# Mathematics and Youth Magazine Problems 2004

### Issue 319

1. Find solutions in positive integers of the equation $$3^{x}+1=(y+1)^{2}.$$
2. Let be given a triangle $A B C$ with $B C=2 A B$. Let $M$ be the midpoint of $B C$ and $D$ be the midpoint of $B M$. Prove that $A C=2 A D$.
3. Find the prime divisor $p<300$ of the number $2^{37}-1=137438953471$.
4. Solve the equation $$\left(16 x^{4 n}+1\right)\left(y^{4 n}+1\right)\left(z^{4 n}+1\right)=32 x^{2 n} y^{2 n} z^{2 n}$$ where $n$ is a given positive integer.
5. Let $a, b, c$ be real numbers satisfying the conditions $$a b c>0,\quad |a b+b c+c a|=2 \sqrt{2004 a b c}.$$ Prove that $$(a+b-2004)(b+c-2004)(c+a-2004) \leq 0 .$$
6. $A B C$ is an arbitrary not acute triangle. Put $A B=c$, $B C=a$, $C A=b$. Find the least value of the expression $\dfrac{(a+b)(b+c)(c+a)}{a b c}$.
7. The circle $(O, R)$ with center $O$, radius $R$ cuts the circle $\left(O^{\prime}, R\right)$ with center $O^{\prime}$, radius $R^{\prime}$ at $A$ and $B$. From a point $C$ on the opposite ray of the ray $A B$, draw the tangents $C D$ and $C E$ to the circle $(O, R)$ ($D$ and $E$ are tangent points, $E$ lies inside $\left(O^{\prime}, R^{\prime}\right)$. $A D$ and $A E$ cut again the circle $\left(O^{\prime}, R\right)$ respectively at $M$ and $N$. Prove that the line $D E$ passes through the midpoint of $M N$.
8. Consider the colourings of a rectangular board of size $m \times n(m+n \geq 3)$ such that $k$ little squares of the board are coloured and each not coloured little squares has at least a common point with a coloured little square. Find the least value of $k$.
9. Prove that $$x^{2} y+y^{2} z+z^{2} x \leq x^{3}+y^{3}+z^{3} \leq 1+\frac{1}{2}\left(x^{4}+y^{4}+z^{4}\right)$$ for non negative real numbers $x, y, z$ satisfying the condition $x+y+z=2$.
10. Find all functions $f: \mathbb N^{*} \rightarrow \mathbb N^{*}$ satisfying the condition $$2\left(f\left(m^{2}+n^{2}\right)\right)^{3}=f^{2}(m) \cdot f(n)+f^{2}(n) \cdot f(m)$$ for all distinct $m, n \in \mathbb N^{*}$.
11. Let $A D$, $B E$, $C F$ be the altitudes of an acute triangle $A B C$. Let $M$, $N$, $P$ be respectively the points of intersection of the segments $A D$ and $E F$, $B E$ and $F D$, $C F$ and $D E$. Let $S$ denote the area of triangle. Prove that $$\frac{1}{S_{A B C}} \leq \frac{S_{M N P}}{S_{D E F}^{2}} \leq \frac{1}{8 \cos A \cdot \cos B \cdot \cos C \cdot S_{A B C}}$$
12. Consider arbitrary regular quadrilateral pyramids $S A B C D$, the measure $\alpha$ of the flat angle $\widehat{A S B}$ of which satisfies $0<\alpha \leq$ $60^{\circ}$. Let $\varphi$ be the measure of the bihedral angle of side $S A$. Determine $\alpha$ so that the expression $P=\cos 3 \varphi-9 \cos \varphi$ attains its greatest value.

