Country $\mathcal{A}$ has $c\%$ of the world's population and owns $d\%$ of the world's wealth. Country $\mathcal{B}$ has $e\%$ of the world's population and $f\%$ of its wealth. Assume that the citizens of $\mathcal{A}$ share the wealth of $\mathcal{A}$ equally, and assume that those of $\mathcal{B}$ share the wealth of $\mathcal{B}$ equally. Find the ratio of the wealth of a citizen of $\mathcal{A}$ to the wealth of a citizen of $\mathcal{B}$. $ \textbf{(A)}\ \frac{cd}{ef} \qquad\textbf{(B)}\ \frac{ce}{df} \qquad\textbf{(C)}\ \frac{cf}{de} \qquad\textbf{(D)}\ \frac{de}{cf} \qquad\textbf{(E)}\ \frac{df}{ce} $

In a triangle $ABC$, the bisector of angle $BAC$ intersects $BC$ at $D$. The circle $\Gamma$ through $A$ which is tangent to $BC$ at $D$ meets $AC$ again at $M$. Line $BM$ meets $\Gamma$ again at $P$. Prove that line $AP$ is a median of $\triangle ABD.$

Given [i]Fibonacci[/i] sequence $(F_n),$ and a positive integer $m$, denote $k(m)$ by the smallest positive integer satisfying $F_{n+k(m)}\equiv F_n(\bmod m),$ for all natural numbers $n$, $p$ is an odd prime such that $p \equiv \pm 1(\bmod 5)$. Prove that: a) ${F_{p + 1}} \equiv 0(\bmod p).$ b) $k(p)|2p+2.$ c) $k(p)$ is divisible by $4.$

A house has an even number of lamps distributed among its rooms in such a way that there are at least three lamps in every room. Each lamp shares a switch with exactly one other lamp, not necessarily from the same room. Each change in the switch shared by two lamps changes their states simultaneously. Prove that for every initial state of the lamps there exists a sequence of changes in some of the switches at the end of which each room contains lamps which are on as well as lamps which are off. [i]Proposed by Australia[/i]

Let $S$ be the set of positive integers. For any $a$ and $b$ in the set we have $GCD(a, b)>1$. For any $a$, $b$ and $c$ in the set we have $GCD(a, b, c)=1$. Is it possible that $S$ has $2012$ elements? [i]Proposed by Ognjen Stipetić.[/i]

Let $a_1,b_1,c_1,a_2,b_2,c_2$ be positive real number and $F,G:(0,\infty)\to(0,\infty)$ be to differentiable and positive functions that satisfy the identities: $$\frac{x}{F} = 1 + a_1x+ b_1y + c_1G$$ $$\frac{y}{G} = 1 + a_2x+ b_2y + c_2F.$$ Prove that if $0 < x_1 \leq x_2$ and $0 < y_2 \leq y_1$, then $F(x_1,x_2) \leq F(x_2,y_2)$ and $G(x_1,y_1) \geq G(x_2,y_2).$

Let $ ABC$ be a triangle. The $ A$-excircle of triangle $ ABC$ has center $ O_{a}$ and touches the side $ BC$ at the point $ A_{a}$. The $ B$-excircle of triangle $ ABC$ touches its sidelines $ AB$ and $ BC$ at the points $ C_{b}$ and $ A_{b}$. The $ C$-excircle of triangle $ ABC$ touches its sidelines $ BC$ and $ CA$ at the points $ A_{c}$ and $ B_{c}$. The lines $ C_{b}A_{b}$ and $ A_{c}B_{c}$ intersect each other at some point $ X$. Prove that the quadrilateral $ AO_{a}A_{a}X$ is a parallelogram. [i]Remark.[/i] The $ A$[i]-excircle[/i] of a triangle $ ABC$ is defined as the circle which touches the segment $ BC$ and the extensions of the segments $ CA$ and $ AB$ beyound the points $ C$ and $ B$, respectively. The center of this circle is the point of intersection of the interior angle bisector of the angle $ CAB$ and the exterior angle bisectors of the angles $ ABC$ and $ BCA$. Similarly, the $ B$-excircle and the $ C$-excircle of triangle $ ABC$ are defined. [hide="Source of the problem"][i]Source of the problem:[/i] Theorem (88) in: John Sturgeon Mackay, [i]The Triangle and its Six Scribed Circles[/i], Proceedings of the Edinburgh Mathematical Society 1 (1883), pages 4-128 and drawings at the end of the volume.[/hide]

Let $V$ be the set of vertices of a regular $25$ sided polygon with center at point $C.$ How many triangles have vertices in $ V$ and contain the point $C$ in the interior of the triangle?

Determine all non-negative real numbers $a$, for which $f(a)=0$ for all functions $f: \mathbb{R}_{\ge 0}\to \mathbb{R}_{\ge 0} $, who satisfy the equation $f(f(x) + f(y)) = yf(1 + yf(x))$ for all non-negative real numbers $x$ and $y$.

