Let $A$ be a $5\times10$ matrix with real entries, and let $A^{\text T}$ denote its transpose. Suppose every $5\times1$ matrix with real entries can be written in the form $A\mathbf u$ where $\mathbf u$ is a $10\times1$ matrix with real entries. Prove that every $5\times1$ matrix with real entries can be written in the form $AA^{\text T}\mathbf v$ where $\mathbf v$ is a $5\times1$ matrix with real entries.

Find the number of polynomials $P(x)$ of degree $6$ whose coefficients are in the set $\{1,2,\ldots,1999\}$ and which are divisible by $x^3+x^2+x+1$.

Pete claims that he can draw $3$ segments of length $1$ and a circle of radius less than $\sqrt3 / 3$ on a piece of paper, such that all segments would lie inside the circle and there would be no line that intersects each of $3 $ segments. Is Pete right?

The increasing geometric sequence $x_{0},x_{1},x_{2},\ldots$ consists entirely of integral powers of $3.$ Given that \[\sum_{n=0}^{7}\log_{3}(x_{n}) = 308\qquad\text{and}\qquad 56 \leq \log_{3}\left ( \sum_{n=0}^{7}x_{n}\right ) \leq 57,\] find $\log_{3}(x_{14}).$

The streets of a town are arranged in three directions , dividing the town into blocks which are equilateral triangles of equal area. Traffic , when reaching an intersection, may only go straight ahead, or turn left or right through $120^0$ , as shown in the diagram. [img]https://cdn.artofproblemsolving.com/attachments/3/6/a100a5c39bf15116582bc0bceb76fcbae28af9.png[/img] No turns are permitted except the ones at intersections . One car departs for a certain nearby intersection and when it reaches it a second car starts moving toward it. From then on both cars continue travelling at the same speed (but do not necessarily take the same turns). Is it possible that there will be a time when they will encounter each other somewhere? ( N . N . Konstantinov , Moscow )

Prove that for any positive integer $n$ there is an equiangular hexagon whose sidelengths are $n + 1, n + 2 ,..., n + 6$ in some order.

A square is inscribed into an acute-angled triangle: two vertices of this square lie on the same side of the triangle and two remaining vertices lies on two remaining sides. Two similar squares are constructed for the remaining sides. Prove that three segments congruent to the sides of these squares can be the sides of an acute-angled triangle.

Let $ A_0\equal{}(0,0)$. Distinct points $ A_1,A_2,\ldots$ lie on the $ x$-axis, and distinct points $ B_1,B_2,\ldots$ lie on the graph of $ y\equal{}\sqrt{x}$. For every positive integer $ n$, $ A_{n\minus{}1}B_nA_n$ is an equilateral triangle. What is the least $ n$ for which the length $ A_0A_n\ge100$? $ \textbf{(A)}\ 13\qquad \textbf{(B)}\ 15\qquad \textbf{(C)}\ 17\qquad \textbf{(D)}\ 19\qquad \textbf{(E)}\ 21$

There are $n$ points in the page such that no three of them are collinear.Prove that number of triangles that vertices of them are chosen from these $n$ points and area of them is 1,is not greater than $\frac23(n^2-n)$.

A cylindrical hole of radius $r$ is bored through a cylinder of radiues $R$ ($r\leq R$) so that the axes intersect at right angles. i) Show that the area of the larger cylinder which is inside the smaller one can be expressed in the form $$S=8r^2\int_{0}^{1} \frac{1-v^{2}}{\sqrt{(1-v^2)(1-m^2 v^2)}}\;dv,\;\; \text{where} \;\; m=\frac{r}{R}.$$ ii) If $K=\int_{0}^{1} \frac{1}{\sqrt{(1-v^2)(1-m^2 v^2)}}\;dv$ and $E=\int_{0}^{1} \sqrt{\frac{1-m^2 v^2}{1-v^2 }}dv$. show that $$S=8[R^2 E - (R^2 - r^2 )K].$$

For each positive integer $n$, let $S(n)$ be the sum of the digits of $n$. Determines the smallest positive integer $a$ such that there are infinite positive integers $n$ for which you have $S (n) -S (n + a) = 2018$.

For any positive integer $n$, define $$c_n=\min_{(z_1,z_2,...,z_n)\in\{-1,1\}^n} |z_1\cdot 1^{2018} + z_2\cdot 2^{2018} + ... + z_n\cdot n^{2018}|.$$ Is the sequence $(c_n)_{n\in\mathbb{Z}^+}$ bounded?

