Navvye/Olympiad-Problems-CSV huggingface dataset:
1991 Mexico National Olympiad
The sum of squares of two consecutive integers can be a square, as in $3^2+4^2 =5^2$. Prove that the sum of squares of $m$ consecutive integers cannot be a square for $m = 3$ or $6$ and find an example of $11$ consecutive integers the sum of whose squares is a square.
1991 Mexico National Olympiad
Given an $n$-gon ($n\ge 4$), consider a set $T$ of triangles formed by vertices of the polygon having the following property: Every two triangles in T have either two common vertices, or none. Prove that $T$ contains at most $n$ triangles.
1990 Mexico National Olympiad
How many paths are there from$ A$ to the line $BC$ if the path does not go through any vertex twice and always moves to the left?
1990 Mexico National Olympiad
$ABC$ is a triangle with $\angle B = 90^o$ and altitude $BH$. The inradii of $ABC, ABH, CBH$ are $r, r_1, r_2$. Find a relation between them.
1990 Mexico National Olympiad
Show that $n^{n-1}-1$ is divisible by$ (n-1)^2$ for $n > 2$.
1990 Mexico National Olympiad
Find $0/1 + 1/1 + 0/2 + 1/2 + 2/2 + 0/3 + 1/3 + 2/3 + 3/3 + 0/4 + 1/4 + 2/4 + 3/4 + 4/4 + 0/5 + 1/5 + 2/5 + 3/5 + 4/5 + 5/5 + 0/6 + 1/6 + 2/6 + 3/6 + 4/6 + 5/6 + 6/6$
1990 Mexico National Olympiad
Given $19$ points in the plane with integer coordinates, no three collinear, show that we can always find three points whose centroid has integer coordinates.
1990 Mexico National Olympiad
$ABC$ is a triangle with $\angle C = 90^o$. $E$ is a point on $AC$, and $F$ is the midpoint of $EC$. $CH$ is an altitude. $I$ is the circumcenter of $AHE$, and $G$ is the midpoint of $BC$. Show that $ABC$ and $IGF$ are similar.
1989 Mexico National Olympiad
In a triangle $ABC$ the area is $18$, the length $AB$ is $5$, and the medians from $A$ and $B$ are orthogonal. Find the lengths of the sides $BC,AC$.
1989 Mexico National Olympiad
Find two positive integers $a,b$ such that $a | b^2, b^2 | a^3, a^3 | b^4, b^4 | a^5$, but $a^5$  does not divide $b^6$
1989 Mexico National Olympiad
Prove that there is no $1989$-digit natural number at least three of whose digits are equal to $5$ and such that the product of its digits equals their sum.
1989 Mexico National Olympiad
Find the smallest possible natural number $n = \overline{a_m ...a_2a_1a_0} $ (in decimal system) such that the number $r =  \overline{a_1a_0a_m ..._20} $ equals $2n$.
1989 Mexico National Olympiad
Let $C_1$ and $C_2$ be two tangent unit circles inside a circle $C$ of radius $2$. Circle $C_3$ inside $C$ is tangent to the circles $C,C_1,C_2$, and circle $C_4$ inside $C$ is tangent to $C,C_1,C_3$. Prove that the centers of $C,C_1,C_3$ and $C_4$ are vertices of a rectangle.
1989 Mexico National Olympiad
Determine the number of paths from $A$ to $B$ on the picture that go along gridlines only, do not pass through any