In triangle $ ABC$ the bisector of $ \angle ACB$ intersects $ AB$ at $ D$. Consider an arbitrary circle $ O$ passing through $ C$ and $ D$, so that it is not tangent to $ BC$ or $ CA$. Let $ O\cap BC \equal{} \{M\}$ and $ O\cap CA \equal{} \{N\}$. a) Prove that there is a circle $ S$ so that $ DM$ and $ DN$ are tangent to $ S$ in $ M$ and $ N$, respectively. b) Circle $ S$ intersects lines $ BC$ and $ CA$ in $ P$ and $ Q$ respectively. Prove that the lengths of $ MP$ and $ NQ$ do not depend on the choice of circle $ O$.

At the start of the PUMaC opening ceremony in McCosh auditorium, the speaker counts $90$ people in the audience. Every minute afterwards, either one person enters the auditorium (due to waking up late) or leaves (in order to take a dreadful math contest). The speaker observes that in this time, exactly $100$ people enter the auditorium, $100$ leave, and $100$ was the largest audience size he saw. Find the largest integer $m$ such that $2^m$ divides the number of different possible sequences of entries and exits given the above information.

Determine the smallest positive integer that has all its digits equal to $4$, and is a multiple of $169$.

Let $ABC$ be atriangle with $\overline{AC}> \overline{BC}$ and incircle $k$. Let $M,W,L$ be the intersections of the median, angle bisector and altitude from point $C$ respectively. The tangent to $k$ passing through $M$, that is different from $AB$, touch $k$ in $T$. Prove that the angles $\angle MTW$ and $\angle TLM$ are equal.

$x, y, z$ are positive reals which their product is not smaller then their sum. Prove the inequality: $$\sqrt{2x^2+yz}+\sqrt{2y^2+zx}+\sqrt{2z^2+xy} \geq 9$$

Point $H$ is the foot of the altitude from $A$ of triangle $ABC$. On the lines $AB$ and $AC$ points $X$ and $Y$ are marked such that the circumcircles of triangles $BXH$ and $CYH$ are tangent, call this circles $w_B$ and $w_C$ respectively. Tangent lines to circles $w_B$ and $w_C$ at $X$ and $Y$ intersect at $Z$. Prove that $ZA=ZH$. [i]Vadzim Kamianetski[/i]

Let $P_1 , P_2 , ...$ be arbitrary points and A be a connected compact set in the plane with a diameter greater than 4. Show that for some point P in A , $\overline {PP_1} \cdot \overline {PP_2} \cdots \overline {PP_n}>1$. Furthermore, prove that this is no longer necessarily true for compact sets of diameter 4.

