Find all positive integers $m$ such that there exist positive integers $a_1,a_2,\ldots,a_{1378}$ such that: \[ m=\sum_{k=1}^{1378}{\frac{k}{a_k}}. \]

Fourteen coins are submitted to the judge. An expert knows, that the coins from number one to seven are false, and from $8$ to $14$ -- normal. The judge is sure only that all the true coins have the same weight and all the false coins weights equal each other, but are less then the weight of the true coins. The expert has the scales without weights. a) The expert wants to prove, that the coins $1--7$ are false. How can he do it in three weighings? b) How can he prove, that the coins $1--7$ are false and the coins $8--14$ are true in three weighings?

Evan the ant lives on a right hexagonal pyramid $ABCDEFP$ whose base is regular hexagon $ABCDEF$, and $PA=PB=PC=PD=PE=PF=38\sqrt{3}$. Let $M$ and $N$ be the midpoints of sides $AB$ and $CD$, respectively. Let $X$ be the point on segment $MP$ and $Y$ be the point on segment $NP$ such that $MX=NY=\sqrt{3}$. Given that $PM=37\sqrt{3}$, the length of the shortest path from $X$ to $Y$ that Evan can take by crawling along the surface of $ABCDEFP$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Lightning 4.4[/i]

Let $a, b, c$ be positive real numbers such that $abc=8$. Prove that \[ \frac{a^2}{\sqrt{(1+a^3)(1+b^3)}} +\frac{b^2}{\sqrt{(1+b^3)(1+c^3)}} +\frac{c^2}{\sqrt{(1+c^3)(1+a^3)}} \geq \frac{4}{3} \]

How many rearrangements of the letters of "$HMMTHMMT$" do not contain the substring "$HMMT$"? (For instance, one such arrangement is $HMMHMTMT$.)

[b]Problem Section #2 c). Denote by $\mathbb{Q^+}$ the set of all positive rational numbers. Determine all functions $f:\mathbb{Q^+}\to\mathbb{Q^+}$ which satisfy the following equation for all $x,y \in \mathbb{Q^+} : f(f(x)^2.y)=x^3.f(xy)$.

A jar contains $2$ yellow candies, $4$ red candies, and $6$ blue candies. Candies are randomly drawn out of the jar one-by-one and eaten. The probability that the $2$ yellow candies will be eaten before any of the red candies are eaten is given by the fraction $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

The function $f$ defined by $\displaystyle f(x)= \frac{ax+b}{cx+d}$. where $a,b,c$ and $d$ are nonzero real numbers, has the properties $f(19)=19, f(97)=97$ and $f(f(x))=x$ for all values except $\displaystyle \frac{-d}{c}$. Find the unique number that is not in the range of $f$.

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.