$ (a)$ A group of people attends a party. Each person has at most three acquaintances in the group, and if two people do not know each other, then they have a common acquaintance in the group. What is the maximum possible number of people present? $ (b)$ If, in addition, the group contains three mutual acquaintances, what is the maximum possible number of people?

Place all numbers from 1 to 10 to the boxes such that every number except the uppermost is equal to the difference between the two numbers on its top. [asy] unitsize(-4); draw((0,0)--(5,0)--(5,5)--(0,5)--cycle); draw((10,0)--(15,0)--(15,5)--(10,5)--cycle); draw((20,0)--(25,0)--(25,5)--(20,5)--cycle); draw((30,0)--(35,0)--(35,5)--(30,5)--cycle); draw((5,10)--(10,10)--(10,15)--(5,15)--cycle); draw((15,10)--(20,10)--(20,15)--(15,15)--cycle); draw((25,10)--(30,10)--(30,15)--(25,15)--cycle); draw((10,20)--(15,20)--(15,25)--(10,25)--cycle); draw((20,20)--(25,20)--(25,25)--(20,25)--cycle); draw((15,30)--(20,30)--(20,35)--(15,35)--cycle); draw((2.5,5)--(7.5, 10)); draw((12.5,5)--(17.5, 10)); draw((22.5,5)--(27.5, 10)); draw((32.5,5)--(27.5, 10)); draw((22.5,5)--(17.5, 10)); draw((12.5,5)--(7.5, 10)); draw((7.5,15)--(12.5, 20)); draw((17.5,15)--(22.5, 20)); draw((27.5,15)--(22.5, 20)); draw((17.5,15)--(12.5, 20)); draw((12.5,25)--(17.5, 30)); draw((22.5,25)--(17.5, 30)); [/asy]The number in the lower box is at most $\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5$

What is $1-2+3-4$? $\textbf{(A) } {-}2\qquad\textbf{(B) } {-}1\qquad\textbf{(C) } 1\qquad\textbf{(D) } 4\qquad\textbf{(E) } 9$

Let $O$ be the circumcenter of triangle $ABC,$ and $D$ be an arbitrary point on the circumcircle of $ABC.$ Let points $X, Y$ and $Z$ be the orthogonal projections of point $D$ onto lines $OA, OB$ and $OC,$ respectively. Prove that the incenter of triangle $XYZ$ is on the Simson-Wallace line of triangle $ABC$ corresponding to point $D.$

Let $BB_1$ and $CC_1$ be the altitudes of an acute-angled triangle $ABC$, which intersect its angle bisector $AL$ at two different points $P$ and $Q$, respectively. Denote by $F$ such a point that $PF\parallel AB$ and $QF\parallel AC$, and by $T$ the intersection point of the tangents drawn at points $B$ and $C$ to the circumscribed circle of the triangle $ABC$. Prove that the points $A, F$ and $T$ lie on the same line.

We want to choose telephone numbers for a city. The numbers have $10$ digits and $0$ isn’t used in the numbers. Our aim is: We don’t choose some numbers such that every $2$ telephone numbers are different in more than one digit OR every $2$ telephone numbers are different in a digit which is more than $1$. What is the maximum number of telephone numbers which can be chosen? In how many ways, can we choose the numbers in this maximum situation?

All the two-digit numbers from $19$ to $80$ are written in a line without spaces. Is the obtained number $192021....7980$ divisible by $1980$?

Show that $\frac{x^2}{1 - x}+\frac{(1 - x)^2}{x} \ge 1$ for all real numbers $x$, where $0 < x < 1$

Suppose $a,b,c \in \mathbb R^+$and \[\frac1{a^2+1}+\frac1{b^2+1}+\frac1{c^2+1}=2\] Prove that $ab+ac+bc\leq \frac32$

In a triangle $ABC$, it is known that $\angle A=100^{\circ}$ and $AB=AC$. The internal angle bisector $BD$ has length $20$ units. Find the length of $BC$ to the nearest integer, given that $\sin 10^{\circ} \approx 0.174$

Find all triples $(p,q,r)$ of pairwise distinct primes such that \[p\mid q+r, q\mid r+2p, r\mid p+3q.\]

Starting with four given integers $a_1, b_1, c_1, d_1$ is defined recursively for all positive integers $n$: $$a_{n+1} := |a_n - b_n|, b_{n+1} := |b_n - c_n|, c_{n+1} := |c_n - d_n|, d_{n+1} := |d_n - a_n|.$$ Prove that there is a natural number $k$ such that all terms $a_k, b_k, c_k, d_k$ take the value zero.

