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.)

Let f be a polynomial of degree $3$ with integer coefficients such that $f(0) = 3$ and $f(1) = 11$. If f has exactly $2$ integer roots, how many such polynomials $f$ exist?

The sum of the numbers $3a - 4$, $3b - 4$, and $3c - 4$ is $2016$. Find the sum of the numbers $4a - 3$, $4b - 3$, and $4c - 3$.

The number $125$ can be written as a sum of some pairwise coprime integers larger than $1$. Determine the largest number of terms that the sum may have.

Determine all real numbers a for which there exists positive reals $x_{1}, \ldots, x_{5}$ which satisfy the relations $ \sum_{k=1}^{5} kx_{k}=a,$ $ \sum_{k=1}^{5} k^{3}x_{k}=a^{2},$ $ \sum_{k=1}^{5} k^{5}x_{k}=a^{3}.$

The $25$ integers from $-10$ to $14,$ inclusive, can be arranged to form a $5$-by-$5$ square in which the sum of the numbers in each row, the sum of the numbers in each column, and the sum of the numbers along each of the main diagonals are all the same. What is the value of this common sum? $\textbf{(A) }2 \qquad\textbf{(B) } 5\qquad\textbf{(C) } 10\qquad\textbf{(D) } 25\qquad\textbf{(E) } 50$

In Lineland there are $n\geq1$ towns, arranged along a road running from left to right. Each town has a [i]left bulldozer[/i] (put to the left of the town and facing left) and a [i]right bulldozer[/i] (put to the right of the town and facing right). The sizes of the $2n$ bulldozers are distinct. Every time when a left and right bulldozer confront each other, the larger bulldozer pushes the smaller one off the road. On the other hand, bulldozers are quite unprotected at their rears; so, if a bulldozer reaches the rear-end of another one, the first one pushes the second one off the road, regardless of their sizes. Let $A$ and $B$ be two towns, with $B$ to the right of $A$. We say that town $A$ can [i]sweep[/i] town $B$ [i]away[/i] if the right bulldozer of $A$ can move over to $B$ pushing off all bulldozers it meets. Similarly town $B$ can sweep town $A$ away if the left bulldozer of $B$ can move over to $A$ pushing off all bulldozers of all towns on its way. Prove that there is exactly one town that cannot be swept away by any other one.

For all $n \geq 1$, define $a_{n}$ to be the fraction $\frac{k}{2^n}$ such that $a_{n}$ is the closest to $\frac{1}{3}$ over all integer values of $k$. Prove that the sequence $a_{1}, a_{2}, \cdots $satisfies the equation $2a_{i+2} = a_{i+1} + a_{i}$ for all $i \geq 1$.

Prove that if the positive real numbers $a, b, c$ satisfy the equation \[a^4 + b^4 + c^4 + 4a^2b^2c^2 = 2 (a^2b^2 + a^2c^2 + b^2c^2),\] then there is a triangle $ABC$ with internal angles $\alpha, \beta, \gamma$ such that \[\sin \alpha = a, \qquad \sin \beta = b, \qquad \sin \gamma= c.\]

How many positive integers smaller than $1000$ and not divisible by $5$ and by $7$ are there?

Given triangle $ABC$. Points $A_1,B_1$ and $C_1$ are symmetric to its vertices with respect to opposite sides. $C_2$ is the intersection point of lines $AB_1$ and $BA_1$. Points$ A_2$ and $B_2$ are defined similarly. Prove that the lines $A_1 A_2, B_1 B_2$ and $C_1 C_2$ are parallel. (A. Zaslavsky)

Jane and Josh wish to buy a candy. However Jane needs seven more cents to buy the candy, while John needs one more cent. They decide to buy only one candy together, but discover that they do not have enough money. How much does the candy cost?

A line intersects a semicircle with diameter $AB$ and center $O$ at $C$ and $D$, and the line $AB$ at $M$, where $MB < MA$ and $MD < MC.$ If the circumcircles of the triangles $AOC$ and $DOB$ meet again at $K,$ prove that $\angle MKO$ is right.

