An isosceles trapezoid has legs and shorter base of length $1$. Find the maximum possible value of its area

Let $n$ be positive integer,$x_1,x_2,\cdots,x_n$ be positive real numbers such that $x_1x_2\cdots x_n=1 $ . Prove that$$\sum\limits_{i = 1}^{n}x_i\sqrt{x^2_1+x^2_2+\cdots x^2_i}\ge\frac{n+1}{2}\sqrt{n}$$

Let $\alpha$ and $\omega$ be two circles such that $\omega$ goes through the center of $\alpha$.$\omega$ intersects $\alpha$ at $A$ and $B$.Let $P$ any point on the circumference $\omega$.The lines $PA$ and $PB$ intersects $\alpha$ again at $E$ and $F$ respectively.Prove that $AB=EF$.

Find the digits left and right of the decimal point in the decimal form of the number \[ (\sqrt{2} + \sqrt{3})^{1980}. \]

The set of natural numbers is represented as a union of pairwise disjoint subsets, whose elements form infinite arithmetic progressions with positive differences $d_1,d_2,d_3,...$. Is it possible that the sum $\frac{1}{d_1}+\frac{1}{d_1}+\frac{1}{d_3}+... $ does not exceed $0.9$? Consider the cases where (a) the total number of progressions is finite, and (b) the number of progressions is infinite. (In this case the condition that $\frac{1}{d_1}+\frac{1}{d_1}+\frac{1}{d_3}+... $ does not exceed $0.9$ should be taken to mean that the sum of any finite number of terms does not exceed 0.9.) (A. Tolpugo, Kiev)

Define the unit $N$-hypercube to be the set of points $[0, 1]^N \subset R^N$ . For example, the unit $0$-hypercube is a point, and the unit $3$-hypercube is the unit cube. Define a $k$-face of the unit $N$-hypercube to be a copy of the $k$-hypercube in the exterior of the $N$-hypercube. More formally, a $k$-face of the unit $N$-hypercube is a set of the form $$\prod_{i=1}^{N} S_i$$ where $S_i$ is either $\{0\}$, $\{1\}$, or $[0, 1]$ for each $1 \le i \le N$, and there are exactly $k$ indices $i$ such that $S_i = [0, 1]$. The expected value of the dimension of a random face of the unit $ 8$-hypercube (where the dimension of a face can be any value between $0$ and $N$) can be written in the form $p/q$ where $p$ and $q$ are relatively prime positive integers. Find $p + q$.

A binary operation $\diamondsuit$ has the properties that $a\,\diamondsuit\, (b\,\diamondsuit \,c) = (a\,\diamondsuit \,b)\cdot c$ and that $a\,\diamondsuit \,a=1$ for all nonzero real numbers $a, b,$ and $c$. (Here $\cdot$ represents multiplication). The solution to the equation $2016 \,\diamondsuit\, (6\,\diamondsuit\, x)=100$ can be written as $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $p+q?$ $\textbf{(A) }109\qquad\textbf{(B) }201\qquad\textbf{(C) }301\qquad\textbf{(D) }3049\qquad\textbf{(E) }33,601$

Let $m, n \ge 2$ be integers. Carl is given $n$ marked points in the plane and wishes to mark their centroid.* He has no standard compass or straightedge. Instead, he has a device which, given marked points $A$ and $B$, marks the $m-1$ points that divide segment $\overline{AB}$ into $m$ congruent parts (but does not draw the segment). For which pairs $(m,n)$ can Carl necessarily accomplish his task, regardless of which $n$ points he is given? *Here, the [i]centroid[/i] of $n$ points with coordinates $(x_1, y_1), \dots, (x_n, y_n)$ is the point with coordinates $\left( \frac{x_1 + \dots + x_n}{n}, \frac{y_1 + \dots + y_n}{n}\right)$. [i]Proposed by Holden Mui and Carl Schildkraut[/i]

A square has sides of length $ 10$, and a circle centered at one of its vertices has radius $ 10$. What is the area of the union of the regions enclosed by the square and the circle? $ \textbf{(A)}\ 200 \plus{} 25\pi\qquad \textbf{(B)}\ 100 \plus{} 75\pi\qquad \textbf{(C)}\ 75 \plus{} 100\pi\qquad \textbf{(D)}\ 100 \plus{} 100\pi$ $ \textbf{(E)}\ 100 \plus{} 125\pi$

Let $ABCDEF$ be a regular hexagon with side length $1$. Denote by $X$, $Y$, and $Z$ the midpoints of sides $\overline{AB},\overline{CD},\overline{EF}$, respectively. What is the area of the convex hexagon whose interior is the intersection of the interiors of $\triangle{ACE}$ and $\triangle{XYZ}$? $\textbf{(A) }\dfrac{3}{8}\sqrt{3}\qquad\textbf{(B) }\dfrac{7}{16}\sqrt{3}\qquad\textbf{(C) }\dfrac{15}{32}\sqrt{3}\qquad\textbf{(D) }\dfrac{1}{2}\sqrt{3}\qquad\textbf{(E) }\dfrac{9}{16}\sqrt{3}$

Let $ABC$ be an equilateral triangle with side length $1.$ Then, let $M$ be the midpoint of $\overline{BC}.$ The area of all points within $ABC$ that are closer to $M$ than either of $A, B,$ or $C$ can be expressed as the fraction $\tfrac{\sqrt{a}}{b}$ where $a$ is not divisible by the square of any prime and $b$ is a positive integer. Find $a+b.$