### Issue 320

1. Write the sum of $18$ fractions $$1, \frac{1}{2}, \frac{1}{3}, \ldots, \frac{1}{17}, \frac{1}{18}$$ in the form of an irreducible fraction $\dfrac{a}{b}$. Prove that $b$ is divisible by $2431$.
2. Given a triangle $A B C$ with $A B > A C$, the foot of its altitude $A H$ lies inside $B C$. The angled-bisectors of $\widehat{A B C}$ and of $\widehat{A C B}$ cut $A H$ respectively at $E$ and $F$. Prove that $B E>E F+F C$.
3. Find positive integers $a \geq b \geq c$ and $x \geq y \geq z$ so that $$\begin{cases} a+b+c &=x y z \\ x+y+z &=a b c \end{cases}$$
4. Solve the equation $$(x-2) \sqrt{x-1}-\sqrt{2} x+2=0.$$
5. Find the greatest value of the expression $$\sqrt{4 x-x^{3}}+\sqrt{x+x^{3}}$$ where $0 \leq x \leq 2$.
6. The circle $(O)$ with center $O$ cuts the circle $\left(O^{\prime}\right)$ with center $O^{\prime}$ at $P$ and $Q$. Their common tangent (nearer to $P$) touches $(O)$ at $A$, $\left(O^{\prime}\right)$ at $B$. Let $C$ be the point of intersection of the tangents to $(O)$ at $P$ with the circle $\left(O^{\prime}\right)$. Let $D$ be the point of intersection of the tangents to $\left(O^{\prime}\right)$ at $P$ with the circle $(O)$. Let $M$ be the point such that $A B$ and $P M$ have common midpoint. The line $A P$ cuts $B C$ at $E$ and the line $B P$ cuts $A D$ at $F$. Prove that $A M B E Q F$ is a hexagone inscribed in a circle.
7. Construct a triangle $A B C$ with given $P$, $Q$, $R$ so that $B$ is the midpoint of $A P$, $C$ is the midpoint of $B Q$, $A$ is the midpoint of $C R$.
8. Prove the following equalities for positive integer $n$.
a) $\displaystyle \sum_{k=1}^{n} \frac{(-1)^{k-1}}{2 k-1} C_{n}^{k} \cdot C_{n+k-1}^{k-1}=1$.
b) $\displaystyle \sum_{k=1}^{n} \frac{(-1)^{k-1} \cdot k^{n}}{2 k-1} C_{n}^{k}=\frac{(n !)^{2} \cdot 2^{n}}{(2 n) !}$
9. Solve the following system of equations of $n$ unknowns $$\begin{cases}\sqrt{x_{1}}+\sqrt{x_{2}}+\ldots+\sqrt{x_{n}} &=n \\ \sqrt{x_{1}+8}+\sqrt{x_{2}+8}+\ldots+\sqrt{x_{11}+8} &=3 n\end{cases}$$ ($n$ is a given positive integer). Generalize the problem.
10. Find the greatest value of the function $$f(x)=\sqrt{2} \sin x+\sqrt{15-10 \sqrt{2} \cos x}$$
11. Let $r$ and $R$ be respectively the inradius and the circumradius of triangle $A B C$. Let $p$ and $p^{\prime}$ be respectively the perimeter of $\triangle A B C$ and $\triangle A^{\prime} B^{\prime} C^{\prime}$ where $A^{\prime}$, $B^{\prime}$, $C^{\prime}$ are the touching points of $B C$, $C A$, $A B$ with the incircle. Prove that $$\frac{r}{R} \leq \frac{p^{\prime}}{p} \leq \frac{1}{2}$$
12. Let be given a cube $A B C D A_{1} B_{1} C_{1} D_{1}$ with side $A B=a$. From a point $E$ on the side $C D$ ($E$ distinct from $C$, $D$) draw a line cutting the lines $A A_{1}$ and $B_{1} C_{1}$ respectively at $M$ and $N$. From $M$ draw a line cutting the lines $B C$ and $C_{1} D_{1}$ respectively at $F$ and $P$. Determine the position of $E$ so that the perimeter of triangle $M N P$ attains its least value and calculate this least value.

