At the currency exchange of the island of Luck they sell dinars (D), guilders (G), reals (R) and thalers (T). Stock brokers have the right to make a purchase and sale transaction with any pair of currencies no more than once per day. The exchange rates are as follows: $D = 6G$, $D = 25R$, $D = 120T$, $G = 4R$, $G = 21T$, $R = 5T$. For example, the entry $D = 6G$ means that $1$ dinar can be bought for $6$ guilders (or $6$ guilders can be sold for $1$ dinar). In the morning the broker had $80$ dinars, $100$ guilders, $100$ reals and $50,400$ thalers. In the evening he had the same number of dinars and thalers. What is the maximum value of this number?

Let $A$ be a set of $2n$ points in the plane, no three of which are collinear. Suppose that $n$ of them are colored red and the remaining $n$ blue. Prove or disprove: there are $n$ closed straight line segments, no two with a point in common, such that the endpoints of each segment are points of $A$ having different colors.

Consider a sequence $\{a_n\}$ of integers, satisfying $a_1=1, a_2=2$ and $a_{n+1}$ is the largest prime divisor of $a_1+a_2+\ldots+a_n$. Find $a_{100}$.

Let $a,b,c$ be real number greater than 1 satisfying $$\lfloor a\rfloor b = \lfloor b \rfloor c = \lfloor c\rfloor a.$$Prove that $a=b=c$ (Here, $\lfloor x \rfloor$ denotes the laregst integer that is less than or equal to $x$.)

The cells of the $9 \times 9$ table are colored black and white. It turned out, that there were $k$ rows, in each of which there are more black cells than white ones white, and there were $k$ columns, each of which contained more than black. At what highest $ k$ is this possible?

The three roots of $P(x) = x^3 - 2x^2 - x + 1$ are $a>b>c \in \mathbb{R}$. Find the value of $a^2b+b^2c+c^2a$. :D

$n$ students, numbered from $1$ to $n$ are sitting next to each other in a class. In the beginning the $1$st student has $n$ pieces of paper in one pile. The goal of the students is to distribute the $n$ pieces in a way that everyone gets exactly one. The teacher claps once in a minute and for each clap the students can choose one of the following moves (or do nothing): $\bullet$ They divide one of their piles of paper into two smaller piles. $\bullet$ They give one of their piles of paper to the student with the next number. At least how many times does the teacher need to clap in order to make it possible for the students to distribute all the pieces of paper amongst themselves?

In the following diagram, let $ABCD$ be a square and let $M,N,P$ and $Q$ be the midpoints of its sides. Prove that \[S_{A'B'C'D'} = \frac 15 S_{ABCD}.\] [asy] import graph; size(200); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen qqttzz = rgb(0,0.2,0.6); pen qqzzff = rgb(0,0.6,1); draw((0,4)--(4,4),qqttzz+linewidth(1.6pt)); draw((4,4)--(4,0),qqttzz+linewidth(1.6pt)); draw((4,0)--(0,0),qqttzz+linewidth(1.6pt)); draw((0,0)--(0,4),qqttzz+linewidth(1.6pt)); draw((0,4)--(2,0),qqzzff+linewidth(1.2pt)); draw((2,4)--(4,0),qqzzff+linewidth(1.2pt)); draw((0,2)--(4,4),qqzzff+linewidth(1.2pt)); draw((0,0)--(4,2),qqzzff+linewidth(1.2pt)); dot((0,4),ds); label("$A$", (0.07,4.12), NE*lsf); dot((0,0),ds); label("$D$", (-0.27,-0.37), NE*lsf); dot((4,0),ds); label("$C$", (4.14,-0.39), NE*lsf); dot((4,4),ds); label("$B$", (4.08,4.12), NE*lsf); dot((2,4),ds); label("$M$", (2.08,4.12), NE*lsf); dot((4,2),ds); label("$N$", (4.2,1.98), NE*lsf); dot((2,0),ds); label("$P$", (1.99,-0.49), NE*lsf); dot((0,2),ds); label("$Q$", (-0.48,1.9), NE*lsf); dot((0.8,2.4),ds); label("$A'$", (0.81,2.61), NE*lsf); dot((2.4,3.2),ds); label("$B'$", (2.46,3.47), NE*lsf); dot((3.2,1.6),ds); label("$C'$", (3.22,1.9), NE*lsf); dot((1.6,0.8),ds); label("$D'$", (1.14,0.79), NE*lsf); clip((-4.44,-11.2)--(-4.44,6.41)--(16.48,6.41)--(16.48,-11.2)--cycle); [/asy] [$S_{X}$ denotes area of the $X.$]

