Find the least positive integer $n$ ($n\geq 3$), such that among any $n$ points (no three are collinear) in the plane, there exist three points which are the vertices of a non-isoscele triangle.

The natural numbers $x_1, x_2, \ldots , x_n$ are such that all their $2^n$ partial sums are distinct. Prove that: $$ {x_1}^2 + {x_2}^2 + \ldots + {x_n}^2 \geq \frac{4^n – 1}{3}. $$

Consider the function $ f(x) \equal{} \frac {2x^3 \minus{} 3}{3x^2 \minus{} 1}$. $ 1.$ Prove that there is a continuous function $ g(x)$ on $ \mathbb{R}$ satisfying $ f(g(x)) \equal{} x$ and $ g(x) > x$ for all real $ x$. $ 2.$ Show that there exists a real number $ a > 1$ such that the sequence $ \{a_n\}$, $ n \equal{} 1, 2, \ldots$, defined as follows $ a_0 \equal{} a$, $ a_{n \plus{} 1} \equal{} f(a_n)$, $ \forall n\in\mathbb{N}$ is periodic with the smallest period $ 1995$.

Let $ABCD$ be an isosceles trapezoid with $AB=5$, $CD = 8$, and $BC = DA = 6$. There exists an angle $\theta$ such that there is only one point $X$ satisfying $\angle AXD = 180^{\circ} - \angle BXC = \theta$. Find $\sin(\theta)^2$.

Find all integer solutions of \[ x^{4}\plus{}y^{2}\equal{}z^{4}\]

In the coordinate plane a rectangle with vertices $ (0, 0),$ $ (m, 0),$ $ (0, n),$ $ (m, n)$ is given where both $ m$ and $ n$ are odd integers. The rectangle is partitioned into triangles in such a way that [i](i)[/i] each triangle in the partition has at least one side (to be called a “good” side) that lies on a line of the form $ x \equal{} j$ or $ y \equal{} k,$ where $ j$ and $ k$ are integers, and the altitude on this side has length 1; [i](ii)[/i] each “bad” side (i.e., a side of any triangle in the partition that is not a “good” one) is a common side of two triangles in the partition. Prove that there exist at least two triangles in the partition each of which has two good sides.

Let $AB,AC$ and $AD$ be the edges of a cube, $AB=\alpha$. Point $E$ was marked on the ray $AC$ so that $AE=\lambda \alpha$, and point $F$ was marked on the ray $AD$ so that $AF=\mu \alpha$ ($\mu> 0, \lambda >0$). Find (characterize) pairs of numbers $\lambda$ and $\mu$ such that the cross-sectional area of ​​a cube by any plane parallel to the plane $BCD$ is equal to the cross-sectional area of ​​the tetrahedron $ABEF$ by the same plane.

A scanning code consists of a $7 \times 7$ grid of squares, with some of its squares colored black and the rest colored white. There must be at least one square of each color in this grid of $49$ squares. A scanning code is called [i]symmetric[/i] if its look does not change when the entire square is rotated by a multiple of $90 ^{\circ}$ counterclockwise around its center, nor when it is reflected across a line joining opposite corners or a line joining midpoints of opposite sides. What is the total number of possible symmetric scanning codes? $\textbf{(A)} \text{ 510} \qquad \textbf{(B)} \text{ 1022} \qquad \textbf{(C)} \text{ 8190} \qquad \textbf{(D)} \text{ 8192} \qquad \textbf{(E)} \text{ 65,534}$

At the beginning of the school year, Lisa’s goal was to earn an A on at least $ 80\%$ of her $ 50$ quizzes for the year. She earned an A on $ 22$ of the first $ 30$ quizzes. If she is to achieve her goal, on at most how many of the remaining quizzes can she earn a grade lower than an A? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$

