UA-UMO 1997

There are four apples on a plate, of weights $600,400,300,$ and $250$ grammes. Two pupils take an apple each (one by one) and start eating simultaneously. A pupil can take another apple only after he has eaten the previous apple. It is assumed that they equaly fast (measured in grammes per second). How should the first pupil do in order to eat as large amount as possible?

Solve in real numbers the equation $9^x+4^x+1=6^x+3^x+2^x.$

The incircle of a triangle $ABC$ is tangent to its sides $AB,BC,CA$ at $M,N,K,$ respectively. A line $l$ through the midpoint $D$ of $AC$ is parallel to $MN$ and intersects the lines $BC$ and $BD$ at $T$ and $S$ , respectively. Prove that $TC=KD=AS.$

Does there exist a positvie integer $a$ such that all the numbers $a,2a,3a,...,1997a$ are perfect powers (i.e. numbers of the form $m^k$ , where $m,k \in \mathbb{N}$ , $k \ge 2$ )?

Find the smallest natural number which can be represented in the form $19m+97n$ , where $m,n$ are positive integers, in at least two different ways.

The cells of a rectangular board are colored in the chess manner. Every cell is filled with an integer. Assume that the sum of numbers in any row and in any column is even. Prove that the sum of numbers in the black cells is even.

Triangles $ABC$ and $A_1 B_1 C_1$ are non-congruent, but $AC=A_1 C_1=b,$ $BC=B_1 C_1=a$ , and $BH=B_1 H_1$ , where $BH$ and $B_1 H_1$ are the altitudes. Prove the inequality:

$a \cdot AB+b \cdot A_1 B_1 \le \sqrt{2}(a^2+b^2).$

For every natural number $n$ prove the inequality:

$\sqrt{2+\sqrt{2+\sqrt{2+...+\sqrt{2}}}}+\sqrt{6+\sqrt{6+\sqrt{6+...+\sqrt{6}}}}<5$

where there are $n$ twos and $n$ sixs on the left hand side.

Find the maximum value of the function:

$f(x)=\frac{x}{x^2+9}+\frac{1}{x^2-6x+21}+\cos {2 \pi x}, x>0.$

Each side of an equilateral triangle is divided into $1997$ equal parts, and each vertex of the triangle is joined by a segment to each of the division points on the opposite side. All the intersection points of these segments inside the triangle are marked. How many marked points are there?

A sequence is defined by $a_1=a_2=a_3=1$ and $a_{n+3}=-a_n-a_{n+1}$ for all $n \in \mathbb{N}$ . Prove that this sequence is not bounded , i.e. that for every $M \in \mathbb{R}$ there is an $n$ for which $|a_n|>M.$

Two regular pentagons $ABCDE$ and $AEKPL$ are placed in space so that $\angle DAK=60^{\circ}$ . Prove that the planes $ACK$ and $BAL$ are perpendicular.

Solve in the real numbers the system:

$x_1+x_2+...+x_{1997}=1997,$

$x_1^4+x_2^4+...+x_{1997}^4=x_1^3+x_2^3+...+x_{1997}^3.$

In a parallelogram $ABCD$ , $M$ is the midpoint of $BC$ and $N$ an arbitrary point on the side $AD$ . Let $P$ be the intersection of $MN$ and $AC$ , and $Q$ the intersection of $AM$ and $BN$ . Prove that the triangles $BDQ$ and $DMP$ have equal areas.

Let $d(n)$ denote the largest odd divisor of a positive integer $n$ . The function $f: \mathbb{N} \rightarrow \mathbb{N}$ is defined by $f(2n-1)=2^n$ and $f(2n)=n+\frac{2n}{d(n)}$ for all $n \in \mathbb{N}$ . Find all natural numbers $k$ such that: $f(f(...f(1)...))=1997.$ (where the paranthesis appear $k$ times)

Let $ABCD$ be a parallelogram with $AB=1$ . Suppose that $K$ is a point on the side $AD$ such that $KD=1, \angle ABK=90^{\circ}$ and $\angle DBK=30^{\circ}$ . Determine $AD$ .

Prove that among any four distinct numbers from the interval $(0,\frac{\pi}{2})$ there are two, say $x,y,$ such that:

$8 \cos x \cos y \cos (x-y)+1>4(\cos ^2 x+\cos ^2 y).$

In space are given $1997$ points. Let $M$ and $m$ be the largest and smallest distance between two of the given points. Prove that $\frac{M}{m}>9$ .

The equation $ax^3+bx^2+cx+d=0$ has three distinct solutions. How many distinct solutions does the following equation have:

$4(ax^3+bx^2+cx+d)(3ax+b)=(3ax^2+2bx+c)^2?$

Find all real solutions to the equation:

$(2+\sqrt{3})^x+1=\left( 2\sqrt{2+\sqrt{3}} \right)^x$ .

Find all functions $f: \mathbb{Q}^{+} \rightarrow \mathbb{Q}^{+}$ which satisfy the conditions:

$(i)$ $f(x+1)=f(x)+1$ for all $x \in \mathbb{Q}^{+}$

$(ii)$ $f(x^2)=f(x)^2$ for all $x \in \mathbb{Q}^{+}$ .

Determine the smallest natural number $n$ such that among any $n$ integers one can choose $18$ integers whose sum is divisible by $18$ .

On the edges $AB,BC,CD,DA$ of a parallelepiped $ABCDA_1 B_1 C_1 D_1$ points $K,L,M,N$ are selected, respectively. Prove that the circumcenters of the tetrahedra $A_1 AKN, B_1 BKL, C_1 CLM, D_1 DMN$ are vertices of a parallelogram.