Navvye/Olympiad-Problems-CSV huggingface dataset:
2018 Regional Competition For Advanced Students
If $a, b$ are positive reals such that $a+b<2$. Prove that $$\frac{1}{1+a^2}+\frac{1}{1+b^2} \le \frac{2}{1+ab}$$and determine all $a, b$ yielding equality.



Proposed by Gottfried Perz
2018 Regional Competition For Advanced Students
Let $k$ be a circle with radius $r$ and $AB$ a chord of $k$ such that $AB > r$. Furthermore, let $S$ be the point on the chord $AB$ satisfying $AS = r$. The perpendicular bisector of $BS$ intersects $k$ in the points $C$ and $D$. The line through $D$ and $S$ intersects $k$ for a second time in point $E$. Show that the triangle $CSE$ is equilateral.



Proposed by Stefan Leopoldseder
2018 Regional Competition For Advanced Students
Let $n \ge 3$ be a natural number.

Determine the number $a_n$ of all subsets of $\{1, 2,...,n\}$ consisting of three elements such that one of them is the arithmetic mean of the other two.



Proposed by Walther Janous
2018 Regional Competition For Advanced Students
Let $d(n)$ be the number of all positive divisors of a natural number $n \ge 2$.

Determine all natural numbers $n \ge 3$ such that  $d(n -1) + d(n) + d(n + 1) \le  8$.



Proposed by Richard Henner
2017 Regional Competition For Advanced Students
Let $x_1, x_2, \dots, x_n$ be non-negative real numbers such that

$$x_1^2+x_2^2 + \dots x_9^2 \ge 25.$$Prove that one can choose three of these numbers such that their sum is at least $5$.



Proposed by Karl Czakler
2017 Regional Competition For Advanced Students
Let $ABCD$ be a cyclic quadrilateral with perpendicular diagonals and circumcenter $O$. Let $g$ be the line obtained by reflection of the diagonal $AC$ along the angle bisector of $\angle BAD$. Prove that the point $O$ lies on the line $g$.



Proposed by Theresia Eisenkölbl
2017 Regional Competition For Advanced Students
The nonnegative integers $2000$, $17$ and $n$ are written on the blackboard. Alice and Bob play the following game: Alice begins, then they play in turns. A move consists in replacing one of the three numbers by the absolute difference of the other two. No moves are allowed, where all three numbers remain unchanged. The player who is in turn and cannot make an allowed move loses the game.



Prove that the game will end for every number $n$.

Who wins the game in the case $n = 2017$?





Proposed by Richard Henner
2017 Regional Competition For Advanced Students
Determine all integers $n \geq 2$, satisfying

$$n=a^2+b^2,$$where $a$ is the smallest divisor of $n$ different from $1$ and $b$ is an arbitrary divisor of $n$.

Proposed by Walther Janous
2016 Regional Competition For Advanced Students
Determine all positive integers $k$ and $n$ satisfying the equation



$$k^2 - 2016 = 3^n$$

(Stephan Wagner)
2016 Regional Competition For Advanced Students
Let $a$, $b$, $c$ and $d$ be real numbers with $a^2 + b^2 + c^2 + d^2 = 4$.  Prove that the inequality

$$(a+2)(b+2) \ge cd$$holds and give four numbers $a$, $b$, $c$ and $d$ such that equality holds.



(Walther Janous)
2016 Regional Competition For Advanced Students
On the occasion of the 47th Mathematical Olympiad 2016 the numbers 47 and 2016 are written on the blackboard. Alice and Bob play the following game: Alice begins and in turns they choose two numbers $a$ and $b$ with $a > b$ written on the blackboard, whose difference $a-b$ is not yet written on the blackboard and write this difference additionally on the board. The game ends when no further move is possible. The winner is the player who made the last move.



Prove that Bob wins, no matter how they play.



(Richard Henner)
2016 Regional Competition For Advanced Students
Let $ABC$ be a triangle with $AC > AB$ and circumcenter $O$.  The tangents to the circumcircle at $A$ and $B$ intersect at $T$.  The perpendicular bisector of the side $BC$ intersects side $AC$ at $S$.



(a) Prove that the points $A$, $B$, $O$, $S$, and $T$ lie on a common circle.