Let a.b.c be positive reals such that their sum is 1. Prove that $ \frac{a^{2}b^{2}}{c^{3}(a^{2}\minus{}ab\plus{}b^{2})}\plus{}\frac{b^{2}c^{2}}{a^{3}(b^{2}\minus{}bc\plus{}c^{2})}\plus{}\frac{a^{2}c^{2}}{b^{3}(a^{2}\minus{}ac\plus{}c^{2})}\geq \frac{3}{ab\plus{}bc\plus{}ac}$

In an acute-angled triangle $ABC$ is $AB > BC$ , and the points $A_1$ and $C_1$ are the feet of the altitudes of from the vertices $A$ and $C$. Let $D$ be the second intersection of the circumcircles of triangles $ABC$ and $A_1BC_1$ (different of $B$). Let $Z$ be the intersection of the tangents to the circumcircle of the triangle ABC at the points $A$ and $C$ , and let the lines $ZA$ and $A_1C_1$ intersect at the point $X$, and the lines $ZC$ and $A_1C_1$ intersect at the point $Y$. Prove that the point $D$ lies on the circumcircle of the triangle $XYZ$.

If $ \sqrt{x^2+2y+1} +\sqrt[3]{y^3+3x^2+3x+1} $ is rational, then $ x=y. $

Let $M$ be the set of real numbers of the form $\frac{m+n}{\sqrt{m^2+n^2}}$, where $m$ and $n$ are positive integers. Prove that for every pair $x \in M, y \in M$ with $x < y$, there exists an element $z \in M$ such that $x < z < y.$

Determine the smallest possible value of the expression $$\frac{ab+1}{a+b}+\frac{bc+1}{b+c}+\frac{ca+1}{c+a}$$ where $a,b,c \in \mathbb{R}$ satisfy $a+b+c = -1$ and $abc \leqslant -3$

Let $n$ be a positive integer. Show that there exist positive integers $a$ and $b$ such that \[ \frac{a^2 + a + 1}{b^2 + b + 1} = n^2 + n + 1. \]

[b]p1.[/b] Find all positive integers $n$ such that $n^3 - 14n^2 + 64n - 93$ is prime. [b]p2.[/b] Let $a, b, c$ be real numbers such that $0 \le a, b, c \le 1$. Find the maximum value of $$\frac{a}{1 + bc}+\frac{b}{1 + ac}+\frac{c}{1 + ab}$$ [b]p3.[/b] Find the maximum value of the function $f(x, y, z) = 4x + 3y + 2z$ on the ellipsoid $16x^2 + 9y^2 + 4z^2 = 1$ [b]p4.[/b] Let $x_1,..., x_n$ be numbers such that $x_1+...+x_n = 2009$. Find the minimum value of $x^2_1+...+x^2_n$ (in term of $n$). [b]p5.[/b] Find the number of odd integers between $1000$ and $9999$ that have at least 3 distinct digits. [b]p6.[/b] Let $A_1,A_2,...,A_{2^n-1}$ be all the possible nonempty subsets of $\{1, 2, 3,..., n\}$. Find the maximum value of $a_1 + a_2 + ... + a_{2^n-1}$ where $a_i \in A_i$ for each $i = 1, 2,..., 2^n - 1$. [b]p7.[/b] Find the rightmost digit when $41^{2009}$ is written in base $7$. [b]p8.[/b] How many integral ordered triples $(x, y, z)$ satisfy the equation $x+y+z = 2009$, where $10 \le x < 31$, $100 < z < 310$ and $y \ge 0$. [b]p9.[/b] Scooby has a fair six-sided die, labeled $1$ to $6$, and Shaggy has a fair twenty-sided die, labeled $1$ to $20$. During each turn, they both roll their own dice at the same time. They keep rolling the die until one of them rolls a 5. Find the probability that Scooby rolls a $5$ before Shaggy does. [b]p10.[/b] Let $N = 1A323492110877$ where $A$ is a digit in the decimal expansion of $N$. Suppose $N$ is divisible by $7$. Find $A$. [b]p11.[/b] Find all solutions $(x, y)$ of the equation $\tan^4(x+y)+\cot^4(x+y) = 1-2x-x^2$, where $-\frac{\pi}{2} \le x; y \le \frac{\pi}{2}$ [b]p12.[/b] Find the remainder when $\sum^{50}_{k=1}k!(k^2 + k - 1)$ is divided by $1008$. [b]p13.[/b] The devil set of a positive integer $n$, denoted $D(n)$, is defined as follows: (1) For every positive integer $n$, $n \in D(n)$. (2) If $n$ is divisible by $m$ and $m < n$, then for every element $a \in D(m)$, $a^3$ must be in $D(n)$. Furthermore, call a set $S$ scary if for any $a, b \in S$, $a < b$ implies that $b$ is divisible by $a$. What is the least positive integer $n$ such that $D(n)$ is scary and has at least $2009$ elements? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the coefficient of $x^2$ in the following polynomial $$(1 -x)^2(1 + 2x)^2(1 - 3x)^2... (1 -11x)^2.$$

