Prove that for any positive integer $n$ there exists a matrix of the form $$A=\begin{pmatrix}1&a&b&c\\0&1&a&b\\0&0&1&a\\0&0&0&1\end{pmatrix},$$ (a) with nonzero entries, (b) with positive entries, such that the entries of $A^n$ are all perfect squares.

The fifth and eighth terms of a geometric sequence of real numbers are $ 7!$ and $ 8!$ respectively. What is the first term? $ \textbf{(A)}\ 60\qquad \textbf{(B)}\ 75\qquad \textbf{(C)}\ 120\qquad \textbf{(D)}\ 225\qquad \textbf{(E)}\ 315$

There are a total of $n$ boys and girls sitting in a big circle. Now, Dave wants to walk around the circle. For a start point, if at any time, one of the following two conditions holds: 1. he doesn't see any girl 2. the number of boys he saw $\ge$ the number of girls he saw $+k$ Then we say this point is [i]good[/i]. What is the maximum of $r$ with the property that there is at least one good point whenever the number of girls is $r$?

Consider a checkered $3m\times 3m$ square, where $m$ is an integer greater than $1.$ A frog sits on the lower left corner cell $S$ and wants to get to the upper right corner cell $F.$ The frog can hop from any cell to either the next cell to the right or the next cell upwards. Some cells can be [i]sticky[/i], and the frog gets trapped once it hops on such a cell. A set $X$ of cells is called [i]blocking[/i] if the frog cannot reach $F$ from $S$ when all the cells of $X$ are sticky. A blocking set is [i] minimal[/i] if it does not contain a smaller blocking set.[list=a][*]Prove that there exists a minimal blocking set containing at least $3m^2-3m$ cells. [*]Prove that every minimal blocking set containing at most $3m^2$ cells.

There are $5$ accents in French, each applicable to only specific letters as follows: [list] [*] The cédille: ç [*] The accent aigu: é [*] The accent circonflexe: â, ê, î, ô, û [*] The accent grave: à, è, ù [*] The accent tréma: ë, ö, ü [/list] Cédric needs to write down a phrase in French. He knows that there are $3$ words in the phrase and that the letters appear in the order: \[cesontoiseaux.\] He does not remember what the words are and which letters have what accents in the phrase. If $n$ is the number of possible phrases that he could write down, then determine the number of distinct primes in the prime factorization of $n$.

How many natural numbers $n$ are there so that $n^2 + 2014$ is a perfect square? (A): $1$, (B): $2$, (C): $3$, (D): $4$, (E) None of the above.

Given a triangle $ABC$ with $M$ the midpoint of minor arc $BC$. Let $H$ be the feet of altitude from $A$ to $BC$. Let $S$ and $T$ be the reflections of $B$ and $C$ with respect to line $AM$. Suppose the circle $(HST)$ meets $BC$ again at a point $P$. Prove that $\angle AMP = 90^\circ$. [i](Proposed by Tan Rui Xuen)[/i]

Given are positive reals $x_1, x_2,..., x_n$ such that $\sum\frac {1}{1+x_i^2}=1$. Find the minimal value of the expression $\frac{\sum x_i}{\sum \frac{1}{x_i}}$ and find when it is achieved.

Find all real numbers $a,b,c,d,e$ in the interval $[-2,2]$, that satisfy: \begin{align*}a+b+c+d+e &= 0\\ a^3+b^3+c^3+d^3+e^3&= 0\\ a^5+b^5+c^5+d^5+e^5&=10 \end{align*}

You are somewhere on a ladder with $5$ rungs. You have a fair coin and an envelope that contains either a double-headed coin or a double-tailed coin, each with probability $1/2$. Every minute you flip a coin. If it lands heads you go up a rung, if it lands tails you go down a rung. If you move up from the top rung you win, if you move down from the bottom rung you lose. You can open the envelope at any time, but if you do then you must immediately flip that coin once, after which you can use it or the fair coin whenever you want. What is the best strategy (i.e. on what rung(s) should you open the envelope)?

What is the last three digits of the sum 11! + 12! + 13! +


+ 2006!

Is it possible to make a $100 \times 100$ table of numbers such that the sum of numbers in each column is positive while the sum of numbers in each row is negative?

Find the number of representations of the number $180$ in the form $180 =x+y+z$, where $x, y, z$ are positive integers that are proportional with some three consecutive positive integers

Let $ABC$ be a triangle, and let $D$, $A$, $B$, $E$ be points on line $AB$, in that order, such that $AC=AD$ and $BE=BC$. Let $\omega_1, \omega_2$ be the circumcircles of $\triangle ABC$ and $\triangle CDE$, respectively, which meet at a point $F \neq C$. If the tangent to $\omega_2$ at $F$ cuts $\omega_1$ again at $G$, and the foot of the altitude from $G$ to $FC$ is $H$, prove that $\angle AGH=\angle BGH$. [i]Proposed by David Stoner[/i]

Let $a,b,c>0$ the sides of a right triangle. Find all real $x$ for which $a^x>b^x+c^x$, with $a$ is the longest side.