(b) Prove that the line $ST$ is parallel to the side $BC$.



(Karl Czakler)
2015 Regional Competition For Advanced Students
Determine all triples $(a,b,c)$ of positive integers satisfying the conditions



$$\gcd(a,20) = b$$$$\gcd(b,15) = c$$$$\gcd(a,c) = 5$$

(Richard Henner)
2015 Regional Competition For Advanced Students
Let $x$, $y$, and $z$ be positive real numbers with $x+y+z = 3$.  Prove that at least one of the three numbers

$$x(x+y-z)$$$$y(y+z-x)$$$$z(z+x-y)$$is less or equal $1$.



(Karl Czakler)
2015 Regional Competition For Advanced Students
Let $n \ge 3$ be a fixed integer.  The numbers $1,2,3, \cdots , n$ are written on a board.  In every move one chooses two numbers and replaces them by their arithmetic mean.  This is done until only a single number remains on the board.



Determine the least integer that can be reached at the end by an appropriate sequence of moves.



(Theresia Eisenkölbl)
2015 Regional Competition For Advanced Students
Let $ABC$ be an isosceles triangle with $AC = BC$ and $\angle ACB < 60^\circ$.  We denote the incenter and circumcenter by $I$ and $O$, respectively.  The circumcircle of triangle $BIO$ intersects the leg $BC$ also at point $D \ne B$.



(a) Prove that the lines $AC$ and $DI$ are parallel.

(b) Prove that the lines $OD$ and $IB$ are mutually perpendicular.



(Walther Janous)
2014 Regional Competition For Advanced Students
Show that there are no positive real numbers $x, y, z$ such $(12x^2+yz)(12y^2+xz)(12z^2+xy)= 2014x^2y^2z^2$ .
2014 Regional Competition For Advanced Students
You can determine all 4-ples $(a,b, c,d)$ of real numbers, which solve the following equation system $\begin{cases} ab + ac = 3b + 3c \\
bc + bd = 5c + 5d \\
ac + cd = 7a + 7d \\
ad + bd = 9a + 9b \end{cases} $
2014 Regional Competition For Advanced Students
The sequence $(a_n)$ is defined with the recursion $a_{n + 1} = 5a^6_n + 3a^3_{n-1} + a^2_{n-2}$ for $n\ge 2$ and the set of initial values $\{a_0, a_1, a_2\} = \{2013, 2014, 2015\}$. (That is, the initial values are these three numbers in any order.)

Show that the sequence contains no sixth power of a natural number.
2014 Regional Competition For Advanced Students
For a point $P$ in the interior of a triangle $ABC$ let $D$ be the intersection of $AP$ with $BC$, let $E$ be the intersection of $BP$ with $AC$ and let $F$ be the intersection of $CP$ with $AB$.Furthermore let $Q$ and $R$ be the intersections of the parallel to $AB$ through $P$ with the sides $AC$ and $BC$, respectively. Likewise, let $S$ and $T$ be the intersections of the

parallel to $BC$ through $P$ with the sides $AB$ and $AC$, respectively.In a given triangle $ABC$, determine all points $P$ for which the triangles $PRD$, $PEQ$and $PTE$ have the same area.
2013 Regional Competition For Advanced Students
For which integers between $2000$ and $2010$ (including) is the probability that a random divisor is smaller or equal $45$ the largest?
2013 Regional Competition For Advanced Students
Determine all integers $x$ satisfying

$$ \left[\frac{x}{2}\right] \left[\frac{x}{3}\right] \left[\frac{x}{4}\right] = x^2. $$($[y]$ is the largest integer which is not larger than $y.$)
2013 Regional Competition For Advanced Students
For non-negative real numbers $a,$ $b$ let $A(a, b)$ be their arithmetic mean and $G(a, b)$ their geometric mean. We consider the sequence $\langle a_n \rangle$ with $a_0 = 0,$ $a_1 = 1$ and $a_{n+1} = A(A(a_{n-1}, a_n), G(a_{n-1}, a_n))$ for $n > 0.$

(a) Show that each $a_n = b^2_n$ is the square of a rational number (with $b_n \geq 0$).

