezium with bases $AD$ and $BC$, let $P$ and $Q$ be the middles of diagonals $AC$ and $BD$ respectively. Prove that if $\angle DAQ = \angle CAB$ then $\angle PBA = \angle DBC$.

Five bricklayers working together finish a job in $3$ hours. Working alone, each bricklayer takes at most $36$ hours to finish the job. What is the smallest number of minutes it could take the fastest bricklayer to complete the job alone? [i]Author: Ray Li[/i]

A swimmer is standing at a corner of a square swimming pool. She swims at a fixed speed and runs at a fixed speed (possibly different). No time is taken entering or leaving the pool. What path should she follow to reach the opposite corner of the pool in the shortest possible time?

Given regular pentagon $ABCDE$, a circle can be drawn that is tangent to $\overline{DC}$ at $D$ and to $\overline{AB}$ at $A$. The number of degrees in minor arc $AD$ is $\textbf{(A)}\ 72 \qquad \textbf{(B)}\ 108 \qquad \textbf{(C)}\ 120 \qquad \textbf{(D)}\ 135 \qquad \textbf{(E)}\ 144$ [asy] size(100); defaultpen(linewidth(0.7)); draw(rotate(18)*polygon(5)); real x=0.6180339887; draw(Circle((-x,0), 1)); int i; for(i=0; i<5; i=i+1) { dot(origin+1*dir(36+72*i)); } label("$B$", origin+1*dir(36+72*0), dir(origin--origin+1*dir(36+72*0))); label("$A$", origin+1*dir(36+72*1), dir(origin--origin+1*dir(36+72))); label("$E$", origin+1*dir(36+72*2), dir(origin--origin+1*dir(36+144))); label("$D$", origin+1*dir(36+72*3), dir(origin--origin+1*dir(36+72*3))); label("$C$", origin+1*dir(36+72*4), dir(origin--origin+1*dir(36+72*4)));[/asy]

Let $ABCD$ be a quadrilateral that can be inscribed in a circle. Denote by $P$ the intersection point of lines $AD$ and $BC$, and by $Q$ the intersection point of lines $AB$ and $DC$. Let $E$ be the fourth vertex of the parallelogram $ABCE$, and $F$ the intersection of lines $CE$ is $PQ$. Prove that the points $D,E, F$, and $Q$ lie on the same circle.

Let $S$ be the set of points whose coordinates $x,$ $y,$ and $z$ are integers that satisfy $0\le x\le2,$ $0\le y\le3,$ and $0\le z\le4.$ Two distinct points are randomly chosen from $S.$ The probability that the midpoint of the segment they determine also belongs to $S$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$