There are $101$ chess players who participated in several tournaments. There was no tournament in which all of them participated. Each pair of these $101$ players met exactly once during these tournaments. Prove that one of them participated in no less than $11$ tournaments. (Assume that each pair of participants in each tournament plays each other once in that tournament). (A Andjans, Riga)

A sequence ($a_n$) satisfies the following conditions: (i) For each $m \in N$ it holds that $a_{2^m} = 1/m$. (ii) For each natural $n \ge 2$ it holds that $a_{2n-1}a_{2n} = a_n$. (iii) For all integers $m,n$ with $2m > n \ge 1$ it holds that $a_{2n}a_{2n+1} = a_{2^m+n}$. Determine $a_{2000}$. You may assume that such a sequence exists.

$ \forall n > 0, n \in \mathbb{Z},$ there exists uniquely determined integers $ a_n, b_n, c_n \in \mathbb{Z}$ such \[ \left(1 \plus{} 4 \cdot \sqrt[3]{2} \minus{} 4 \cdot \sqrt[3]{4} \right)^n \equal{} a_n \plus{} b_n \cdot \sqrt[3]{2} \plus{} c_n \cdot \sqrt[3]{4}.\] Prove that $ c_n \equal{} 0$ implies $ n \equal{} 0.$

Find the smallest positive integer $n$ with the following property: For any integer $m$ with $0 < m < 2004$, there exists an integer $k$ such that \[\frac{m}{2004}<\frac{k}{n}<\frac{m+1}{2005}.\]

The integers from $1$ to $36$ are written on a “mathlotto” ticket. When you buy a “mathlotto” ticket, you choose $6$ of these $36$ numbers. Then $6$ of the integers from $1$ to $36$ are drawn, and a winning ticket is one which does not contain any of them. Prove that (a) if you buy $9$ tickets, you can choose your numbers so that regardless of which numbers are drawn, you are guaranteed to have at least one winning ticket; (b) if you buy only $8$ tickets, it is possible for you not to have any winning tickets, regardless of how you choose your numbers. (S Tokarev)

Show that the perimeter of an arbitrary planar section of a tetrahedron is less than the perimeter of one of the faces of the tetrahedron. [Gy. Hajos]

Several hikers travel at fixed speeds along a straight road. It is known that over some period of time, the sum of their pairwise distances is monotonically decreasing. Show that there is a hiker, the sum of whose distances to the other hikers is monotonically decreasing over the same period. [i]A. Shapovalov[/i]

Estimate the number of $1$s in the hexadecimal representation of $2020!$. If $E$ is your estimate and $A$ is the correct answer, you will receive $\max (25 - 0.5|A - E|, 0)$ points, rounded to the nearest integer.

A bag contains two red beads and two green beads. You reach into the bag and pull out a bead, replacing it with a red bead regardless of the color you pulled out. What is the probability that all beads in the bag are red after three such replacements? $ \textbf{(A)}\ \frac{1}{8} \qquad \textbf{(B)}\ \frac{5}{32} \qquad \textbf{(C)}\ \frac{9}{32} \qquad \textbf{(D)}\ \frac{3}{8} \qquad \textbf{(E)}\ \frac{7}{16}$

Define a sequence $\left( a_{n}\right) _{n\in\mathbb{N}}$ by $a_{1}=a_{2}=a_{3}=1$ and $a_{n+1}=\dfrac{a_{n}^{2}+a_{n-1}^{2}}{a_{n-2}}$ for every integer $n\geq3$. Show that all elements $a_{i}$ of this sequence are integers. (L. J. Mordell and apparently Dana Scott, see also http://oeis.org/A064098)

Let $n$ be a positive integer, and let $f(n)$ denote the last nonzero digit in the decimal expansion of $n!$. $(\text a)$ Show that if $a_1,a_2,\ldots,a_k$ are distinct nonnegative integers, then $f(5^{a_1}+5^{a_2}+\ldots+5^{a_k})$ depends only on the sum $a_1+a_2+\ldots+a_k$. $(\text b)$ Assuming part $(\text a)$, we can define $$g(s)=f(5^{a_1}+5^{a_2}+\ldots+5^{a_k}),$$where $s=a_1+a_2+\ldots+a_k$. Find the least positive integer $p$ for which $$g(s)=g(s+p),\enspace\text{for all }s\ge1,$$or show that no such $p$ exists.

Find all triangles such that its angles form an arithmetic sequence and the corresponding sides form a geometric sequence.

Let $ p$ be a prime number and let $ 0\leq a_{1}< a_{2}<\cdots < a_{m}< p$ and $ 0\leq b_{1}< b_{2}<\cdots < b_{n}< p$ be arbitrary integers. Let $ k$ be the number of distinct residues modulo $ p$ that $ a_{i}\plus{}b_{j}$ give when $ i$ runs from 1 to $ m$, and $ j$ from 1 to $ n$. Prove that a) if $ m\plus{}n > p$ then $ k \equal{} p$; b) if $ m\plus{}n\leq p$ then $ k\geq m\plus{}n\minus{}1$.