(b) Show that the inequality $\left|b_n - \frac{2}{3}\right| < \frac{1}{2^n}$ holds for all $n > 0.$
2013 Regional Competition For Advanced Students
We call a pentagon distinguished  if either all side lengths or all angles are equal. We call it very distinguished if in addition two of the other parts are equal. i.e. $5$ sides and $2$ angles or $2$ sides and $5$ angles.Show that every very  distinguished pentagon has an axis of symmetry.
2012 Regional Competition For Advanced Students
Prove that the inequality $$ a + a^3 - a^4 - a^6 < 1$$ holds for all real numbers $a$.
2012 Regional Competition For Advanced Students
Determine all integer solutions $(x, y)$ of the equation $$(x - 1)x(x + 1) + (y - 1)y(y + 1) = 24 - 9xy\mbox{.}$$
2012 Regional Competition For Advanced Students
In an arithmetic sequence, the difference of consecutive terms in constant. We consider sequences of integers in which the difference of consecutive terms equals the sum of the differences of all preceding consecutive terms.

Which of these sequences with $a_0 = 2012$ and $1\leqslant d = a_1-a_0 \leqslant 43$ contain square numbers?
2012 Regional Competition For Advanced Students
In a triangle $ABC$, let $H_a$, $H_b$ and $H_c$ denote the base points of the altitudes on the sides $BC$, $CA$ and $AB$, respectively.

Determine for which triangles $ABC$ two of the lengths $H_aH_b$, $H_bH_c$ and $H_aH_c$ are equal.
2011 Regional Competition For Advanced Students
Let $p_1, p_2, \ldots, p_{42}$ be $42$ pairwise distinct prime numbers. Show that the sum $$\sum_{j=1}^{42}\frac{1}{p_j^2+1}$$ is not a unit fraction $\frac{1}{n^2}$ of some integer square number.
2011 Regional Competition For Advanced Students
Determine all triples $(x,y,z)$ of real numbers such that the following system of equations holds true:

\begin{align*}2^{\sqrt[3]{x^2}}\cdot 4^{\sqrt[3]{y^2}}\cdot 16^{\sqrt[3]{z^2}}&=128\\
\left(xy^2+z^4\right)^2&=4+\left(xy^2-z^4\right)^2\mbox{.}\end{align*}
2011 Regional Competition For Advanced Students
Let $k$ be a circle centered at $M$ and let $t$ be a tangentline to $k$ through some point $T\in k$. Let $P$ be a point on $t$ and let $g\neq t$ be a line through $P$ intersecting $k$ at $U$ and $V$. Let $S$ be the point on $k$ bisecting the arc $UV$ not containing $T$ and let $Q$ be the the image of $P$ under a reflection over $ST$.

Prove that $Q$, $T$, $U$ and $V$ are vertices of a trapezoid.
2011 Regional Competition For Advanced Students
Define the sequence $(a_n)_{n=1}^\infty$ of positive integers by $a_1=1$ and the condition that $a_{n+1}$ is the least integer such that $$\mathrm{lcm}(a_1, a_2, \ldots, a_{n+1})>\mathrm{lcm}(a_1, a_2, \ldots, a_n)\mbox{.}$$

Determine the set of elements of $(a_n)$.
2010 Regional Competition For Advanced Students
Let $0 \le a$, $b \le 1$ be real numbers. Prove the following inequality:

$$\sqrt{a^3b^3}+ \sqrt{(1-a^2)(1-ab)(1-b^2)} \le 1.$$

(41th Austrian Mathematical Olympiad, regional competition, problem 1)
2010 Regional Competition For Advanced Students
Solve the following in equation in $\mathbb{R}^3$:

$$x^4-x^2(4y^4+4z^4-1)-2xyz+y^8+2y^4z^4+y^2z^2+z^8=0.$$

(41st Austrian Mathematical Olympiad, regional competition, problem 2)
2010 Regional Competition For Advanced Students
Let $\triangle ABC$ be a triangle and let $D$ be a point on side $\overline{BC}$. Let $U$ and $V$ be the circumcenters of triangles $\triangle ABD$ and $\triangle ADC$, respectively. Show, that $\triangle ABC$ and $\triangle AUV$ are similar.