Suppose that $ p$ is a prime number. Prove that for each $ k$, there exists an $ n$ such that: \[ \left(\begin{array}{c}n\\ \hline p\end{array}\right)\equal{}\left(\begin{array}{c}n\plus{}k\\ \hline p\end{array}\right)\]

Let $p > 3$ be a prime number, and let $F_p$ denote the (finite) set of residue classes modulo $p$. Let $S_d$ denote the set of $2$-variable polynomials $P(x, y)$ with coefficients in $F_p$, total degree $\le d$, and satisfying $P(x, y) = P(y,- x -y)$. Show that $$|S_d| = p^{\lceil (d+1)(d+2)/6 \rceil}$$. [i]The total degree of a $2$-variable polynomial $P(x, y)$ is the largest value of $i + j$ among monomials $x^iy^j$ [/i] appearing in $P$.

a) Find two sets $X,Y$ such that $X\cap Y =\emptyset$, $X\cup Y = \mathbb Q^{\star}_{+}$ and $Y = \{a\cdot b \mid a,b \in X \}$. b) Find two sets $U,V$ such that $U\cap V =\emptyset$, $U\cup V = \mathbb R$ and $V = \{x+y \mid x,y \in U \}$.

Let $\alpha, \beta$ and $\gamma$ denote the angles of the triangle $ABC$. The perpendicular bisector of $AB$ intersects $BC$ at the point $X$, the perpendicular bisector of $AC$ intersects it at $Y$. Prove that $\tan(\beta) \cdot \tan(\gamma) = 3$ implies $BC= XY$ (or in other words: Prove that a sufficient condition for $BC = XY$ is $\tan(\beta) \cdot \tan(\gamma) = 3$). Show that this condition is not necessary, and give a necessary and sufficient condition for $BC = XY$.

Find the number of integers $n > 1$ which divide $a^{25} - a$ for every integer $a$.

Consider $120$ unit squares arbitrarily situated in a $20\times 25$ rectangle. Prove that one can place a circle with unit diameter in the rectangle without intersecting any of the squares.

In how many ways can one arrange seven white and five black balls in a line in such a way that there are no two neighboring black balls?

Let $p$ be a prime number and let $m, n$ be integers greater than $1$ such that $n | m^{p(n-1)} - 1$. Prove that $gcd(m^{n-1} - 1, n) > 1$.

Let the number $x$. Using multiply and division operations of any 2 given or already given numbers we can obtain powers with natural exponent of the number $x$ (for example, $x\cdot x=x^{2}$, $x^{2}\cdot x^{2}=x^{4}$, $x^{4}: x=x^{3}$, etc). Determine the minimal number of operations needed for calculating $x^{2006}$.

Let the numbers $a, b,c$ satisfy the relation $a^2+b^2+c^2+d^2 \le 12$. Determine the maximum value of $M = 4(a^3 + b^3 + c^3+d^3) - (a^4 + b^4 + c^4+d^4)$

In a classroom of at least four students, when any four of them take seats around a round table, there is always someone who either knows both of his neighbors, or does not know either of his neighbors. Prove that it is possible to divide the students into two groups so that in one of them, all students knows one another, and in the other, none of the students know each other. [i]Note: If $A$ knows $B$, then $B$ knows $A$ as well.[/i]

Find the biggest natural number $m$ that has the following property: among any five 500-element subsets of $\{ 1,2,\dots, 1000\}$ there exist two sets, whose intersection contains at least $m$ numbers.

Let $ \mathcal{F}$ be a family of subsets of a ground set $ X$ such that $ \cup_{F \in \mathcal{F}}F=X$, and (a) if $ A,B \in \mathcal{F}$, then $ A \cup B \subseteq C$ for some $ C \in \mathcal{F};$ (b) if $ A_n \in \mathcal{F} \;(n=0,1,...)\ , B \in \mathcal{F},$ and $ A_0 \subset A_1 \subset...,$ then, for some $ k \geq 0, \;A_n \cap B=A_k \cap B$ for all $ n \geq k$. Show that there exist pairwise disjoint sets ${ X_{\gamma} \;( \gamma \in \Gamma}\ )$, with $ X= \cup \{ X_{\gamma} : \;\gamma \in \Gamma \ \},$ such that every $ X_{\gamma}$ is contained in some member of $ \mathcal{F}$, and every element of $ \mathcal{F}$ is contained in the union of finitely many $ X_{\gamma}$'s. [i]A. Hajnal[/i]

$ n$ sets $ S_1$, $ S_2$ $ \cdots$, $ S_n$ consists of non-negative numbers. $ x_i$ is the sum of all elements of $ S_i$, prove that there is a natural number $ k$, $ 1<k<n$, and: \[ \sum_{i\equal{}1}^n x_i < \frac{1}{k\plus{}1} \left[ k \cdot \frac{n(n\plus{}1)(2n\plus{}1)}{6} \minus{} (k\plus{}1)^2 \cdot \frac{n(n\plus{}1)}{2} \right]\] and there exists subscripts $ i$, $ j$, $ t$, and $ l$ (at least $ 3$ of them are distinct) such that $ x_i \plus{} x_j \equal{} x_t \plus{} x_l$.