point twice, and never go upwards?
1988 Mexico National Olympiad
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?
1988 Mexico National Olympiad
If $a$ and $b$ are positive integers, prove that $11a+2b$ is a multiple of $19$ if and only if so is $18a+5b$ .
1988 Mexico National Olympiad
Two externally tangent circles with different radii are given. Their common tangents form a triangle. Find the area of this triangle in terms of the radii of the two circles.
1988 Mexico National Olympiad
In how many ways can one select eight integers $a_1,a_2, ... ,a_8$, not necesarily distinct, such that $1 \le  a_1 \le ... \le a_8 \le 8$?
1988 Mexico National Olympiad
If $a$ and $b$ are coprime positive integers and $n$ an integer, prove that the greatest common divisor of $a^2+b^2-nab$ and $a+b$ divides $n+2$.
1988 Mexico National Olympiad
Consider two fixed points $B,C$ on a circle $w$. Find the locus of the incenters of all triangles $ABC$ when point $A$ describes $w$.
1988 Mexico National Olympiad
Two disjoint subsets of the set $\{1,2, ... ,m\}$ have the same sums of elements. Prove that each of the subsets $A,B$ has less than $m / \sqrt2$ elements.
1988 Mexico National Olympiad
Compute the volume of a regular octahedron circumscribed about a sphere of radius $1$.
1987 Mexico National Olympiad
Prove that if the sum of two irreducible fractions is an integer then the two fractions have the same denominator.
1987 Mexico National Olympiad
How many positive divisors does number $20!$ have?
1987 Mexico National Olympiad
Consider two lines $\ell$ and $\ell ' $ and a fixed point $P$ equidistant from these lines. What is the locus of projections $M$ of $P$ on $AB$, where $A$ is on  $\ell $, $B$ on  $\ell ' $, and angle $\angle APB$ is right?
1987 Mexico National Olympiad
Calculate the product of all positive integers less than $100$ and having exactly three positive divisors. Show that this product is a square.
1987 Mexico National Olympiad
In a right triangle $ABC$, M is a point on the hypotenuse $BC$ and $P$ and $Q$ the projections of $M$ on $AB$ and $AC$ respectively. Prove that for no such point $M$ do the triangles $BPM, MQC$ and the rectangle $AQMP$ have the same area.
1987 Mexico National Olympiad
Prove that for every positive integer n the number $(n^3 -n)(5^{8n+4} +3^{4n+2})$ is a multiple of $3804$.
1987 Mexico National Olympiad
Show that the fraction $ \frac{n^2+n-1}{n^2+2n}$ is irreducible for every positive integer n.
1987 Mexico National Olympiad
(a) Three lines $l,m,n$ in space pass through point $S$. A plane perpendicular to $m$ intersects $l,m,n $ at $A,B,C$ respectively. Suppose that $\angle ASB = \angle BSC = 45^o$ and $\angle ABC = 90^o$. Compute $\angle ASC$.

(b) Furthermore, if a plane perpendicular to $l$ intersects  $l,m,n$  at $P,Q,R$ respectively and $SP = 1$, find the sides of triangle $PQR$.
2008 Moldova MO 11-12
Consider the equation $ x^4 - 4x^3 + 4x^2 + ax + b = 0$, where $ a,b\in\mathbb{R}$. Determine the largest value $ a + b$ can take, so that the given equation has two distinct positive roots $ x_1,x_2$ so that $ x_1 + x_2 = 2x_1x_2$.
2008 Moldova MO 11-12
Find the exact value of $ E=\displaystyle\int_0^{\frac\pi2}\cos^{1003}x\text{d}x\cdot\int_0^{\frac\pi2}\cos^{1004}x\text{d}x\cdot$.
2008 Moldova MO 11-12
In the usual coordinate system $ xOy$ a line $ d$ intersect the circles $ C_1:$ $ (x+1)^2+y^2=1$ and $ C_2:$ $ (x-2)^2+y^2=4$ in the points $ A,B,C$ and $ D$ (in this order). It is known that $ A\left(-\frac32,\frac{\sqrt3}2\right)$ and $ \angle{BOC}=60^{\circ}$. All the $ Oy$ coordinates of these $ 4$ points are positive. Find the slope of $ d$.
2008 Moldova MO 11-12
Define the sequence $ (a_p)_{p\ge0}$ as follows: $ a_p=\displaystyle\frac{\binom p0}{2\cdot 4}-\frac{\binom p1}{3\cdot5}+\frac{\binom p2}{4\cdot6}-\ldots+(-1)^p\cdot\frac{\binom pp}{(p+2)(p+4)}$.