Let $n$ be an integer greater than $1$, $\alpha$ is a real, $0<\alpha < 2$, $a_1,\ldots ,a_n,c_1,\ldots ,c_n$ are all positive numbers. For $y>0$, let $$f(y)=\left(\sum_{a_i\le y} c_ia_i^2\right)^{\frac{1}{2}}+\left(\sum_{a_i>y} c_ia_i^{\alpha} \right)^{\frac{1}{\alpha}}.$$ If positive number $x$ satisfies $x\ge f(y)$ (for some $y$), prove that $f(x)\le 8^{\frac{1}{\alpha}}\cdot x$.

Suppose $\alpha, \beta$ are two positive rational numbers. Assume for some positive integers $m,n$, it is known that $\alpha^{\frac 1n}+\beta^{\frac 1m}$ is a rational number. Prove that each of $\alpha^{\frac 1n}$ and $\beta^{\frac 1m}$ is a rational number.

Consider the $5\times 5$ grid $Z^2_5 = \{(a, b) : 0 \le a, b \le 4\}$. Say that two points $(a, b)$,$(x, y)$ are adjacent if $a - x \equiv -1, 0, 1$ (mod $5$) and $b - y \equiv -1, 0, 1$ (mod $5$) . For example, in the diagram, all of the squares marked with $\cdot$ are adjacent to the square marked with $\times$. [img]https://cdn.artofproblemsolving.com/attachments/2/6/c49dd26ab48ddff5e1beecfbd167d5bb9fe16d.png[/img] What is the largest number of $\times$ that can be placed on the grid such that no two are adjacent?

Find all the functions $ f: \mathbb{N}\rightarrow \mathbb{N}$ such that \[ 3f(f(f(n))) \plus{} 2f(f(n)) \plus{} f(n) \equal{} 6n, \quad \forall n\in \mathbb{N}.\]

In the $3\times5$ grid shown, fill in each empty box with a two-digit positive integer such that: [list][*]no number appears in more than one box, and [*] for each of the $9$ lines in the grid consisting of three boxes connected by line segments, the box in the middle of the line contains the least common multiple of the numbers in the two boxes on the line.[/list] You do not need to prove that your answer is the only one possible; you merely need to find an answer that satisfies the constraints above. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.) [asy] import graph; size(7cm); real labelscalefactor = 0.5; pen dps = linewidth(0.8) + fontsize(14); defaultpen(dps); draw((0,0)--(1,0)--(1,1)--(0,1)--cycle); draw((2,0)--(3,0)--(3,1)--(2,1)--cycle); draw((4,0)--(5,0)--(5,1)--(4,1)--cycle); draw((6,0)--(7,0)--(7,1)--(6,1)--cycle); draw((8,0)--(9,0)--(9,1)--(8,1)--cycle); draw((0,2)--(1,2)--(1,3)--(0,3)--cycle); draw((0,4)--(1,4)--(1,5)--(0,5)--cycle); draw((2,2)--(3,2)--(3,3)--(2,3)--cycle); draw((2,4)--(3,4)--(3,5)--(2,5)--cycle); draw((4,4)--(5,4)--(5,5)--(4,5)--cycle); draw((4,2)--(5,2)--(5,3)--(4,3)--cycle); draw((6,2)--(7,2)--(7,3)--(6,3)--cycle); draw((6,4)--(7,4)--(7,5)--(6,5)--cycle); draw((8,4)--(9,4)--(9,5)--(8,5)--cycle); draw((8,2)--(9,2)--(9,3)--(8,3)--cycle); draw((0.5,1)--(0.5,2)); draw((0.5,3)--(0.5,4)); draw((1,4)--(2,3)); draw((2.5,1)--(2.5,2)); draw((2.5,3)--(2.5,4)); draw((3,4)--(4,3)); draw((3,2)--(4,1)); draw((4.5,1)--(4.5,2)); draw((4.5,3)--(4.5,4)); draw((5,4.5)--(6,4.5)); draw((7,4.5)--(8,4.5)); draw((5,4)--(6,3)); draw((7,2)--(8,1)); draw((5,2)--(6,1)); draw((5,0.5)--(6,0.5)); draw((7,0.5)--(8,0.5)); draw((8.5,1)--(8.5,2)); draw((8.5,3)--(8.5,4)); label("$4$",(4.5, 0.5)); label("$9$",(8.5, 4.5)); [/asy]

Every cell of a $m \times n$ chess board, $m\ge 2,n\ge 2$, is colored with one of four possible colors, e.g white, green, red, blue. We call such coloring good if the four cells of any $2\times 2$ square of the chessboard are colored with pairwise different colors. Determine the number of all good colorings of the chess board. [i]Proposed by N. Beluhov[/i]

Let $x_1, x_2, ..., x_{2004}$ be a sequence of integer numbers such that $x_{k+3}=x_{k+2}+x_{k}x_{k+1}$, $\forall 1 \le k \le 2001$. Is it possible that more than half of the elements are negative?