(41th Austrian Mathematical Olympiad, regional competition, problem 3)
2010 Regional Competition For Advanced Students
Let $(b_n)_{n \ge 0}=\sum_{k=0}^{n} (a_0+kd)$ for positive integers $a_0$ and $d$. We consider all such sequences containing an element $b_i$ which equals $2010$. Determine the greatest possible value of $i$ and for this value the integers $a_0$ and $d$.



(41th Austrian Mathematical Olympiad, regional competition, problem 4)
2009 Regional Competition For Advanced Students
Find the largest interval $ M \subseteq \mathbb{R^ + }$, such that for all $ a$, $ b$, $ c$, $ d \in M$ the inequality

$$ \sqrt {ab} + \sqrt {cd} \ge \sqrt {a + b} + \sqrt {c + d}$$

holds. Does the inequality

$$ \sqrt {ab} + \sqrt {cd} \ge \sqrt {a + c} + \sqrt {b + d}$$

hold too for all $ a$, $ b$, $ c$, $ d \in M$?

($ \mathbb{R^ + }$ denotes the set of positive reals.)
2009 Regional Competition For Advanced Students
How many integer solutions $ (x_0$, $ x_1$, $ x_2$, $ x_3$, $ x_4$, $ x_5$, $ x_6)$ does the equation

$$ 2x_0^2+x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2=9$$

have?
2009 Regional Competition For Advanced Students
Let $ D$, $ E$, $ F$ be the feet of the altitudes wrt sides $ BC$, $ CA$, $ AB$ of acute-angled triangle $ \triangle ABC$, respectively. In triangle $ \triangle CFB$, let $ P$ be the foot of the altitude wrt side $ BC$. Define $ Q$ and $ R$ wrt triangles $ \triangle ADC$ and $ \triangle BEA$ analogously. Prove that lines $ AP$, $ BQ$, $ CR$ don't intersect in one common point.
2009 Regional Competition For Advanced Students
Two infinite arithmetic progressions are called considerable different if the do not only differ by the absence of finitely many members at the beginning of one of the sequences.

How many pairwise considerable different non-constant arithmetic progressions of positive integers that contain an infinite non-constant geometric progression $ (b_n)_{n\ge0}$ with $ b_2=40 \cdot 2009$ are there?
2008 Regional Competition For Advanced Students
Show: For all real numbers $ a,b,c$ with $ 0<a,b,c<1$ is:

$$ \sqrt{a^2bc+ab^2c+abc^2}+\sqrt{(1-a)^2(1-b)(1-c)+(1-a)(1-b)^2(1-c)+(1-a)(1-b)(1-c)^2}<\sqrt{3}.$$
2008 Regional Competition For Advanced Students
For a real number $ x$ is $ [x]$ the next smaller integer to $ x$, that is the integer $ g$ with $ g\leqq<g+1$, and $ \{x\}=x-[x]$ is the “decimal part” of $ x$.

Determine all triples $ (a,b,c)$ of real numbers, which fulfil the following system of equations:

$$ \{a\}+[b]+\{c\}=2,9$$$$ \{b\}+[c]+\{a\}=5,3$$$$\{c\}+[a]+\{b\}=4,0$$
2008 Regional Competition For Advanced Students
Given is an acute angled triangle $ ABC$. Determine all points $ P$ inside the triangle with

$$1\leq\frac{\angle APB}{\angle ACB},\frac{\angle BPC}{\angle BAC},\frac{\angle CPA}{\angle CBA}\leq2$$
2008 Regional Competition For Advanced Students
For every positive integer $ n$ let

$$ a_n=\sum_{k=n}^{2n}\frac{(2k+1)^n}{k}$$

Show that there exists no $ n$, for which $ a_n$ is a non-negative integer.
2007 Regional Competition For Advanced Students
Let $ 0<x_0,x_1, \dots , x_{669}<1$ be pairwise distinct real numbers. Show that there exists a pair $ (x_i,x_j)$ with

$ 0<x_ix_j(x_j-x_i)<\frac{1}{2007}$
2007 Regional Competition For Advanced Students
Find all tuples $ (x_1,x_2,x_3,x_4,x_5)$ of positive integers with $ x_1>x_2>x_3>x_4>x_5>0$ and