[u]Round 6 [/u] [b]p16.[/b] Triangle $ABC$ with $AB < AC$ is inscribed in a circle. Point $D$ lies on the circle and point $E$ lies on side $AC$ such that $ABDE$ is a rhombus. Given that $CD = 4$ and $CE = 3$, compute $AD^2$. [b]p17.[/b] Wam and Sang are walking on the coordinate plane. Both start at the origin. Sang walks to the right at a constant rate of $1$ m/s. At any positive time $t$ (in seconds),Wam walks with a speed of $1$ m/s with a direction of $t$ radians clockwise of the positive $x$-axis. Evaluate the square of the distance betweenWamand Sang in meters after exactly $5\pi$ seconds. [b]p18.[/b] Mawile is playing a game against Salamance. Every turn,Mawile chooses one of two moves: Sucker Punch or IronHead, and Salamance chooses one of two moves: Dragon Dance or Earthquake. Mawile wins if the moves used are Sucker Punch and Earthquake, or Iron Head and Dragon Dance. Salamance wins if the moves used are Iron Head and Earthquake. If the moves used are Sucker Punch and Dragon Dance, nothing happens and a new turn begins. Mawile can only use Sucker Punch up to $8$ times. All other moves can be used indefinitely. Assuming bothMawile and Salamance play optimally, find the probability thatMawile wins. [u]Round 7 [/u] [b]p19.[/b] Ephram is attempting to organize what rounds everyone is doing for the NEAML competition. There are $4$ rounds, of which everyone must attend exactly $2$. Additionally, of the 6 people on the team(Ephram,Wam, Billiam, Hacooba,Matata, and Derke), exactly $3$ must attend every round. In how many different ways can Ephram organize the teams like this? [b]p20.[/b] For some $4$th degree polynomial $f (x)$, the following is true: $\bullet$ $f (-1) = 1$. $\bullet$ $f (0) = 2$. $\bullet$ $f (1) = 4$. $\bullet$ $f (-2) = f (2) = f (3)$. Find $f (4)$. [b]p21.[/b] Find the minimum value of the expression $\sqrt{5x^2-16x +16}+\sqrt{5x^2-18x +29}$ over all real $x$. [u]Round 8 [/u] [b]p22.[/b] Let $O$ and $I$ be the circumcenter and incenter, respectively, of $\vartriangle ABC$ with $AB = 15$, $BC = 17$, and $C A = 16$. Let $X \ne A$ be the intersection of line $AI$ and the circumcircle of $\vartriangle ABC$. Find the area of $\vartriangle IOX$. [b]p23.[/b] Find the sum of all integers $x$ such that there exist integers $y$ and $z$ such that $$x^2 + y^2 = 3(2016^z )+77.$$ [b]p24.[/b] Evaluate $$ \left \lfloor \sum^{2022}_{i=1} \frac{1}{\sqrt{i}} \right \rfloor = \left \lfloor \frac{1}{\sqrt{1}} +\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+ \frac{1}{\sqrt{2022}}\right \rfloor$$ [u]Round 9[/u] [b]p25.[/b]Either: 1. Submit $-2$ as your answer and you’ll be rewarded with two points OR 2. Estimate the number of teams that choose the first option. If your answer is within $1$ of the correct answer, you’ll be rewarded with three points, and if you are correct, you’ll receive ten points. [b]p26.[/b] Jeff is playing a turn-based game that starts with a positive integer $n$. Each turn, if the current number is $n$, Jeff must choose one of the following: 1. The number becomes the nearest perfect square to $n$ 2. The number becomes $n-a$, where $a$ is the largest digit in $n$ Let $S(k)$ be the least number of turns Jeff needs to get from the starting number $k$ to $0$. Estimate $$\sum^{2023}_{k=1}S(k).$$ If your estimation is $E$ and the actual answer is $A$, you will receive $\max \left( \left \lfloor 10 - \left| \frac{E-A}{6000} \right| \right \rfloor , 0 \right)$ points. [b]p27.[/b] Estimate the smallest positive integer n such that if $N$ is the area of the $n$-sided regular polygon with circumradius $100$, $10000\pi -N < 1$ is true. If your estimation is $E$ and the actual answer is $A$, you will receive $ \max \left \lfloor \left( 10 - \left| 10 \cdot \log_3 \left( \frac{A}{E}\right) \right|\right| ,0\right \rfloor.$ points. PS. You should use hide for answers. Rounds 1-5 have been posted [url=https://artofproblemsolving.com/community/c3h3167360p28825713]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ A$ be an $ n \times n$ matrix whose elements are non-negative real numbers. Assume that $ A$ is a non-singular matrix and all elements of $ A^{\minus{}1}$ are non-negative real numbers. Prove that every row and every column of $ A$ has exactly one non-zero element.

Find all pairs of nonnegative integers $(x, p)$, where $p$ is prime, that verify $$x(x+1)(x+2)(x+3)=1679^{p-1}+1680^{p-1}+1681^{p-1}.$$

For a 3-digit-number $n=\overline{abc}$, if $a,b,c$ can be three sides of an isosceles triangle (regular triangle included), then the number of such numbers is $\text{(A)}45\qquad\text{(B)}81\qquad\text{(C)}165\qquad\text{(D)}216$

The line $\ell$ parallel to the side $BC$ of the triangle $ABC$, intersects its sides $AB,AC$ at the points $D,E$, respectively. The circumscribed circle of triangle $ABC$ intersects line $\ell$ at points $F$ and $G$, such that points $F,D,E,G$ lie on line $\ell$ in this order. The circumscribed circles of the triangles $FEB$ and $DGC$ intersect at points $P$ and $Q$. Prove that points $A, P$ and $Q$ are collinear.

Three lines $m$, $n$, and $\ell$ lie in a plane such that no two are parallel. Lines $m$ and $n$ meet at an acute angle of $14^{\circ}$, and lines $m$ and $\ell$ meet at an acute angle of $20^{\circ}$. Find, in degrees, the sum of all possible acute angles formed by lines $n$ and $\ell$. [i]Ray Li[/i]

Let $P$ and $A$ denote the perimeter and area respectively of a right triangle with relatively prime integer side-lengths. Find the largest possible integral value of $\frac{P^2}{A}$ [color = red]The official statement does not have the final period.[/color]

$ k$ black pieces are placed on $ k$ consecutive squares of top row starting from upper left of a $ 2\times 5$ board. We are placing white pieces on empty squares one by one in arbitrary order. Two squares is said to adjacent if they have common vertex. When a white piece is placed on a square, the pieces on adjacent squares change their color. For which $ k$, when all the squares are filled, it is possible that color of every piece is white? $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{None}$

Find all $ f:N\rightarrow N$, such that $\forall m,n\in N $ $ 2f(mn) \geq f(m^2+n^2)-f(m)^2-f(n)^2 \geq 2f(m)f(n) $