Prove that for every nonnegative integer $ k$ there is a product $$(k + 1)(k + 2)...(k + 1980)$$ divisible by $ 1980^{197}$.

Circle $\Gamma$ touches the circumcircle of triangle $ABC$ at point $R$, and it touches the sides $AB$ and $AC$ at points $P$ and $Q$, respectively. Rays $PQ$ and $BC$ intersect at point $X$. The tangent line at point $R$ to the circle $\Gamma$ meets the segment $QX$ at point $Y$. The line segment $AX$ intersects the circumcircle of triangle $APQ$ at point $Z$. Prove that the circumscribed circles of triangles $ABC$ and $XY Z$ are tangent.

An integer $n \geq 3$ is given. We call an $n$-tuple of real numbers $(x_1, x_2, \dots, x_n)$ [i]Shiny[/i] if for each permutation $y_1, y_2, \dots, y_n$ of these numbers, we have $$\sum \limits_{i=1}^{n-1} y_i y_{i+1} = y_1y_2 + y_2y_3 + y_3y_4 + \cdots + y_{n-1}y_n \geq -1.$$ Find the largest constant $K = K(n)$ such that $$\sum \limits_{1 \leq i < j \leq n} x_i x_j \geq K$$ holds for every Shiny $n$-tuple $(x_1, x_2, \dots, x_n)$.

On each face of two dice some positive integer is written. The two dice are thrown and the numbers on the top face are added. Determine whether one can select the integers on the faces so that the possible sums are $2,3,4,5,6,7,8,9,10,11,12,13$, all equally likely?

Several figures can be made by attaching two equilateral triangles to the regular pentagon $ ABCDE$ in two of the five positions shown. How many non-congruent figures can be constructed in this way? [asy]unitsize(2cm); pair A=dir(306); pair B=dir(234); pair C=dir(162); pair D=dir(90); pair E=dir(18); draw(A--B--C--D--E--cycle,linewidth(.8pt)); draw(E--rotate(60,D)*E--D--rotate(60,C)*D--C--rotate(60,B)*C--B--rotate(60,A)*B--A--rotate(60,E)*A--cycle,linetype("4 4")); label("$A$",A,SE); label("$B$",B,SW); label("$C$",C,WNW); label("$D$",D,N); label("$E$",E,ENE);[/asy]$ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$

The altitudes of the triangle ${ABC}$ meet in the point ${H}$. You know that ${AB = CH}$. Determine the value of the angle $\widehat{BCA}$.

Let $P$ be a point on the circumcircle of acute triangle $ABC$. Let $D,E,F$ be the reflections of $P$ in the $A$-midline, $B$-midline, and $C$-midline. Let $\omega$ be the circumcircle of the triangle formed by the perpendicular bisectors of $AD, BE, CF$. Show that the circumcircles of $\triangle ADP, \triangle BEP, \triangle CFP,$ and $\omega$ share a common point.

(1) 2009 Kansai University entrance exam Calculate $ \int \frac{e^{\minus{}2x}}{1\plus{}e^{\minus{}x}}\ dx$. (2) 2009 Rikkyo University entrance exam/Science Evaluate $ \int_0^ 1 \frac{2x^3}{1\plus{}x^2}\ dx$.

For $n$ distinct positive integers all their $n(n-1)/2$ pairwise sums are considered. For each of these sums Ivan has written on the board the number of original integers which are less than that sum and divide it. What is the maximum possible sum of the numbers written by Ivan?

Let $T_1$ and $T_2$ internally tangent circumferences at $P$, with radius $R$ and $2R$, respectively. Find the locus traced by $P$ as $T_1$ rolls tangentially along the entire perimeter of $T_2$

A [i]$n$-trønder walk[/i] is a walk starting at $(0, 0)$, ending at $(2n, 0)$ with no self intersection and not leaving the first quadrant, where every step is one of the vectors $(1, 1)$, $(1, -1)$ or $(-1, 1)$. Find the number of $n$-trønder walks.

On the sides $ AB,BC,CD,DA $ of the parallelogram $ ABCD, $ consider the points $ M,N,P, $ respectively, $ Q, $ such that $ \overrightarrow{MN} +\overrightarrow{QP} =\overrightarrow{AC} . $ Show that $ \overrightarrow{PN} +\overrightarrow{QM} = \overrightarrow{DB} . $

3. Show that there do not exist positive integers $a$, $b$, $c$ and $d$, pairwise co-prime, such that $ab+cd$, $ac+bd$ and $ad+bc$ are odd divisors of the number $(a+b-c-d)(a-b+c-d)(a-b-c+d)$.