$ {\left \lfloor \frac{x_1+x_2}{3} \right \rfloor }^2 + {\left \lfloor \frac{x_2+x_3}{3} \right \rfloor }^2 + {\left \lfloor \frac{x_3+x_4}{3} \right \rfloor }^2 + {\left \lfloor \frac{x_4+x_5}{3} \right \rfloor }^2 = 38.$
2007 Regional Competition For Advanced Students
Let $ a$ be a positive real number and $ n$ a non-negative integer. Determine $ S-T$, where

$ S= \sum_{k=-2n}^{2n+1} \frac{(k-1)^2}{a^{| \lfloor \frac{k}{2} \rfloor |}}$  and  $ T= \sum_{k=-2n}^{2n+1} \frac{k^2}{a^{| \lfloor \frac{k}{2} \rfloor |}}$
2007 Regional Competition For Advanced Students
Let $ M$ be the intersection of the diagonals of a convex quadrilateral $ ABCD$. Determine all such quadrilaterals for which there exists a line $ g$ that passes through $ M$ and intersects the side $ AB$ in $ P$ and the side $ CD$ in $ Q$, such that the four triangles $ APM$, $ BPM$, $ CQM$, $ DQM$ are similar.
2006 Regional Competition For Advanced Students
Let $ 0 < x <y$ be real numbers. Let

$ H=\frac{2xy}{x+y}$ , $ G=\sqrt{xy}$ , $ A=\frac{x+y}{2}$ , $ Q=\sqrt{\frac{x^2+y^2}{2}}$

be the harmonic, geometric, arithmetic and root mean square (quadratic mean) of $ x$ and $ y$. As generally known $ H<G<A<Q$. Arrange the intervals $ [H,G]$ , $ [G,A]$ and $ [A,Q]$ in ascending order by their length.
2006 Regional Competition For Advanced Students
Let $ n>1$ be a positive integer an $ a$ a real number. Determine all real solutions $ (x_1,x_2,\dots,x_n)$ to following system of equations:

$ x_1+ax_2=0$

$ x_2+a^2x_3=0$

…

$ x_k+a^kx_{k+1}=0$

…

$ x_n+a^nx_1=0$
2006 Regional Competition For Advanced Students
In a non isosceles triangle $ ABC$ let $ w$ be the angle bisector of the exterior angle at $ C$. Let $ D$ be the point of intersection of $ w$ with the extension of $ AB$. Let $ k_A$ be the circumcircle of the triangle $ ADC$ and analogy $ k_B$ the circumcircle of the triangle $ BDC$. Let $ t_A$ be the tangent line to $ k_A$ in A and $ t_B$ the tangent line to $ k_B$ in B. Let $ P$ be the point of intersection of $ t_A$ and $ t_B$.

Given are the points $ A$ and $ B$. Determine the set of points $ P=P(C )$ over all points $ C$, so that $ ABC$ is a non isosceles, acute-angled triangle.
2006 Regional Competition For Advanced Students
Let $ <h_n>$ $ n\in\mathbb N$ a harmonic sequence of positive real numbers (that means that  every $ h_n$ is the harmonic mean of its two neighbours $ h_{n-1}$ and $ h_{n+1}$ : $ h_n=\frac{2h_{n-1}h_{n+1}}{h_{n-1}+h_{n+1}}$)

Show that: if the sequence includes a member $ h_j$, which is the square of a rational number, it includes infinitely many members $ h_k$, which are squares of rational numbers.
2005 Regional Competition For Advanced Students
Show for all integers $ n \ge 2005$ the following chaine of inequalities:

$ (n+830)^{2005}<n(n+1)\dots(n+2004)<(n+1002)^{2005}$
2005 Regional Competition For Advanced Students
Construct the semicircle $ h$ with the diameter $ AB$ and the midpoint $ M$. Now construct the semicircle $ k$ with the diameter $ MB$ on the same side as $ h$. Let $ X$ and $ Y$ be points on $ k$, such that the arc $ BX$ is $ \frac{3}{2}$ times the arc $ BY$. The line $ MY$ intersects the line $ BX$ in $ D$ and the semicircle $ h$ in $ C$.

