$8$ people participate in a party. (1) Among any $5$ people there are $3$ who pairwise know each other. Prove that there are $4$ people who paiwise know each other. (2) If Among any $6$ people there are $3$ who pairwise know each other, then can we find $4$ people who pairwise know each other?

Let $a$ and $b$ be positive integers with $a>b$. Suppose that $$\sqrt{\sqrt{a}+\sqrt{b}}+\sqrt{\sqrt{a}-\sqrt{b}}$$ is an integer. (a) Must $\sqrt{a}$ be an integer? (b) Must $\sqrt{b}$ be an integer?

Let $A, B, C,$ be points in the plane, such that we can draw $3$ equal circumferences in which the first one passes through $A$ and $B$, the second one passes through $B$ and $C$, the last one passes through $C$ and $A$, and all $3$ circumferences share a common point $P$. Show that the radius of each of these circumferences is equal to the circumradius of triangle $ABC$, and that $P$ is the orthocenter of triangle $ABC$.

The War of $1812$ started with a declaration of war on Thursday, June $18$, $1812$. The peace treaty to end the war was signed $919$ days later, on December $24$, $1814$. On what day of the week was the treaty signed? $\textbf{(A)}\ \text{Friday} \qquad \textbf{(B)}\ \text{Saturday} \qquad \textbf{(C)}\ \text{Sunday} \qquad \textbf{(D)}\ \text{Monday} \qquad \textbf{(E)}\ \text{Tuesday} $

The number $2.5252525...$ can be written as a fraction. When reduced to lowest terms the sum of the numerator and denominator of this fraction is: $ \textbf{(A) }7\qquad\textbf{(B)} 29\qquad\textbf{(C) }141\qquad\textbf{(D) }349\qquad\textbf{(E) }\text{none of these} $

Find all functions $f : R \to R$ which satisfy $f(x)f(y) = f(xy) + xy$ for all $x, y \in R$.

Let $S$ be the subset of $N$($N$ is the set of all natural numbers) satisfying: i)Among each $2003$ consecutive natural numbers there exist at least one contained in $S$; ii)If $n \in S$ and $n>1$ then $[\frac{n}{2}] \in S$ Prove that:$S=N$ I hope it hasn't posted before. :lol: :lol:

Denote by $ d(n)$ the number of all positive divisors of a natural number $ n$ (including $ 1$ and $ n$). Prove that there are infinitely many $ n$, such that $ n/d(n)$ is an integer.

The numbers of the set $\{1,2,\ldots,n\}$ are colored with $6$ colors. Let $$S:=\{(x,y,z)\in\{1,2,\ldots,n\}^3:x+y+z\equiv0\pmod n\text{ and }x,y,z\text{ have the same color}\}$$and $$D:=\{(x,y,z)\in\{1,2,\ldots,n\}^3:x+y+z\equiv0\pmod n\text{ and }x,y,z\text{ have three different colors}\}.$$Prove that $$|D|\le2|S|+\frac{n^2}2.$$

In how many ways can $n^2$ distinct real numbers be arranged into an $n\times n$ array $(a_{ij })$ such that max$_{j}$ min $_i \,\, a_{ij} $= min$_i$ max$_j \,\, a_{ij}$?

Given a triangle with sides $a, b, c$ that satisfy $\frac{a}{b+c}=\frac{c}{a+b}$. Determine the angles of this triangle, if you know that one of them is equal to $120^0$.

Prove that from a set of ten distinct two-digit numbers, it is always possible to find two disjoint subsets whose members have the same sum.

Let $a,b,c$ be three postive real numbers such that $a+b+c=1$. Prove that $9abc\leq ab+ac+bc < 1/4 +3abc$.

Let $n\geq 2$ and $\mathcal{M}$ be a subset of $S_n$ with at least two elements, and which is closed under composition. Consider a function $f:\mathcal{M}\rightarrow\mathbb{R}$ which satisfies $$|f(\sigma\tau)-f(\sigma)-f(\tau)|\leq 1,$$ for all $\sigma,\tau\in\mathcal{M}$. Prove that $$\max_{\sigma,\tau\in\mathcal{M}}|f(\sigma)-f(\tau)|\leq 2-\frac{2}{|\mathcal{M}|}.$$

Let $ S$ be the set of all points $ (x,y)$ in the coordinate plane such that $ 0\le x\le \frac \pi2$ and $ 0\le y\le \frac \pi2$. What is the area of the subset of $ S$ for which \[ \sin^2 x \minus{} \sin x\sin y \plus{} \sin^2 y\le \frac 34? \]$ \textbf{(A) } \frac {\pi^2}9 \qquad \textbf{(B) } \frac {\pi^2}8 \qquad \textbf{(C) } \frac {\pi^2}6\qquad \textbf{(D) } \frac {3\pi^2}{16} \qquad \textbf{(E) } \frac {2\pi^2}9$

Show that there do not exist any distinct natural numbers $ a$, $ b$, $ c$, $ d$ such that $ a^3\plus{}b^3\equal{}c^3\plus{}d^3$ and $ a\plus{}b\equal{}c\plus{}d$.

