Let $ABC$ be an acute triangle with the perimeter of $2s$. We are given three pairwise disjoint circles with pairwise disjoint interiors with the centers $A, B$, and $C$, respectively. Prove that there exists a circle with the radius of $s$ which contains all the three circles. [i]Proposed by Josef Tkadlec, Czechia[/i]

Place the 21 two-digit prime numbers in the white squares of the grid on the right so that each two-digit prime is used exactly once. Two white squares sharing a side must contain two numbers with either the same tens digit or ones digit. A given digit in a white square must equal at least one of the two digits of that square’s prime number. [asy] size(10cm); real s= 10.0; int[][] x = { {0,0,0,0,0}, {0,0,0,0,0}, {0,0,0,0,0}, {0,0,0,0,0}, {0,0,0,0,0}}; void square(int a, int b) { fill(s*(a,b)--s*(a+1,b)--s*(a+1,b+1)--s*(a,b+1)--cycle); } square(1,2); square(1,3); square(3,1); square(3,2); for(int i = 0; i < 6; ++i) { draw(s*(i,0)--s*(i,5)); } for(int i = 0; i < 6; ++i) { draw(s*(0,i)--s*(5,i)); } for(int k = 0; k<5; ++k){ for(int l = 0; l<5; ++l){ if(x[k][l]!=0){ label(scale(5.0)*string(x[k][l]),s*(l+0.5,-k+4.5)); } } } void sudokuLabel(int p, int q, int r) { label(string(r), s*(p, q) + (1, -1)); } sudokuLabel(1, 1, 4); sudokuLabel(2, 1, 1); sudokuLabel(3, 1, 1); sudokuLabel(4, 1, 3); sudokuLabel(0, 3, 9); sudokuLabel(2, 3, 9); sudokuLabel(4, 3, 5); sudokuLabel(0, 5, 3); sudokuLabel(1, 5, 1); sudokuLabel(2, 5, 3); sudokuLabel(3, 5, 2);[/asy] There is a unique solution, but 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.)

A number $x_n$ of the form 10101...1 has $n$ ones. Find all $n$ such that $x_n$ is prime.

Higher Secondary P4 If the fraction $\dfrac{a}{b}$ is greater than $\dfrac{31}{17}$ in the least amount while $b<17$, find $\dfrac{a}{b}$.

Given integer $ n$ larger than $ 5$, solve the system of equations (assuming $x_i \geq 0$, for $ i=1,2, \dots n$): \[ \begin{cases} \displaystyle x_1+ \phantom{2^2} x_2+ \phantom{3^2} x_3 + \cdots + \phantom{n^2} x_n &= n+2, \\ x_1 + 2\phantom{^2}x_2 + 3\phantom{^2}x_3 + \cdots + n\phantom{^2}x_n &= 2n+2, \\ x_1 + 2^2x_2 + 3^2 x_3 + \cdots + n^2x_n &= n^2 + n +4, \\ x_1+ 2^3x_2 + 3^3x_3+ \cdots + n^3x_n &= n^3 + n + 8. \end{cases} \]

Let $ c > 2,$ and let $ a(1), a(2), \ldots$ be a sequence of nonnegative real numbers such that \[ a(m \plus{} n) \leq 2 \cdot a(m) \plus{} 2 \cdot a(n) \text{ for all } m,n \geq 1, \] and $ a\left(2^k \right) \leq \frac {1}{(k \plus{} 1)^c} \text{ for all } k \geq 0.$ Prove that the sequence $ a(n)$ is bounded. [i]Author: Vjekoslav Kovač, Croatia[/i]

Find all functions $f : \mathbb{Q} \to \mathbb{R}$ such that $f(x)f(y)f(x+y) = f(xy)(f(x) + f(y))$ for all $x,y\in\mathbb{Q}$. [i]Sammy Luo and Alex Zhu.[/i]

A sequence of integers $ a_{1},a_{2},a_{3},\ldots$ is defined as follows: $ a_{1} \equal{} 1$ and for $ n\geq 1$, $ a_{n \plus{} 1}$ is the smallest integer greater than $ a_{n}$ such that $ a_{i} \plus{} a_{j}\neq 3a_{k}$ for any $ i,j$ and $ k$ in $ \{1,2,3,\ldots ,n \plus{} 1\}$, not necessarily distinct. Determine $ a_{1998}$.

Map the half-plane without a segment perpendicular to its boundary conformally onto the half-plane.

In a circular circuit there are petrol stations, so that the total accumulated petrol in them it is exactly enough for a car to go around the circuit. Prove that there is a position from where a car, with the tank of finite capacity and initially empty, can leave and get to go a full loop around the circuit, stopping to refuel at positions. [hide=original wording]En un circuito circular hay puestos de gasolina, de modo que el total de la gasolina acumulada en ellos es exactamente suciente para que un auto de una vuelta completa al circuito. Demostrar que existe un puesto desde donde un auto, con el estanque de capacidad finita e inicialmente vacio, puede partir y conseguir recorrer una vuelta completa al circuito, deteniendose a reabastecerse de gasolina en los puestos.[/hide]

Find all real numbers $ a$ and $ b$ such that \[ 2(a^2 \plus{} 1)(b^2 \plus{} 1) \equal{} (a \plus{} 1)(b \plus{} 1)(ab \plus{} 1). \] [i]Valentin Vornicu[/i]

Beter Pai wants to tell you his fastest $40$-line clear time in Tetris, but since he does not want Qep to realize she is better at Tetris than he is, he does not tell you the time directly. Instead, he gives you the following requirements, given that the correct time is t seconds: $\bullet$ $t < 100$. $\bullet$ $t$ is prime. $\bullet$ $t -1$ has 5 proper factors. $\bullet$ all prime factors of $t +1$ are single digits. $\bullet$ $t -2$ is a multiple of $3$. $\bullet$ $t +2$ has $2$ factors. Find t.

