MD-MO 2002

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?

Let $a,b,c\geq 0$ such that $a+b+c=1$ . Prove that:

$a^2+b^2+c^2\geq 4(ab+bc+ca)-1$

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.

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?

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 $B$ .

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?

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.

Prove that there are infinitely many triplets $(a,b,c)$ that satisfy the following equalities:

$\dfrac{2a-b+6}{4a+c+2}=\dfrac{b-2c}{a-c}=\dfrac{2a+b+2c-2}{6a+2c-2}$

Find all real solutions of the equation: $[x]+[x+\dfrac{1}{2}]+[x+\dfrac{2}{3}]=2002$

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 $9$ .

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 $AB$ .

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 ?

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?

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.

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.

Let $x\in \mathbb R$ . Find the minimum and maximum values of the expresion:

$E=\dfrac{(1+x)^8+16x^4}{(1+x^2)^4}$

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 $|a| + |b| + |c|$ .

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 $3$ ?

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.

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 $DA$ .

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 $1$ .

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.

Let $a,b,c>0$ . Prove that:

$\dfrac{a}{2a+b}+\dfrac{b}{2b+c}+\dfrac{c}{2c+a}\leq 1$

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 $C_2$ .

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?

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.

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.

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.

Find all triplets of primes in the form $(p, 2p+1, 4p+1)$ .

Let $a,b,c\in \mathbb R$ such that $a\ge b\ge c > 1$ . Prove the inequality:

$\log_c\log_c b + \log_b\log_b a + \log_a\log_a c\geq 0$

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.

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 $A$ .

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 $\lim_{n\rightarrow \infty} na_n=1$

For every nonnegative integer $n$ and every real number $x$ prove the inequality:

$|\cos x|+|\cos 2x|+\ldots+|\cos 2^nx|\geq \dfrac{n}{2\sqrt{2}}$

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.

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

$b=\sqrt[2003]{a_1^{2003}+a_2^{2003}+\ldots+a_n^{2003} }$

Solve in $\mathbb R$ the equation $\sqrt{1-x}=2x^2-1+2x\sqrt{1-x^2}$ .

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 $1$ ?

Let $a,b> 0$ such that $a\ne b$ . Prove that:

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

The circumradius of a tetrahedron $ABCD$ is $R$ , and the lenghts of the segments connecting the vertices $A,B,C,D$ with the centroids of the opposite faces are equal to $m_a,m_b,m_c$ and $m_d$ , respectively. Prove that:

$m_a+m_b+m_c+m_d\leq \dfrac{16}{3}R$