Find $ \lim_{n\to\infty}(a_0+a_1+\ldots+a_n)$.
2008 Moldova MO 11-12
Find the least positive integer $ n$ so that the polynomial $ P(X)=\sqrt3\cdot X^{n+1}-X^n-1$ has at least one root of modulus $ 1$.
2008 Moldova MO 11-12
Find $ \lim_{n\to\infty}a_n$ where $ (a_n)_{n\ge1}$ is defined by $ a_n=\frac1{\sqrt{n^2+8n-1}}+\frac1{\sqrt{n^2+16n-1}}+\frac1{\sqrt{n^2+24n-1}}+\ldots+\frac1{\sqrt{9n^2-1}}$.
2008 Moldova MO 11-12
Triangle $ ABC$ has fixed vertices $ B$ and $ C$, so that $ BC = 2$ and $ A$ is variable. Denote by $ H$ and $ G$ the orthocenter and the centroid, respectively, of triangle $ ABC$. Let $ F\in(HG)$ so that $ \frac {HF}{FG} = 3$. Find the locus of the point $ A$ so that $ F\in BC$.
2008 Moldova MO 11-12
Evaluate $ \displaystyle I = \int_0^{\frac\pi4}\left(\sin^62x + \cos^62x\right)\cdot \ln(1 + \tan x)\text{d}x$.
2006 Moldova MO 11-12
Let $n\in\mathbb{N}^*$. Prove that $$ \lim_{x\to 0}\frac{ \displaystyle (1+x^2)^{n+1}-\prod_{k=1}^n\cos kx}{ \displaystyle x\sum_{k=1}^n\sin kx}=\frac{2n^2+n+12}{6n}.  $$
2006 Moldova MO 11-12
Function $f: [a,b]\to\mathbb{R}$, $0<a<b$ is continuous on $[a,b]$ and differentiable on $(a,b)$. Prove that there exists $c\in(a,b)$ such that  $$ f'(c)=\frac1{a-c}+\frac1{b-c}+\frac1{a+b}.  $$
2006 Moldova MO 11-12
On each of the 2006 cards a natural number is written. Cards are placed arbitrarily in a row. 2 players take in turns a card from any end of the row until all the cards are taken. After that each player calculates sum of the numbers written of his cards. If the sum of the first player is not less then the sum of the second one then the first player wins. Show that there's a winning strategy for the first player.
2006 Moldova MO 11-12
Let $ABCDE$ be a right quadrangular pyramid with vertex $E$ and height $EO$. Point $S$ divides this height in the ratio $ES: SO=m$. In which ratio does the plane $(ABC)$ divide the lateral area of the pyramid.
2006 Moldova MO 11-12
Let $n\in\mathbb{N}^*$. Solve the equation $\sum_{k=0}^n C_n^k\cos2kx=\cos nx$ in $\mathbb{R}$.
2006 Moldova MO 11-12
Sequences $(x_n)_{n\ge1}$, $(y_n)_{n\ge1}$ satisfy the relations $x_n=4x_{n-1}+3y_{n-1}$ and $y_n=2x_{n-1}+3y_{n-1}$ for $n\ge1$. If $x_1=y_1=5$ find $x_n$ and $y_n$.



Calculate $\lim_{n\rightarrow\infty}\frac{x_n}{y_n}$.
2006 Moldova MO 11-12
Let $n\in\mathbb{N}^*$. $2n+3$ points on the plane are given so that no 3 lie on a line and no 4 lie on a circle. Is it possible to find 3 points so that the interior of the circle passing through them would contain exactly $n$ of the remaining points.
2006 Moldova MO 11-12
Given an alfabet of $n$ letters. A sequence of letters such that between any 2 identical letters there are no 2 identical letters is called a word.



a) Find the maximal possible length of a word.



b) Find the number of the words of maximal length.
2008 Moldova National Olympiad
Let $ f_m: \mathbb R \to \mathbb R$, $ f_m(x)=(m^2+m+1)x^2-2(m^2+1)x+m^2-m+1,$ where $ m \in \mathbb R$.



1) Find the fixed common point of all this parabolas.



2) Find $ m$ such that the distance from that fixed point to $ Oy$ is minimal.
2008 Moldova National Olympiad
Find $ f(x): (0,+\infty) \to \mathbb R$ such that $$ f(x)\cdot f(y) + f(\frac{2008}{x})\cdot f(\frac{2008}{y})=2f(x\cdot y)$$ and $ f(2008)=1$ for $ \forall x \in (0,+\infty)$.
2008 Moldova National Olympiad
From the vertex $ A$ of the equilateral triangle $ ABC$ a line is drown that intercepts the segment $ [BC]$ in the point $ E$. The point $ M \in (AE$ is such that $ M$ external to $ ABC$, $ \angle AMB = 20 ^\circ$ and $ \angle AMC = 30 ^ \circ$. What is the measure of the angle $ \angle MAB$?
2008 Moldova National Olympiad
Let $ n$ be a positive integer. Find all $ x_1,x_2,\ldots,x_n$ that satisfy the relation:

