Several farmers have 128 sheep. If one of them has at least half of all sheep, the rest conspire and dispossess him: everyone takes as many sheep as he already has : If two people have 64 sheep, then one of them is dispossessed. There were 7 dispossessions. Prove that all the sheep were gathered from one peasant.

Amy is traveling on the $xy$-plane in a spaceship where her motion is described by the following equation $xe^y = ye^x$. Given that her $x$-component of velocity is a constant $3$ mph , the magnitude of her velocity as she approaches $(1,-1)$ can be expressed as $\sqrt{\frac{a + be^4}{ c + de^2}}$ . Find $\frac{ac}{bd}$ . (You may assume that the initial conditions do allow her to approach $(1,-1)$)

Let $n$ be the smallest positive integer such that there exist integers, $a$, $b$, and $c$, satisfying: $$\frac{n}{2}= a^2 \,\,\, \,\,\, \frac{n}{3}= b^3 \,\,\ , \,\,\ \frac{n}{5}= c^5.$$ Find the number of positive integer factors of $n$.

Let $ABC$ be an isosceles triangle, $AB = AC, \angle A = 20^{\circ}$. Let $D$ be a point on $AB$, and $E$ a point on $AC$ such that $\angle ACD = 20^{\circ}$ and $\angle ABE = 30^{\circ}$. What is the measure of the angle $\angle CDE$?

If the parabola $ y \equal{} ax^2 \plus{} bx \plus{} c$ passes through the points $ ( \minus{} 1, 12), (0, 5),$ and $ (2, \minus{} 3),$ the value of $ a \plus{} b \plus{} c$ is: $ \textbf{(A)}\ \minus{} 4 \qquad \textbf{(B)}\ \minus{} 2 \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ 2$

A single elimination tournament is held with $2016$ participants. In each round, players pair up to play games with each other. There are no ties, and if there are an odd number of players remaining before a round then one person will get a bye for the round. Find the minimum number of rounds needed to determine a winner. [i]Proposed by Nathan Ramesh

Points $ A$ and $ B$ are on a circle of radius $ 5$ and $ AB\equal{}6$. Point $ C$ is the midpoint of the minor arc $ AB$. What is the length of the line segment $ AC$? $ \textbf{(A)}\ \sqrt{10} \qquad \textbf{(B)}\ \frac{7}{2} \qquad \textbf{(C)}\ \sqrt{14} \qquad \textbf{(D)}\ \sqrt{15} \qquad \textbf{(E)}\ 4$

Let $x, y, z$ be non-negative numbers, so that $x + y + z = 1$. Prove that $$\sqrt{x+\frac{(y-z)^2}{12}}+\sqrt{y+\frac{(x-z)^2}{12}}+\sqrt{z+\frac{(x-y)^2}{12}}\le \sqrt{3}$$

Find the limit as below. \[\lim_{n\to\infty} \frac{(1^2+2^2+\cdots +n^2)(1^3+2^3+\cdots +n^3)(1^4+2^4+\cdots +n^4)}{(1^5+2^5+\cdots +n^5)^2}\]

$n$ straight lines are drawn in a plane. They divide the plane onto several parts. Some of the parts are painted. Not a pair of painted parts has non-zero length common bound. Prove that the number of painted parts is not more than $\frac{n^2 + n}{3}$.

$2021$ people are sitting around a circular table. In one move, you may swap the positions of two people sitting next to each other. Determine the minimum number of moves necessary to make each person end up $1000$ positions to the left of their original position.

How many positive integers $k$ are there such that \[\dfrac k{2013}(a+b)=lcm(a,b)\] has a solution in positive integers $(a,b)$?

Let $a, b$ and $c$ be positive real numbers. Prove that \[\frac{\sqrt{a^2+bc}}{b+c}+\frac{\sqrt{b^2+ca}}{c+a}+\frac{\sqrt{c^2+ab}}{a+b}\ge\sqrt{\frac{a}{b+c}}+\sqrt{\frac{b}{c+a}}+\sqrt{\frac{c}{a+b}}\]

Let $n$ be a positive integer, and let $S_1,S_2,…,S_n$ be a collection of finite non-empty sets such that $$\sum_{1\leq i<j\leq n}{\frac{|S_i \cap S_j|}{|S_i||S_j|}} <1.$$ Prove that there exist pairwise distinct elements $x_1,x_2,…,x_n$ such that $x_i$ is a member of $S_i$ for each index $i$.

Given infinite sequences $a_1,a_2,a_3,\cdots$ and $b_1,b_2,b_3,\cdots$ of real numbers satisfying $\displaystyle a_{n+1}+b_{n+1}=\frac{a_n+b_n}{2}$ and $\displaystyle a_{n+1}b_{n+1}=\sqrt{a_nb_n}$ for all $n\geq1$. Suppose $b_{2016}=1$ and $a_1>0$. Find all possible values of $a_1$

In the parallelogram $ABCD$, the diagonal $AC$ touches the incircles of triangles $ABC$ and $ADC$ at $W$ and $Y$ respectively, and the diagonal $BD$ touches the incircles of triangles $BAD$ and $BCD$ at $X$ and $Z$ respectively. Prove that either $W,X, Y$ and $Z$ coincide, or $WXYZ$ is a rectangle.

In base $10$ there exist two-digit natural numbers that can be factorized into two natural factors such that the two digits and the two factors form a sequence of four consecutive integers (for example, $12 = 3 \cdot 4$). Determine all such numbers in all bases.

An arbitrary triangle ABC is given. There are 2 lines, p and q, that are not parallel to each other and they are not perpendicular to the sides of the triangle. The perpendicular lines through points A, B and C to line p we denote with $ p_a, p_b, p_c $ and the perpendicular lines to line q we denote with $ q_a, q_b, q_c $. Let the intersection points of the lines $ p_a, q_a, p_b, q_b, p_c $ and $ q_c $ with $ q_b, p_b, q_c, p_c, q_a $ and $ p_a $ are $ K, L, P, Q, N $ and $ M $. Prove that $ KL, MN $ and $ PQ $ intersect in one point.

Hernan wants to paint a $8\times 8$ board such that every square is painted with blue or red. Also wants to every $3\times 3$ subsquare have exactly $a$ blue squares and every $2\times 4$ or $4\times 2$ rectangle have exactly $b$ blue squares. Find all couples $(a,b)$ such that Hernan can do the required.

[list=a] [*] Let $a<b$ and $f:[a,b]\rightarrow\mathbb{R}$ be a strictly monotonous function such that $\int_a^b f(x) dx=0$. Show that $f(a)\cdot f(b)<0$. [*] Find all convergent sequences $(a_n)_{n\geq 1}$ for which there exists a scrictly monotonous function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$\int_{a_{n-1}}^{a_n} f(x)dx = \int_{a_n}^{a_{n+1}} f(x)dx,\text{ for all }n\geq 2.$$

A game consists of pieces of the shape of a regular tetrahedron of side $1$. Each face of each piece is painted in one of $n$ colors, and by this, the faces of one piece are not necessarily painted in different colors. Determine the maximum possible number of pieces, no two of which are identical.

All edges of a polyhedron are painted with red or yellow. For an angle of a facet, if the edges determining it are of different colors, then the angle is called [i]excentric[/i]. The[i] excentricity [/i]of a vertex $A$, namely $S_A$, is defined as the number of excentric angles it has. Prove that there exist two vertices $B$ and $C$ such that $S_B + S_C \leq 4$.

Prove that the inequality \[ a + a^3 - a^4 - a^6 < 1\] holds for all real numbers $a$.

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?