Let $\omega_1$ and $\omega_2$ be two orthogonal circles, and let the center of $\omega_1$ be $O$. Diameter $AB$ of $\omega_1$ is selected so that $B$ lies strictly inside $\omega_2$. The two circles tangent to $\omega_2$, passing through $O$ and $A$, touch $\omega_2$ at $F$ and $G$. Prove that $FGOB$ is cyclic. [i]Proposed by Eric Chen[/i]

Let $d$ be a real number such that $d^2=r^2+s^2$, where $r$ and $s$ are rational numbers. Prove that we can color all points of the plane with rational coordinates with two different colors such that the points with distance $d$ have different colors.

Let $ ABCD$ be a convex quadrilateral such that $ AB$ and $ CD$ are not parallel and $ AB\equal{}CD$. The midpoints of the diagonals $ AC$ and $ BD$ are $ E$ and $ F$, respectively. The line $ EF$ meets segments $ AB$ and $ CD$ at $ G$ and $ H$, respectively. Show that $ \angle AGH \equal{} \angle DHG$.

All naturals from $1$ to $2002$ are placed in a row. Can the signs: $+$ and $-$ be placed between each consecutive pair of them so that the corresponding algebraic sum is $0$?

Prove that in every $16$-digit number there is a chain of one or more consecutive digits such that the product of those digits is a perfect square. For example, if the original number is $7862328578632785$ we can take the digits $6$, $2$ and $3$ whose product is $6^2$ (note that these appear consecutively in the number).

Let $ p,q $ be the two most distant points (in the Euclidean sense) of a closed surface $ M $ embedded in the Euclidean space. [b]a)[/b] Show that the tangent planes of $ M $ at $ p $ and $ q $ are parallel. [b]b)[/b] What happened if $ M $ would be a closed curve of $ \mathcal{C}^{\infty } \left(\mathbb{R}^3\right) $ class, instead?

Let $d(n)$ denote the number of positive integers that divide $n$, including $1$ and $n$. For example, $d(1)=1,d(2)=2,$ and $d(12)=6$. (This function is known as the [i]divisor function[/i].) Let \[f(n)=\frac{d(n)}{\sqrt[3]{n}}.\] There is a unique positive integer $N$ such that $f(N)>f(n)$ for all positive integers $n\ne N$. What is the sum of the digits of $N?$ $\textbf{(A) }5 \qquad \textbf{(B) }6 \qquad \textbf{(C) }7 \qquad \textbf{(D) }8\qquad \textbf{(E) }9$

Given finitely many points in a plane, it is known that the area of the triangle formed by any three points of the set is less than 1. Show that all points of the set lie inside or on boundary of a triangle with area less than 4.

A $5 \times 5$ Latin Square is a $5 \times 5$ grid of squares in which each square contains one of the numbers $1$ through $5$ such that every number appears exactly once in each row and column. A partially completed grid (with numbers in some of the squares) is puzzle-ready if there is a unique way to fill in the remaining squares to complete a Latin Square. Below is a partially completed grid with seven squares filled in and an additional three squares shaded. Determine what numbers must be filled into the shaded squares to make the grid (now with ten squares filled in) puzzle-ready, and then complete the Latin Square. 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.) [asy] unitsize(1.5cm); defaultpen(font("OT1","cmss","m","n")); defaultpen(fontsize(48pt)); for (int i=0; i<6; ++i) { draw((i,0)--(i,5)); draw((0,i)--(5,i)); } label(scale(2)*"1",(0.5,4.5)); label(scale(2)*"1",(1.5,3.5)); label(scale(2)*"3",(2.5,3.5)); label(scale(2)*"2",(0.5,2.5)); label(scale(2)*"3",(1.5,2.5)); label(scale(2)*"5",(4.5,2.5)); label(scale(2)*"5",(3.5,1.5)); path p = (0,0)--(1,0)--(1,1)--(0,1)--cycle; filldraw(shift(0,1)*p,gray,black); filldraw(shift(4,1)*p,gray,black); filldraw(shift(2,2)*p,gray,black); [/asy]

Let $ n$ and $ k$ be given natural numbers, and let $ A$ be a set such that \[ |A| \leq \frac{n(n+1)}{k+1}.\] For $ i=1,2,...,n+1$, let $ A_i$ be sets of size $ n$ such that \[ |A_i \cap A_j| \leq k \;(i \not=j)\ ,\] \[ A= \bigcup_{i=1}^{n+1} A_i.\] Determine the cardinality of $ A$. [i]K. Corradi[/i]