### Issue 321

1. Write the number $2003^{2004}$ as a sum of positive integers. What is the remainder of the division by $3$ of the sum of the cubes of these integers?
2. Simplify the expression $$\frac{(a-2)(a-1002)}{a(a-b)(a-c)}+\frac{(b-2)(b-1002)}{b(b-a)(b-c)}+\frac{(c-2)(c-1002)}{c(c-a)(c-b)}$$ where $a, b, c$ are distinct numbers such that $a b c \neq 0$.
3. Let $a, b, c, d$ be positive numbers. Prove that
a) $\displaystyle \frac{a}{b}+\frac{b}{c}+\frac{c}{a} \geq \frac{a+b+c}{\sqrt{a b c}}$.
b) $\displaystyle \frac{a^{2}}{b^{2}}+\frac{b^{2}}{c^{2}}+\frac{c^{2}}{d^{2}}+\frac{d^{2}}{a^{2}} \geq \frac{a+b+c+d}{\sqrt{a b c d}}$.
4. Find a necessary and sufficient condition on the number $m$ so that the following system of equations has a unique solution $$\begin{cases} x^{2} &=(2+m) y^{3}-3 y^{2}+m y \\ y^{2} &=(2+m) z^{3}-3 z^{2}+m z \\ z^{2} &=(2+m) x^{3}-3 x^{2}+m x\end{cases}$$
5. Let $A B C D$ be a trapezoid inscribed in a circle with radius $R=3cm$ such that $B C=2 cm$, $A D=4cm$. Let $M$ be the point on side $A B$ such that $M B=3 M A$. Let $N$ be the midpoint of $C D$. The line $M N$ cuts $A C$ at $P$. Calculate the area of the quadrilateral $A P N D$.
6. Let be given three positive integers $m$, $n, p$ such that $n+1$ is divisible by $m$. Find a formula to calculate the number of $p$-uples of positive integers $\left(x_{1}, x_{2}, \ldots, x_{p}\right)$ satisfying the conditions: the sum $x_{1}+x_{2}+\ldots+x_{p}$ is divisible by $m$ and $x_{1}, x_{2}, \ldots, x_{p}$ are not greater than $n$.
7. $a$, $b$ are arbitrary positive numbers such that the equation $x^{3}-a x^{2}+b x-a=0$ has three roots greater than $1$. Determine $a$, $b$ so that the expression $\dfrac{b^{n}-3^{n}}{a^{n}}$ attains its least value and find this value.
8. The incircle of triangle $A B C$ touches the sides $B C$, $C A$, $A B$ respectively at $D$, $E$, $F$. Prove that $$\frac{D E}{\sqrt{B C . C A}}+\frac{E F}{\sqrt{C A \cdot A B}}+\frac{F D}{\sqrt{A B \cdot B C}} \leq \frac{3}{2}.$$

### Issue 322

1. Find all integers $x$ satisfying $$|x-3|+|x-10|+|x+101|+|x+990|+|x+1000|=2004.$$
2. Let $A B C$ be a triangle with its median $A M$. Let $O_{1}$, $O_{2}$. be the incenters of triangles $A B M$, $A C M$ respectively. Prove that $M O_{1}=M O_{2}$ when and only when $A B=A C$.
3. Consider the six pairs of marbles selected from a set of four given marbles, and consider the sum of masses of two marbles of each pair. Let $a, b, c, d, e, f$ be these sums. Determine the mass of each marble, known that $$a+b+c+d+e+f=a^{3}+b^{3}+c^{3}+d^{3}+e^{3}+f^{3}=6.$$
4. Find the least value of the expressions $$\frac{a^{6}}{b^{3}+c^{3}}+\frac{b^{6}}{c^{3}+a^{3}}+\frac{c^{6}}{a^{3}+b^{3}}$$ where $a, b, c, d$ are positive real numbers satisfying the condition $a+b+c=1$.
5. The circumcircle of triangle $A B C$ has center $O$ and diameter $A D$. Let $I$ be the incenter of triangle $A B C$. The lines $A I$, $D I$ cut again the circumcircle at $H, K$ respectively. Draw the line IJ perpendicular to $B C$ at $J$. Prove that $H$, $K$, $J$ are collinear.
6. Prove the inequality $$\frac{1}{2}\left(\sum_{i=1}^{n} x_{i}+\sum_{i=1}^{n} \frac{1}{x_{i}}\right) \geq n-1+\frac{n}{\sum_{i=1}^{n} x_{i}}$$ where $x_{i}(i=1,2, \ldots, n)$ are positive real numbers satisfying $\sum_{i=1}^{n} x_{i}^{2}=n$ and $n$ is an integer greater than $1$.
7. The sequence of numbers $\left(u_{n}\right)(n=$ $1,2,3, \ldots)$ is defined by $$u_{n}=\sum_{k=1}^{n} \frac{1}{(k !)^{2}},\,\forall n=1,2,3, \ldots.$$ Prove that this sequence has a limit and this limit is an irrational.
8. Let $S A B C$ be a tetrahedron. The points $M$, $N$, $P$ lie respectively on the sides $S A$, $S B$, $S C$ so that $A M=B N=C P$ ($M$, $N$, $P$ are distinct from the vertices $S$, $A$, $B$, $C)$. Let $G$ be the centroid of triangle $M N P$. Prove that $G$ lies on a fixed line when $M$, $N$, $P$ move on $S A$, $S B$, $SC$ respectively.