Determine all injective functions defined on the set of positive integers into itself satisfying the following condition: If $S$ is a finite set of positive integers such that $\sum\limits_{s\in S}\frac{1}{s}$ is an integer, then $\sum\limits_{s\in S}\frac{1}{f\left( s\right) }$ is also an integer.

In an island shaped like a regular polygon of $n$ sides, there are airports at each vertex of the island. The island would like to add $k$ new airports into the interior of the island, but it must follow the following rules:\\ $1$. It must be in the interior of the island (none on borders).\\ $2$. No two airports can be at the exact same location.\\ $3$. Every triple of $1$ new and $2$ old airports must form an isoceles triangle.\\ $4$. No three airports can be collinear.\\ Find the maximum value of $k$ for each $n$ [hide=Harder Version]Replace $1$ new and $2$ old with $1$ old and $2$ new.[/hide]

A store normally sells windows at $ \$100$ each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How many dollars will they save if they purchase the windows together rather than separately? $ \textbf{(A)}\ 100 \qquad \textbf{(B)}\ 200 \qquad \textbf{(C)}\ 300 \qquad \textbf{(D)}\ 400 \qquad \textbf{(E)}\ 500$

Evaluate $ \int_0^{2(2\plus{}\sqrt{3})} \frac{16}{(x^2\plus{}4)^2}\ dx$.

Jingyuan is designing a bucket hat for BMT merchandise. The hat has the shape of a cylinder on top of a truncated cone, as shown in the diagram below. The cylinder has radius $9$ and height $12$. The truncated cone has base radius $15$ and height $4$, and its top radius is the same as the cylinder’s radius. Compute the total volume of this bucket hat. [img]https://cdn.artofproblemsolving.com/attachments/a/9/467d19889d08a6081f9dcd3f4d9df60582f244.png[/img]

Of the following (1) $ a(x\minus{}y)\equal{}ax\minus{}ay$ (2) $ a^{x\minus{}y}\equal{}a^x\minus{}a^y$ (3) $ \log (x\minus{}y)\equal{}\log x\minus{}\log y$ (4) $ \frac {\log x}{\log y}\equal{} \log{x}\minus{} \log{y}$ (5) $ a(xy)\equal{}ax\times ay$ $\textbf{(A)}\ \text{Only 1 and 4 are true} \qquad\\ \textbf{(B)}\ \text{Only 1 and 5 are true} \qquad\\ \textbf{(C)}\ \text{Only 1 and 3 are true} \qquad\\ \textbf{(D)}\ \text{Only 1 and 2 are true} \qquad\\ \textbf{(E)}\ \text{Only 1 is true}$

The functions $f(x)-x$ and $f(x^2)-x^6$ are defined for all positive $x$ and increase. Prove that the function $f(x^3) -\frac{\sqrt3}{2} x^6$ also increases for all positive $x$.

Let $ a,b,c$ be positive real numbers such that $ a^3 \plus{} b^3 \equal{} c^3$.Prove that: $ a^2 \plus{} b^2 \minus{} c^2 > 6(c \minus{} a)(c \minus{} b)$.

Let $m, n \ge 2$ be integers. Carl is given $n$ marked points in the plane and wishes to mark their centroid.* He has no standard compass or straightedge. Instead, he has a device which, given marked points $A$ and $B$, marks the $m-1$ points that divide segment $\overline{AB}$ into $m$ congruent parts (but does not draw the segment). For which pairs $(m,n)$ can Carl necessarily accomplish his task, regardless of which $n$ points he is given? *Here, the [i]centroid[/i] of $n$ points with coordinates $(x_1, y_1), \dots, (x_n, y_n)$ is the point with coordinates $\left( \frac{x_1 + \dots + x_n}{n}, \frac{y_1 + \dots + y_n}{n}\right)$. [i]Proposed by Holden Mui and Carl Schildkraut[/i]

Let $0<k<1$ be a given real number and let $(a_n)_{n\ge1}$ be an infinite sequence of real numbers which satisfies $a_{n+1}\le\left(1+\frac kn\right)a_n-1$. Prove that there is an index $t$ such that $a_t<0$.

Find all $f : \mathbb{N} \to \mathbb{N} $ such that $f(a) + f(b)$ divides $2(a + b - 1)$ for all $a, b \in \mathbb{N}$. Remark: $\mathbb{N} = \{ 1, 2, 3, \ldots \} $ denotes the set of the positive integers.

Determine all triplets of real numbers $ x,y,z$ satisfying \[1\plus{}x^4\leq 2(y\minus{}z)^2,\quad 1\plus{}y^4\leq 2(x\minus{}z)^2,\quad 1\plus{}z^4\leq 2(x\minus{}y)^2.\]

Prove that the simultaneous equations $$x^4 -x^2 =y^4 -y^2 =z^4 -z^2$$ are satisfied by the points of $4$ straight lines and $6$ ellipses, and by no other points.