Let $\omega$ be the circumcircle of an acute angled tirangle $ABC.$ The line tangent to $\omega$ at $A$ intersects the line $BC$ at the point $T.$ Let the midpoint of segment $AT$ be $N,$ and the centroid of $\triangle ABC$ be the point $G.$ The other tangent line drawn from $N$ to $\omega$ intersects $\omega$ at the point $L.$ The line $LG$ meets $\omega$ at $S\neq L.$ Prove that $AS\parallel BC.$

Let $ABC$ be any triangle and $D, E, F$ the midpoints of $BC, CA, AB$. The medians $AD, BE$ and $CF$ intersect at point $S$. At least two of the quadrilaterals $AF SE, BDSF, CESD$ are cyclic. Show that the triangle $ABC$ is equilateral.

Consider all trapezoids in a coordinate plane with interior angles of $90^o, 90^o, 45^o$ and $135^o$ whose bases are parallel to a coordinate axis and whose vertices have integer coordinates. Define the [i]size [/i] of such a trapezoid as the total number of points with integer coordinates inside and on the boundary of the trapezoid. (a) How many pairwise non-congruent such trapezoids of size $2001$ are there? (b) Find all positive integers not greater than $50$ that do not appear as sizes of any such trapezoid.

You have a four-liter jug and a six-liter pot (both of cylindrical shape), and a big barrel of water. Can you measure exactly one liter of water?

One side of an equilateral triangle of height $24$ lies on line $\ell.$ A circle of radius $12$ is tangent to $\ell$ and is externally tangent to the triangle. The area of the region exterior to the triangle and the circle and bounded by the triangle, the circle, and line $\ell$ can be written as $a\sqrt{b} - c\pi,$ where $a,$ $b,$ and $c$ are positive integers and $b$ is not divisible by the square of any prime. What is $a+b+c\,?$ $\phantom{boo}$ $\displaystyle \textbf{(A)}\; 72 \quad \textbf{(B)}\; 73 \quad \textbf{(C)}\; 74 \quad \textbf{(D)}\; 75 \quad \textbf{(E)}\; 76 $

In the figure below, you see three half-circles. The circle $C$ is tangent to two of the half-circles and to the line $PQ$ perpendicular to the diameter $AB$. The area of the shaded region is $39\pi$, and the area of the circle $C$ is $9\pi$. Find the length of the diameter $AB$.

For arbitrary positive integers $a, b$, denote $a @ b =\frac{a-b}{gcd(a,b)}$ Let $n$ be a positive integer. Prove that the following conditions are equivalent: (i) $gcd(n, n @ m) = 1$ for every positive integer $m < n$, (ii) $n = p^k$ where $p$ is a prime number and $k$ is a non-negative integer.

Prove that if $a, b, c$ are positive real numbers, then the least possible value of \[6a^3 + 9b^3 + 32c^3 + \frac{1}{4abc}\] is $6$. For which values of $a, b$ and $c$ is equality attained?