Show that $ Y$ ist he midpoint of  $ CD$.
2005 Regional Competition For Advanced Students
For which values of $ k$ and $ d$ has the system $ x^3+y^3=2$ and $ y=kx+d$ no real solutions $ (x,y)$?
2005 Regional Competition For Advanced Students
Prove: if an infinte arithmetic sequence ($ a_n=a_0+nd$) of positive real numbers contains two different powers of an integer $ a>1$, then the sequence contains an infinite geometric sequence ($ b_n=b_0q^n$) of real numbers.
2004 Regional Competition For Advanced Students
Determine all integers $ a$ and $ b$, so that $ (a^3+b)(a+b^3)=(a+b)^4$
2004 Regional Competition For Advanced Students
Solve the following equation for real numbers:

$ \sqrt{4-x\sqrt{4-(x-2)\sqrt{1+(x-5)(x-7)}}}=\frac{5x-6-x^2}{2}$

(all square roots are non negative)
2004 Regional Competition For Advanced Students
Given is a convex quadrilateral $ ABCD$ with $ \angle ADC=\angle BCD>90^{\circ}$.

Let $ E$ be the point of intersection of the line $ AC$ with the parallel line to $ AD$ through $ B$ and $ F$ be the point of intersection of the line $ BD$ with the parallel line to $ BC$ through $ A$. Show that $ EF$ is parallel to $ CD$
2004 Regional Competition For Advanced Students
The sequence $ < x_n >$ is defined through:

$ x_{n + 1} = \left(\frac {n}{2004} + \frac {1}{n}\right)x_n^2 - \frac {n^3}{2004} + 1$ for $ n > 0$

Let $ x_1$ be a non-negative integer smaller than $ 204$ so that all members of the sequence are non-negative integers.

Show that there exist infinitely many prime numbers in this sequence.
2003 Regional Competition For Advanced Students
Find the minimum value of the expression $ \frac{a+1}{a(a+2)}+\frac{b+1}{b(b+2)}+\frac{c+1}{c(c+2)}$, where $ a,b,c$ are positive real numbers with $ a+b+c \le 3$.
2003 Regional Competition For Advanced Students
Find all prime numbers $ p$ with $ 5^p+4p^4$ is the square of an integer.
2003 Regional Competition For Advanced Students
Given are two parallel lines $ g$ and $ h$ and a point $ P$, that lies outside of the corridor bounded by $ g$ and $ h$. Construct three lines $ g_1$, $ g_2$ and $ g_3$ through the point $ P$. These lines intersect $ g$ in $ A_1,A_2, A_3$ and $ h$ in $ B_1, B_2, B_3$ respectively. Let $ C_1$ be the intersection of the lines $ A_1B_2$ and $ A_2B_1$, $ C_2$ be the intersection of the lines $ A_1B_3$ and $ A_3B_1$ and let $ C_3$ be the intersection of the lines $ A_2B_3$ and $ A_3B_2$. Show that there exists exactly one line $ n$, that contains the points $ C_1,C_2,C_3$ and that $ n$ is parallel to $ g$ and $ h$.
2003 Regional Competition For Advanced Students
For every real number $ b$ determine all real numbers $ x$ satisfying $ x-b= \sum_{k=0}^{\infty}x^k$.
2002 Regional Competition For Advanced Students
Find the smallest natural number $x> 0$ so that all following fractions are simplified

$\frac{3x+9}{8},\frac{3x+10}{9},\frac{3x+11}{10},...,\frac{3x+49}{48}$ , i.e. numerators and denominators are relatively prime.
2002 Regional Competition For Advanced Students
Solve the following system of equations over the real numbers:

$2x_1 = x_5 ^2 - 23$

$4x_2 = x_1 ^2 + 7$

$6x_3 = x_2 ^2 + 14$

$8x_4 = x_3 ^2 + 23$

$10x_5 = x_4 ^2 + 34$
2002 Regional Competition For Advanced Students
In the convex  $ABCDEF$ (has all interior angles less than $180^o$) with the perimeter $s$ the triangles $ACE$ and $BDF$ have perimeters $u$ and $v$ respectively.