Regular hexagon $ABCDEF$ has side length $1$. Given that $P$ is a point inside $ABCDEF$, compute the minimum of $AP \sqrt3 + CP + DP + EP\sqrt3$.

A cube is constructed from $4$ white unit cubes and $4$ black unit cubes. How many different ways are there to construct the $2 \times 2 \times 2$ cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.) $\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 11$

Let $n$ be a positive integer such that $24\mid n+1$. Prove that the sum of the positive divisors of $n$ is divisble by 24.

Let $n $ be a positive integer. Let $f $ be a function from nonnegative integers to themselves. Let $f (0,i)=f (i,0)=0$, $f (1, 1)=n $, and $ f(i, j)= [\frac {f(i-1,j)}{2}]+ [\frac {f(i, j-1)}{2}] $ for positive integers $i, j$ such that $i*j>1$. Find the number of pairs $(i,j) $ such that $f (i, j) $ is an odd number.( $[x]$ is the floor function).

City of Mar del Plata is a square shaped $WSEN$ land with $2(n + 1)$ streets that divides it into $n \times n$ blocks, where $n$ is an even number (the leading streets form the perimeter of the square). Each block has a dimension of $100 \times 100$ meters. All streets in Mar del Plata are one-way. The streets which are parallel and adjacent to each other are directed in opposite direction. Street $WS$ is driven in the direction from $W$ to $S$ and the street $WN$ travels from $W$ to $N$. A street cleaning car starts from point $W$. The driver wants to go to the point $E$ and in doing so, he must cross as much as possible roads. What is the length of the longest route he can go, if any $100$-meter stretch cannot be crossed more than once? (The figure shows a plan of the city for $n=6$ and one of the possible - but not the longest - routes of the street cleaning car. See http://goo.gl/maps/JAzD too.) [img]http://s14.postimg.org/avfg7ygb5/CPS_2012_P5.jpg[/img]

For positive integer $n$, prove that at least one of the numbers $$A=2n-1 , B=5n-1, C=13n-1$$ is not perfect square

Let $n$ be a positive integer. Let $a_1$, $a_2$, $\ldots\,$, $a_n$ be real numbers such that $-1 \le a_i \le 1$ (for all $1 \le i \le n$). Let $b_1$, $b_2$, $\ldots$, $b_n$ be real numbers such that $-1 \le b_i \le 1$ (for all $1 \le i \le n$). Prove that \[ \left| \prod_{i=1}^n a_i - \prod_{i=1}^n b_i \right| \le \sum_{i = 1}^n \left| a_i - b_i \right| \, . \]

A circular track with diameter $500$ is externally tangent at a point A to a second circular track with diameter $1700.$ Two runners start at point A at the same time and run at the same speed. The first runner runs clockwise along the smaller track while the second runner runs clockwise along the larger track. There is a first time after they begin running when their two positions are collinear with the point A. At that time each runner will have run a distance of $\frac{m\pi}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n. $

line $l$ through the point $A$ from triangle $ABC$ . Point $X$ is on line $l$.${\omega}_b$ and ${\omega}_c$ are circles that through points $X,A$ and respectively tanget to $AB$ adn $AC$. tangets from $B,C$ respectively to ${\omega}_b$ and ${\omega}_c$ meet them in $Y,Z$. Prove that by changing $X$, the circumcircle of the circle $XYZ$ passes through two fixed points. [i]Proposed by Ali Zamani [/i]

A tetrahedron $ABCD$ satisfies $AB=6$, $CD=8$, and $BC=DA=5$. Let $V$ be the maximum value of $ABCD$ possible. If we can write $V^4=2^n3^m$ for some integers $m$ and $n$, find $mn$.

Various points $x_1,..., x_n$ ($n \ge 3$) are randomly located on the $Ox$ axis. Construct all parabolas defined by the monic square trinomials and intersecting the Ox axis at these points (and not intersecting axis at other points). Let$ y = f_1$, $...$ , $y = f_m$ are functions that define these parabolas. Prove that the parabola $y = f_1 +...+ f_m$ intersects the $Ox$ axis at two points.

15 of the cells of a chessboard 8x8 are chosen. We draw the segments which unite the centers of every two of the chosen squares. Prove that among these segments there are four segments which have the same length.

Let $n$ be a natural number. Prove that, \[ \left\lfloor \frac{n}{1} \right\rfloor+ \left\lfloor \frac{n}{2} \right\rfloor + \cdots + \left\lfloor \frac{n}{n} \right\rfloor + \left\lfloor \sqrt{n} \right\rfloor \] is even.