$$ \sqrt{x_1-1}+2\cdot \sqrt{x_2-4}+3\cdot \sqrt{x_3-9}+\cdots+n\cdot\sqrt{x_n-n^2}=\frac{1}{2}(x_1+x_2+x_3+\cdots+x_n).$$
2007 Moldova National Olympiad
Define the sequence $(x_{n})$: $x_{1}=\frac{1}{3}$ and $x_{n+1}=x_{n}^{2}+x_{n}$. Find $\left[\frac{1}{x_{1}+1}+\frac{1}{x_{2}+1}+\dots+\frac{1}{x_{2007}+1}\right]$, wehere $[$ $]$ denotes the integer part.
2007 Moldova National Olympiad
Define $a_{n}$ as satisfying: $\left(1+\frac{1}{n}\right)^{n+a_{n}}=e$.

Find $\lim_{n\rightarrow\infty}a_{n}$.
2007 Moldova National Olympiad
$ABCDA_{1}B_{1}C_{1}D_{1}$ is a cube with side length $4a$. Points $E$ and $F$ are taken on $(AA_{1})$ and $(BB_{1})$ such that $AE=B_{1}F=a$. $G$ and $H$ are midpoints of $(A_{1}B_{1})$ and $(C_{1}D_{1})$, respectively.

Find the minimum value of the $CP+PQ$, where $P\in[GH]$ and $Q\in[EF]$.
2007 Moldova National Olympiad
The function $f: \mathbb{R}\rightarrow\mathbb{R}$ satisfies $f(\textrm{cot}x)=\sin2x+\cos2x$, for any $x\in(0,\pi)$. Find the minimum and maximum value of $g: [-1;1]\rightarrow\mathbb{R}$, $g(x)=f(x)\cdot f(1-x)$.
2007 Moldova National Olympiad
Real numbers $a_{1},a_{2},\dots,a_{n}$ satisfy $a_{i}\geq\frac{1}{i}$, for all $i=\overline{1,n}$. Prove the inequality:

$$\left(a_{1}+1\right)\left(a_{2}+\frac{1}{2}\right)\cdot\dots\cdot\left(a_{n}+\frac{1}{n}\right)\geq\frac{2^{n}}{(n+1)!}(1+a_{1}+2a_{2}+\dots+na_{n}).$$
2007 Moldova National Olympiad
Define $(b_{n})$ to be: $b_{0}=12$, $b_{1}=\frac{\sqrt{3}}{2}$ adn $b_{n+1}+b_{n-1}=b_{n}\cdot\sqrt{3}$. Find $b_{0}+b_{1}+\dots+b_{2007}$.



Note. Maybe this seems too easy, but I want to post all the problems...
2007 Moldova National Olympiad
Given a tetrahedron $VABC$ with edges $VA$, $VB$ and $VC$ perpendicular any two of them. The sum of the lengths of the tetrahedron's edges is $3p$. Find the maximal volume of $VABC$.
2007 Moldova National Olympiad
The continuous function and twice differentiable function $f: \mathbb{R}\rightarrow\mathbb{R}$ satisfies $2007^{2}\cdot f(x)+f''(x)=0$. Prove that there exist two such real numbers $k$ and $l$ such that $f(x)=l\cdot\sin(2007x)+k\cdot\cos(2007x)$.
2006 Moldova National Olympiad
Let $a,b$ be the smaller sides of a right triangle. Let $c$ be the hypothenuse and $h$ be the altitude from the right angle. Fint the maximal value of $\frac{c+h}{a+b}$.
2006 Moldova National Olympiad
Let $n$ be a positive integer, $n\geq 2$. Let $M=\{0,1,2,\ldots n-1\}$. For an integer nonzero number $a$ we define the function $f_{a}: M\longrightarrow M$, such that $f_{a}(x)$ is the remainder when dividing $ax$ at $n$. Find a necessary and sufficient condition such that $f_{a}$ is bijective. And if $f_{a}$ is bijective and $n$ is a prime number, prove that $a^{n(n-1)}-1$ is divisible by $n^{2}$.
2006 Moldova National Olympiad
A convex quadrilateral $ ABCD$ is inscribed in a circle. The tangents to the circle through $ A$ and $ C$ intersect at a point $ P$, such that this point $ P$ does not lie on $ BD$, and such that $ PA^{2}=PB\cdot PD$. Prove that the line $ BD$ passes through the midpoint of $ AC$.
2006 Moldova National Olympiad
Find all real values of the real parameter $a$ such that the equation $$ 2x^{2}-6ax+4a^{2}-2a-2+\log_{2}(2x^{2}+2x-6ax+4a^{2})= $$ $$ =\log_{2}(x^{2}+2x-3ax+2a^{2}+a+1). $$