Consider a real number $a$ that satisfies $a=(a-1)^3$. Prove that there exists an integer $N$ that satisfies $$|a^{2021}-N|<2^{-1000}.$$ [i] (Vlad Robu) [/i]

Find the number of positive integers $n$ that are solutions to the simultaneous system of inequalities \begin{align*}4n-18&<2008,\\7n+17&>2008.\end{align*}

If $ c$ is a positive integer and $ p$ is an odd prime, what is the smallest residue (in absolute value) of \[ \sum_{n=0}^{\frac{p-1}{2}} \binom{2n}{n}c^n \;(\textrm{mod}\;p\ ) \ ?\] J. Suranyi

Find the number of integers $ c$ such that $ \minus{}2007 \leq c \leq 2007$ and there exists an integer $ x$ such that $ x^2 \plus{} c$ is a multiple of $ 2^{2007}$.

Let $f(x)$ be a polynomial of degree $n$ with real coefficients, having $n$ (not necessarily distinct) real roots. Prove that for all real $x$, $$f(x)f''(x)\le f'(x)^2.$$

Let $M>1$ be a natural number. Tom and Jerry play a game. Jerry wins if he can produce a function $f: \mathbb{N} \rightarrow \mathbb{N}$ satisfying [list] [*]$f(M) \ne M$ [/*] [*] $f(k)<2k$ for all $k \in \mathbb{N}$[/*] [*] $f^{f(n)}(n)=n$ for all $n \in \mathbb{N}$. For each $\ell>0$ we define $f^{\ell}(n)=f\left(f^{\ell-1}(n)\right)$ and $f^0(n)=n$[/*] [/list] Tom wins otherwise. Prove that for infinitely many $M$, Tom wins, and for infinitely many $M$, Jerry wins. [i]Proposed by Anant Mudgal[/i]

Suppose that $p,p+d,p+2d,p+3d,p+4d$, and $p+5d$ are six prime numbers, where $p$ and $d$ are positive integers. Show that $d$ must be divisible by $2,3,$ and $5$.

Consider a square. Find the locus of midpoints of the hypothenuses of rightangled triangles with the vertices lying on three different sides of the square and not coinciding with its vertices. (B.Frenkin)

A circle $ C$ with center $ O.$ and a line $ L$ which does not touch circle $ C.$ $ OQ$ is perpendicular to $ L,$ $ Q$ is on $ L.$ $ P$ is on $ L,$ draw two tangents $ L_1, L_2$ to circle $ C.$ $ QA, QB$ are perpendicular to $ L_1, L_2$ respectively. ($ A$ on $ L_1,$ $ B$ on $ L_2$). Prove that, line $ AB$ intersect $ QO$ at a fixed point. [i]Original formulation:[/i] A line $ l$ does not meet a circle $ \omega$ with center $ O.$ $ E$ is the point on $ l$ such that $ OE$ is perpendicular to $ l.$ $ M$ is any point on $ l$ other than $ E.$ The tangents from $ M$ to $ \omega$ touch it at $ A$ and $ B.$ $ C$ is the point on $ MA$ such that $ EC$ is perpendicular to $ MA.$ $ D$ is the point on $ MB$ such that $ ED$ is perpendicular to $ MB.$ The line $ CD$ cuts $ OE$ at $ F.$ Prove that the location of $ F$ is independent of that of $ M.$

Let $ABC$ be a triangle with incentre $I$ and circumcircle $\omega$. Let $N$ denote the second point of intersection of line $AI$ and $\omega$. The line through $I$ perpendicular to $AI$ intersects line $BC$, segment $[AB]$, and segment $[AC]$ at the points $D$, $E$, and $F$, respectively. The circumcircle of triangle $AEF$ meets $\omega$ again at $P$, and lines $PN$ and $BC$ intersect at $Q$. Prove that lines $IQ$ and $DN$ intersect on $\omega$.