Positive real numbers $a_1,a_2,\ldots, a_n$ satisfy the equation $$2a_1+a_2+\ldots+a_{n-1}=a_n+\frac{n^2-3n+2}{2}$$ For every positive integer $n \geq 3$ find the smallest possible value of the sum $$\frac{(a_1+1)^2}{a_2}+\ldots+\frac{(a_{n-1}+1)^2}{a_n}$$ [i]M. Zorka[/i]

In the $8 \times 8$ grid below, label $8$ squares with $X$ and 8 squares with $Y$ such that: 1. No square can be labeled with both an $X$ and a $Y$. 2. Each row and each column must contain exactly one square labeled $X$ and one square labeled $Y$. 3. Any square marked with a $?$ or a $\heartsuit$ cannot be labeled with an $X$ or a $Y$. 4. We say that a square marked with a $?$ or a $\heartsuit$ sees a label ($X$ or $Y$) if one can move in a straight line horizontally or vertically from the marked square to the square with the label, without crossing any other squares with $X$’s or $Y$’s. It is OK to cross other squares marked with a $?$ or $\heartsuit$. Using this definition: (a) Each square marked with a $?$ must see exactly 2 $X$’s and 1 $Y$. (b) Each square marked with a $\heartsuit$ must see exactly 1 $X$ and 2 $Y$’s. \begin{tabular}{ | c | c | c | c | c | c | c | c | } \hline & & $\star$ & $\star$ & $\star$ & $\star$ & $\star$ & \\ \hline & & & & & & & $\star$ \\ \hline & & & & & & & $\star$ \\ \hline & $\heartsuit$ & & & & & & $\star$ \\ \hline & & & & & & & $\star$ \\ \hline & & & & & & & $\heartsuit$ \\ \hline & & & & $\star$ & & & \\ \hline & & & & & & & \\ \hline \end{tabular} 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 conditions of the problem. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.)

It is known that $\frac{7}{13} + \sin \phi = \cos \phi$ for some real $\phi$. What is sin $2\phi$?

Let $ABCD$ be a cyclic quadrilateral with $\angle BAD < \angle ADC$. Let $M$ be the midpoint of the arc $CD$ not containing $A$. Suppose there is a point $P$ inside $ABCD$ such that $\angle ADB = \angle CPD$ and $\angle ADP = \angle PCB$. Prove that lines $AD, PM$, and $BC$ are concurrent.

Prove that every positive rational number $m/n$ can be represented as a sum of reciprocals of distinct positive integers.

Circles with centers $ O$ and $ P$ have radii 2 and 4, respectively, and are externally tangent. Points $ A$ and $ B$ are on the circle centered at $ O$, and points $ C$ and $ D$ are on the circle centered at $ P$, such that $ \overline{AD}$ and $ \overline{BC}$ are common external tangents to the circles. What is the area of hexagon $ AOBCPD$? [asy] unitsize(0.4 cm); defaultpen(linewidth(0.7) + fontsize(11)); pair A, B, C, D; pair[] O; O[1] = (6,0); O[2] = (12,0); A = (32/6,8*sqrt(2)/6); B = (32/6,-8*sqrt(2)/6); C = 2*B; D = 2*A; draw(Circle(O[1],2)); draw(Circle(O[2],4)); draw((0.7*A)--(1.2*D)); draw((0.7*B)--(1.2*C)); draw(O[1]--O[2]); draw(A--O[1]); draw(B--O[1]); draw(C--O[2]); draw(D--O[2]); label("$A$", A, NW); label("$B$", B, SW); label("$C$", C, SW); label("$D$", D, NW); dot("$O$", O[1], SE); dot("$P$", O[2], SE); label("$2$", (A + O[1])/2, E); label("$4$", (D + O[2])/2, E);[/asy] $ \textbf{(A) } 18\sqrt {3} \qquad \textbf{(B) } 24\sqrt {2} \qquad \textbf{(C) } 36 \qquad \textbf{(D) } 24\sqrt {3} \qquad \textbf{(E) } 32\sqrt {2}$

In a certain big city, all the streets go in one of two perpendicular directions. During a drive in the city, a car does not pass through any place twice, and returns to the parking place along a street from which it started. If it has made $100$ left turns, how many right turns must it have made? [i](4 points)[/i]

A computer screen shows a $98 \times 98$ chessboard, colored in the usual way. One can select with a mouse any rectangle with sides on the lines of the chessboard and click the mouse button: as a result, the colors in the selected rectangle switch (black becomes white, white becomes black). Find, with proof, the minimum number of mouse clicks needed to make the chessboard all one color.

Show that the equation $ y^{37}\equiv x^3\plus{}11 \pmod p$ is solvable for every prime $ p$, where $ p\leq100$.

An arbitrary point $P$ is chosen on the lateral side $AB$ of the trapezoid $ABCD$. Straight lines passing through it parallel to the diagonals of the trapezoid intersect the bases at points $Q$ and $R$. Prove that the sides $QR$ of all possible triangles $PQR$ pass through a fixed point.

Show that the solution set of the inequality \[ \sum^{70}_{k \equal{} 1} \frac {k}{x \minus{} k} \geq \frac {5}{4} \] is a union of disjoint intervals, the sum of whose length is 1988.

Does there exist a surjective function $f:\mathbb{R} \rightarrow \mathbb{R}$ for which $f(x+y)-f(x)-f(y)$ takes only 0 and 1 for values for random $x$ and $y$?