has a unique solution.
2006 Moldova National Olympiad
Let $x_{1}$, $x_{2}$, $\ldots$, $x_{n}$ be $n$ real numbers in $\left(\frac{1}{4},\frac{2}{3}\right)$. Find the minimal value of the expression: $$ \log_{\frac 32x_{1}}\left(\frac{1}{2}-\frac{1}{36x_{2}^{2}}\right)+\log_{\frac 32x_{2}}\left(\frac{1}{2}-\frac{1}{36x_{3}^{2}}\right)+\cdots+ \log_{\frac 32x_{n}}\left(\frac{1}{2}-\frac{1}{36x_{1}^{2}}\right). $$
2006 Moldova National Olympiad
Let a triangle $ABC$ satisfy $AC = BC$; in other words, let $ABC$ be an isosceles triangle with base $AB$. Let $P$ be a point inside the triangle $ABC$ such that $\angle PAB = \angle PBC$. Denote by $M$ the midpoint of the segment $AB$. Show that $\angle APM + \angle BPC = 180^{\circ}$.
2006 Moldova National Olympiad
Consider an octogon with equal angles and rational side lengths. Prove that it has a symmetry center.
2006 Moldova National Olympiad
Let $M=\{x^2+x \mid x\in \mathbb N^{\star} \}$. Prove that for every integer $k\geq 2$ there exist elements $a_{1}, a_{2}, \ldots, a_{k},b_{k}$ from $M$, such that $a_{1}+a_{2}+\cdots+a_{k}=b_{k}$.
2002 Moldova National Olympiad
Before going to vacation, each of the $ 7$ pupils decided to send to each of the  $ 3$ classmates one postcard. Is it possible that each student receives postcards only from the classmates he has sent postcards?
2002 Moldova National Olympiad
Let $ a,b,c\geq 0$ such that $ a+b+c=1$. Prove that:



$ a^2+b^2+c^2\geq 4(ab+bc+ca)-1$
2002 Moldova National Olympiad
Consider an angle $ \angle DEF$, and the fixed points $ B$ and $ C$ on the semiline $ (EF$ and the variable point $ A$ on $ (ED$. Determine the position of $ A$ on $ (ED$ such that the sum $ AB+AC$ is minimum.
2002 Moldova National Olympiad
Twelve teams participated in a soccer tournament. According to the rules, one team gets $ 2$ points for a victory, $ 1$ point for a draw and $ 0$ points for a defeat. When the tournament was over, all teams had distinct numbers of points, and the team ranked second had as many points as the teams ranked on the last five places in total. Who won the match between the fourth and the eighth place teams?
2002 Moldova National Olympiad
Volume $ A$ equals one fourth of the sum of the volumes $ B$ and $ C$, while volume $ B$ equals one sixth of the sum of the volumes $ C$ and $ A$.

Find the ratio of the volume $ C$ to the sum of the volumes $ A$ and $ B$.
2002 Moldova National Olympiad
Five parcels of land are given. In each step, we divide one parcel into three or four smaller ones. Assume that, after several steps, the number of obtained parcels equals four times the number of steps made. How many steps were performed?
2002 Moldova National Olympiad
In a triangle $ ABC$, the angle bisector at $ B$ intersects $ AC$ at $ D$ and the circumcircle again at $ E$. The circumcircle of the triangle $ DAE$ meets the segment $ AB$ again at $ F$. Prove that the triangles $ DBC$ and $ DBF$ are congruent.
2002 Moldova National Olympiad
Prove that there are infinitely many triplets $ (a,b,c)$ that satisfy the following equalities:



$ \dfrac{2a-b+6}{4a+c+2}=\dfrac{b-2c}{a-c}=\dfrac{2a+b+2c-2}{6a+2c-2}$
2002 Moldova National Olympiad
Find all real solutions of the equation: $ [x]+[x+\dfrac{1}{2}]+[x+\dfrac{2}{3}]=2002$
2002 Moldova National Olympiad
Given a positive integer $ k$, there is a positive integer $ n$ with the property that one can obtain the sum of the first $ n$ positive integers by writing some $ k$ digits to the right of $ n$. Find the remainder of $ n$ when dividing at $ 9$.
2002 Moldova National Olympiad
Consider a circle $ \Gamma(O,R)$ and a point $ P$ found in the interior of this circle. Consider a chord $ AB$ of $ \Gamma$ that passes through $ P$. Suppose that the tangents to $ \Gamma$ at the points $ A$ and $ B$ intersect at $ Q$. Let $ M\in QA$ and $ N\in QB$ s.t. $ PM\perp QA$ and $ PN\perp QB$. Prove that the value of $ \frac {1}{PN} + \frac {1}{PM}$ doesn't depend of choosing the chord $ AB$.
2002 Moldova National Olympiad
All the internal phone numbers in a certain company have four digits. The director wants the phone numbers of the administration offices to consist of digits $ 1$, $ 2$, $ 3$ only, and that any of these phone numbers coincide in at most one position. What is the maximum number of distinct phone numbers that these offices can have ?
2002 Moldova National Olympiad
Several pupils wrote a solution of a math problem on the blackboard on the break. When the teacher came in, a pupil was just clearing the blackboard, so the teacher could only observe that there was a rectangle with the sides of integer lenghts and a diagonal of lenght $ 2002$. Then the teacher pointed out that there was a computation error in pupils' solution. Why did he conclude that?
2002 Moldova National Olympiad
From a set of consecutive natural numbers one number is excluded so that the aritmetic mean of the remaining numbers is $ 50.55$. Find the initial set of numbers and the excluded number.
2002 Moldova National Olympiad
In a triangle $ ABC$, the bisectors of the angles at $ B$ and $ C$ meet the opposite sides $ B_1$ and $ C_1$, respectively. Let $ T$ be the midpoint $ AB_1$. Lines $ BT$ and $ B_1C_1$ meet at $ E$ and lines $ AB$ and $ CE$ meet at $ L$. Prove that the lines $ TL$ and $ B_1C_1$ have a point in common.
2002 Moldova National Olympiad
Let $ x\in \mathbb R$. Find the minimum and maximum values of the expresion:



$ E=\dfrac{(1+x)^8+16x^4}{(1+x^2)^4}$
2002 Moldova National Olympiad
Consider the real numbers $ a\ne 0,b,c$ such that the function $ f(x) = ax^2 + bx + c$ satisfies $ |f(x)|\le 1$ for all $ x\in [0,1]$. Find the greatest possible value of $ |a| + |b| + |c|$.
2002 Moldova National Olympiad
Does there exist a positive integer $ n>1$ such that $ n$ is a power of $ 2$ and one of the numbers obtained by permuting its (decimal) digits is a power of $ 3$ ?
2002 Moldova National Olympiad
Prove that for any $ n\in \mathbb N$ the number $ 1+\dfrac{1}{3}+\dfrac{1}{5}+\ldots+\dfrac{1}{2n+1}$ is not an integer.
2002 Moldova National Olympiad
Let $ ABCD$ be a convex  quadrilateral and let $ N$ on side $ AD$ and $ M$ on side $ BC$ be points such that $ \dfrac{AN}{ND}=\dfrac{BM}{MC}$. The lines $ AM$ and $ BN$ intersect at $ P$, while the lines $ CN$ and $ DM$ intersect at $ Q$. Prove that if $ S_{ABP}+S_{CDQ}=S_{MNPQ}$, then either $ AD\parallel BC$ or $ N$ is the midpoint of $ DA$.
2002 Moldova National Olympiad
Integers $ a_1,a_2,\ldots a_9$ satisfy the relations $ a_{k+1}=a_k^3+a_k^2+a_k+2$ for $ k=1,2,...,8$. Prove that among these numbers there exist three with a common divisor greater than $ 1$.
2002 Moldova National Olympiad
The coefficients of the equation $ ax^2+bx+c=0$, where $ a\ne 0$, satisfy the inequality $ (a+b+c)(4a-2b+c)<0$. Prove that this equation has $ 2$ real distinct solutions.
2002 Moldova National Olympiad
Let $ a,b,c>0$. Prove that:



$ \dfrac{a}{2a+b}+\dfrac{b}{2b+c}+\dfrac{c}{2c+a}\leq 1$
2002 Moldova National Olympiad
The circles $ C_1$ and $ C_2$ with centers $ O_1$ and $ O_2$ respectively are externally tangent. Their common tangent not intersecting the segment $ O_1O_2$ touches $ C_1$ at $ A$ and $ C_2$ at $ B$. Let $ C$ be the reflection of $ A$ in $ O_1O_2$ and $ P$ be the intersection of $ AC$ and $ O_1O_2$. Line $ BP$ meets $ C_2$ again at $ L$. Prove that line $ CL$ is tangent to the circle $ C_2$.
2002 Moldova National Olympiad
We are given three nuggets of weights $ 1$ kg, $ 2$ kg and $ 3$ kg, containing different percentages of gold, and need to cut each nugget into two parts so that the obtained parts can be alloyed into two ingots of weights $ 1$ kg ande $ 5$ kg containing the same proportion of gold. How we can do that?
2002 Moldova National Olympiad
Let $ n\ge 3$ distinct non-collinear points be given on a plane. Show that there is a closed simple polygonal line passing through each point.
2002 Moldova National Olympiad
The sides $ AB$,$ BC$ and $ CA$ of the triangle $ ABC$ are tangent to the incircle of the triangle $ ABC$ with center $ I$ at the points $ C_1$,$ A_1$ and $ B_1$, respectively.Let $ B_2$ be the midpoint of the side $ AC$.Prove that the lines $ B_1I$, $ A_1C_1$ and $ BB_2$ are concurrent.
2002 Moldova National Olympiad
In each line and column of a table $ (2n + 1)\times (2n + 1)$ are written arbitrarly the numbers $ 1,2,...,2n + 1$. It was constated that the repartition of the numbers is symmetric to the main diagonal of this table. Prove that all the numbers on the main diagonal are distinct.
2002 Moldova National Olympiad
Find all triplets of primes in the form $ (p, 2p+1, 4p+1)$.
2002 Moldova National Olympiad
Let $ a,b,c\in \mathbb R$ such that $ a\ge b\ge c > 1$. Prove the inequality:



$ \log_c\log_c b + \log_b\log_b a + \log_a\log_a c\geq 0$
2002 Moldova National Olympiad
There are $ 16$ persons in a company, each of which likes exactly $ 8$ other persons. Show that there exist two persons who like each other.
2002 Moldova National Olympiad
Let the triangle $ ADB_1$ s.t. $ m(\angle DAB_1)\ne 90^\circ$.On the sides of this triangle externally are constructed the squares $ ABCD$

and $ AB_1C_1D_1$ with centers $ O_1$ and $ O_2$, respectively.Prove that the circumcircles of the triangles $ BAB_1$, $ DAD_1$ and

$ O_1AO_2$ share a common point, that differs from $ A$.
2002 Moldova National Olympiad
The sequence $ (a_n)$ is defined by $ a_1\in (0,1)$ and $ a_{n+1}=a_n(1-a_n)$ for $ n\ge 1$.

Prove that $ \lim_{n\rightarrow \infty} na_n=1$
2002 Moldova National Olympiad
For every nonnegative integer $ n$ and every real number $ x$ prove the inequality:



$ |\cos x|+|\cos 2x|+\ldots+|\cos 2^nx|\geq \dfrac{n}{2\sqrt{2}}$
2002 Moldova National Olympiad
Let $ P$ be a polyhedron whose all edges are congruent and tangent to a sphere. Suppose that one of the facesof $ P$ has an odd number of sides. Prove that all vertices of $ P$ lie on a single sphere.
2002 Moldova National Olympiad
At least two of the nonnegative real numbers $ a_1,a_2,...,a_n$ aer nonzero. Decide whether $ a$ or $ b$ is larger if



$ a=\sqrt[2002]{a_1^{2002}+a_2^{2002}+\ldots+a_n^{2002}}$



and



$ b=\sqrt[2003]{a_1^{2003}+a_2^{2003}+\ldots+a_n^{2003} }$
2002 Moldova National Olympiad
Solve in $ \mathbb R$ the equation $ \sqrt{1-x}=2x^2-1+2x\sqrt{1-x^2}$.
2002 Moldova National Olympiad
Can a square of side $ 1024$ be partitioned into $ 31$ squares?Can a square of side $ 1023$ be partitioned into $ 30$ squares, one of which has a s side lenght not exceeding $ 1$?
2002 Moldova National Olympiad
Let $ a,b> 0$ such that $ a\ne b$. Prove that:



$ \sqrt {ab} < \dfrac{a - b}{\ln a - \ln b} < \dfrac{a + b}{2}$