Joy has $30$ thin rods, one each of every integer length from $1$ cm through $30$ cm. She places the rods with lengths $3$ cm, $7$ cm, and $15$ cm on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How many of the remaining rods can she choose as the fourth rod? $\textbf{(A) }16\qquad\textbf{(B) }17\qquad\textbf{(C) }18\qquad\textbf{(D) }19\qquad\textbf{(E) }20$

p is a prime number, k is a positive integer Find all (p, k): $k!=(p^3-1)(p^3-p)(p^3-p^2)$

Rectangle $ABCD$ has $AB=6$ and $BC=3$. Point $M$ is chosen on side $AB$ so that $\angle AMD = \angle CMD$. What is the degree measure of $\angle AMD$? $ \textbf{(A)}\ 15 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 60 \qquad \textbf{(E)}\ 75 $

a) Given a convex hexagon $ABCDEF$ with all the equal angles. Prove that $$|AB|-|DE| = |EF|-|BC| = |CD|-|FA|$$ b) The opposite problem: Prove that it is possible to construct a convex hexagon with equal angles of six segments $a_1,a_2,...,a_6$, whose lengths satisfy the condition $$a_1-a_4 = a_5-a_2 = a_3-a_6$$

A [i]tile[/i] $T$ is a union of finitely many pairwise disjoint arcs of a unit circle $K$. The [i]size[/i] of $T$, denoted by $|T|$, is the sum of the lengths of the arcs $T$ consists of, divided by $2\pi$. A [i]copy[/i] of $T$ is a tile $T'$ obtained by rotating $T$ about the centre of $K$ through some angle. Given a positive real number $\varepsilon < 1$, does there exist an infinite sequence of tiles $T_1,T_2,\ldots,T_n,\ldots$ satisfying the following two conditions simultaneously: 1) $|T_n| > 1 - \varepsilon$ for all $n$; 2) The union of all $T_n'$ (as $n$ runs through the positive integers) is a proper subset of $K$ for any choice of the copies $T_1'$, $T_2'$, $\ldots$, $T_n', \ldots$? [hide=Note] In the extralist the problem statement had the clause "three conditions" rather than two, but only two are presented, the ones you see. I am quite confident this is a typo or that the problem might have been reformulated after submission.[/hide]

Given a $2015$-tuple $(a_1,a_2,\ldots,a_{2015})$ in each step we choose two indices $1\le k,l\le 2015$ with $a_k$ even and transform the $2015$-tuple into $(a_1,\ldots,\dfrac{a_k}{2},\ldots,a_l+\dfrac{a_k}{2},\ldots,a_{2015})$. Prove that starting from $(1,2,\ldots,2015)$ in finite number of steps one can reach any permutation of $(1,2,\ldots,2015)$.

Given a circle with diameter $AB$. Points $C$ and $D$ are selected on the circumference of the circle such that the chord $CD$ intersects $AB$ inside the circle, at point $P$. The ratio of the arc length $\overarc {AC}$ to the arc length $\overarc {BD}$ is $4 : 1$ , while the ratio of the arc length $\overarc{AD}$ to the arc length $\overarc {BC}$ is $3 : 2$ . Find $\angle{APC}$ , in degrees.

Given unbounded strictly increasing sequence $a_1, a_2, ... , a_n, ...$ of positive numbers. Prove that a) there exists a number $k_0$ such that for all $k>k_0$ the following inequality is valid: $$\frac{a_1}{a_2}+ \frac{a_2}{a_3} + ... + \frac{a_k}{a_{k-1} }< k - 1$$ b) there exists a number $k_0$ such that for all $k>k_0$ the following inequality is valid: $$\frac{a_1}{a_2}+ \frac{a_2}{a_3} + ... + \frac{a_k}{a_{k-1} }< k - 1985$$

Two streams of water flow through the U-shaped tubes shown. The tube on the left has cross-sectional area $A$, and the speed of the water flowing through it is $v$; the tube on the right has cross-sectional area $A'=1/2A$. If the net force on the tube assembly is zero, what must be the speed $v'$ of the water flowing through the tube on the right? Neglect gravity, and assume that the speed of the water in each tube is the same upon entry and exit. [asy] // Code by riben size(300); draw(arc((0,0),10,90,270)); draw(arc((0,0),7,90,270)); draw((0,10)--(25,10)); draw((0,-10)--(25,-10)); draw((0,7)--(25,7)); draw((0,-7)--(25,-7)); draw(ellipse((25,8.5),0.5,1.5)); draw(ellipse((25,-8.5),0.5,1.5)); draw((20,8.5)--(7,8.5),EndArrow(size=7)); draw((7,-8.5)--(20,-8.5),EndArrow(size=7)); draw(arc((-22,0),12,90,-90)); draw(arc((-22,0),7,90,-90)); draw((-22,12)--(-42,12)); draw((-22,-12)--(-42,-12)); draw((-22,7)--(-42,7)); draw((-22,-7)--(-42,-7)); draw(ellipse((-42,9.5),1.5,2.5)); draw(ellipse((-42,-9.5),1.5,2.5)); draw((-38,9.5)--(-23,9.5),EndArrow(size=7)); draw((-23,-9.5)--(-38,-9.5),EndArrow(size=7)); [/asy] (A) $1/2v$ (B) $v$ (C) $\sqrt{2}v$ (D) $2v$ (E) $4v$