Let $n$ be a fixed integer with $n \ge 2$. We say that two polynomials $P$ and $Q$ with real coefficients are [i]block-similar[/i] if for each $i \in \{1, 2, \ldots, n\}$ the sequences \begin{eqnarray*} P(2015i), P(2015i - 1), \ldots, P(2015i - 2014) & \text{and}\\ Q(2015i), Q(2015i - 1), \ldots, Q(2015i - 2014) \end{eqnarray*} are permutations of each other. (a) Prove that there exist distinct block-similar polynomials of degree $n + 1$. (b) Prove that there do not exist distinct block-similar polynomials of degree $n$. [i]Proposed by David Arthur, Canada[/i]

(a) Prove that for any two positive integers a and b the equation $lcm (a, a + 5) = lcm (b, b + 5)$ implies $a = b$. (b) Is it possible that $lcm (a, b) = lcm (a + c, b + c)$ for positive integers $a, b$ and $c$? (A Shapovalov) PS. part (a) for Juniors, both part for Seniors

Compute the number of positive integers $n$ less than $100$ for which $1+2+\dots+n$ is not divisible by $n.$

Simplify: $ \sum\limits_{k \equal{} 10}^{2006} \binom{k}{10}$ (Your answer should contain no summations but may still contain binomial coefficients/combinations.)

Construct a triangle given its height and median — both from the same vertex — and the radius of the circumscribed circle.

An angle with vertex $ A$ and measure $ \alpha$ and a point $ P_0$ on one of its rays are given so that $ AP_0\equal{}2$. Point $ P_1$ is chose on the other ray. The sequence of points $ P_1,P_2,P_3,...$ is defined so that $ P_n$ lies on the segment $ AP_{n\minus{}2}$ and the triangle $ P_n P_{n\minus{}1} P_{n\minus{}2}$ is isosceles with $ P_n P_{n\minus{}1}\equal{}P_n P_{n\minus{}2}$ for all $ n \ge 2$. $ (a)$ Prove that for each value of $ \alpha$ there is a unique point $ P_1$ for which the sequence $ P_1,P_2,...,P_n,...$ does not terminate. $ (b)$ Suppose that the sequence $ P_1,P_2,...$ does not terminate and that the length of the polygonal line $ P_0 P_1 P_2 ... P_k$ tends to $ 5$ when $ k \rightarrow \infty$. Compute the length of $ P_0 P_1$.

Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.) [i]Proposed by Hong Kong[/i]

The arithmetic function $\nu$ is defined by $$\nu (n) = \begin{cases}0, \,\,\,\,\, n=1 \\ k, \,\,\,\,\, n=p_1^{a_1} p_2^{a_2} ... p_k^{a_k}\end{cases}$$, where $n=p_1^{a_1} p_2^{a_2} ... p_k^{a_k}$ represents the prime factorization of the number. Prove that for any naturals $m,n$, $$\tau (n^m) = \sum_{d | n} m^{\nu (d)}.$$ [i](PP-nine)[/i]

Let I be the incenter of triangle $ABC$ and let $k$ be a circle through the points $A$ and $B$. The circle intersects * the line $AI$ in points $A$ and $P$ * the line $BI$ in points $B$ and $Q$ * the line $AC$ in points $A$ and $R$ * the line $BC$ in points $B$ and $S$ with none of the points $A,B,P,Q,R$ and $S$ coinciding and such that $R$ and $S$ are interior points of the line segments $AC$ and $BC$, respectively. Prove that the lines $PS$, $QR$, and $CI$ meet in a single point. (Stephan Wagner)

A point $P$ is given in the square $ABCD$ such that $\overline{PA}=3$, $\overline{PB}=7$ and $\overline{PD}=5$. Find the area of the square.

All positive divisors of a positive integer $N$ are written on a blackboard. Two players $A$ and $B$ play the following game taking alternate moves. In the firt move, the player $A$ erases $N$. If the last erased number is $d$, then the next player erases either a divisor of $d$ or a multiple of $d$. The player who cannot make a move loses. Determine all numbers $N$ for which $A$ can win independently of the moves of $B$. [i](4th Middle European Mathematical Olympiad, Individual Competition, Problem 2)[/i]

To celebrate $2019$, Faraz gets four sandwiches shaped in the digits $2$, $0$, $1$, and $9$ at lunch. However, the four digits get reordered (but not ipped or rotated) on his plate and he notices that they form a $4$-digit multiple of $7$. What is the greatest possible number that could have been formed?