a) Show the inequalities $\frac{1}{2} \le   \frac{s}{u+v}\le 1$

b) Check whether $1$ is replaced by a smaller number or $1/2$ by a larger number can the inequality remains valid for all convex hexagons.
2002 Regional Competition For Advanced Students
Let $a_0, a_1, ..., a_{2002}$ be real numbers.

a) Show  that the smallest of the values $a_k (1-a_{2002-k})$ ($0 \le k \le 2002$) the following applies:

it is smaller or equal to $1/4$.

b) Does this statement always apply to the smallest of the values $a_k (1-a_{2003-k})$ ($1 \le k \le 2002$) ?

c) Show for positive real numbers $a_0, a_1, ..., a_{2002}$ :

the smallest of the values  $a_k (1-a_{2003-k})$ ($1 \le k \le 2002$) is less than or equal to $1/4$.
2001 Regional Competition For Advanced Students
Let $n$ be an integer. We consider $s (n)$, the sum of the $2001$ powers of $n$ with the exponents $0$ to $2000$. So $s (n) = \sum_{k=0}^{2000}n ^k$ . What is the unit digit of $s (n)$ in the decimal system?
2001 Regional Competition For Advanced Students
Find all real solutions to the equation

$(x+1)^{2001}+(x+1)^{2000}(x-2)+(x+1)^{1999}(x-2)^2+...+(x+1)^2(x-2)^{1999}+(x+1)(x-2)^{2000}+(x-2)^{2001}=0$
2001 Regional Competition For Advanced Students
In a convex pentagon $ABCDE$, the area of the triangles $ABC, ABD, ACD$ and $ADE$  are equal and have the value $F$. What is the area of the triangle $BCE$ ?
2001 Regional Competition For Advanced Students
Let $A_o =\{1, 2\}$ and for $n> 0, A_n$ results from $A_{n-1}$ by adding the natural numbers to $A_{n-1}$ which can be represented as the sum of two different numbers from $A_{n-1}$. Let  $a_n = |A_n |$ be the number of numbers in $A_n$. Determine $a_n$ as a function of  $n$.
1970 Regional Competition For Advanced Students
Let $x,y,z$ be positive real numbers such that $x+y+z=1$  Prove that always $\left( 1+\frac1x\right)\times\left(1+\frac1y\right)\times\left(1 +\frac1z\right)\ge 64$

When does equality hold?
1970 Regional Competition For Advanced Students
In the plane seven different points $P_1, P_2, P_3, P_4, Q_1, Q_2, Q_3$ are given.  The points $P_1, P_2, P_3, P_4$ are on the straight line $p$, the points $Q_1, Q_2, Q_3$ are not on $p$.  By each of the three points $Q_1, Q_2, Q_3$ the perpendiculars are drawn on the straight lines connecting points different of them. Prove that the maximum's number of the possibles intersections of all perpendiculars is to 286, if the points $Q_1, Q_2, Q_3$ are taken in account as intersections.
1970 Regional Competition For Advanced Students
$E_1$ and $E_2$ are parallel planes and their distance is $p$.

(a) How long is the seitenkante of the regular octahedron such that a side lies in $E_1$ and another in $E_2$?

(b) $E$ is a plane between $E_1$ and $E_2$, parallel to $E_1$ and $E_2$, so that its distances from $E_1$ and $E_2$ are in ratio $1:2$

Draw the intersection figure of $E$ and the octahedron for $P=4\sqrt{\frac32}$ cm and justifies, why the that figure must look in such a way
1970 Regional Competition For Advanced Students
Find all real solutions of the following set of equations:

$$72x^3+4xy^2=11y^3$$

$$27x^5-45x^4y-10x^2y^3=\frac{-143}{32}y^5$$
2015 Azerbaijan National Olympiad
Let $a,b$ and $c$ be positive reals such that $abc=\frac{1}{8}$.Then prove that $$a^2+b^2+c^2+a^2b^2+a^2c^2+b^2c^2\ge\frac{15}{16}$$
2015 Azerbaijan National Olympiad
Let $a,b$ and $c$ be the length of sides of a triangle.Then prove that $S\le\frac{a^2+b^2+c^2}{6}$ where $S$ is the area of triangle.