### Issue 323

1. Compare $\dfrac{1}{16}$ with the following sum $A$ of 11 numbers $$A=\frac{1}{5^{2}}+\frac{2}{5^{3}}+\ldots+\frac{n}{5^{n+1}}+\ldots+\frac{11}{5^{12}}$$
2. Prove that $$\frac{a}{2 a+b+c}+\frac{b}{2 b+c+a}+\frac{c}{2 c+a+b} \leq \frac{3}{4}$$ where $a, b, c$ are positive integers.
3. Consider the following sum $A$ of $50$ numbers $$A=\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\ldots+\frac{1}{\sqrt{2 n-1}+\sqrt{2 n}}+\ldots+\frac{1}{\sqrt{99}+\sqrt{100}}.$$ Find the greatest integer not exceeding $A$.
4. Prove that $$\frac{a}{\sqrt{b^{3}+c^{3}}}+\frac{b}{\sqrt{c^{3}+a^{3}}}+\frac{c}{\sqrt{a^{3}+b^{3}}}<2 \sqrt{4}$$ where $a, b, c$ are the lengths of the sides of a triangle.
5. Let $A B C D$ be a given convex quadrilateral. On the lines $B C$, $A D$, take respectively the points $E$, $F$ so that $A E || C D$, $C F || A B$. Prove that the quadrilateral $A B C D$ circumscribes about a circle when and only when the quadrilateral $A E C F$ circumscribes about a circle.
6. Prove that $$a^{\log _{b} c}+b^{\log _{c} a}+c^{\log _{a} b} \geq 3 \sqrt{a b c}$$ where $a, b, c$ are numbers greater than $1$.
7. Prove that for every acute triangle $A B C$, we have $$\frac{1}{3}(\cos 3 A+\cos 3 B)+\cos A+\cos B+\cos C \geq \frac{5}{6}$$
8. In space, let be given a fixed line $d$ and a fixed point $A$ not lying on $d$. A right angle $x M y$ moves so that its side $M x$ passes through $A$ and its side $M y$ cuts orthogonally $d$. Find the locus of $M$.

### Issue 324

1. How many digits does contain the decimal representation of the number $2^{100}$ ? What is the first digit on the left in this representation?
2. Let $A B C$ be a triangle with $\widehat{A B C}=70^{\circ}$, $\widehat{A C B}=50^{\circ}$. On the side $A C$, take $M$ so that $\widehat{A B M}=20^{\circ}$, on the side $A B$, take $N$ so that $\widehat{A C N}=10^{\circ}$. Let $P$ be the point of intersection of $B M$ and $C N$. Prove that $M N=2 P M$.
3. Solve the system of equations $$\begin{cases}x^{3}+y &=2 \\ y^{3}+x &=2\end{cases}$$
4. Prove the inequality $$\sqrt{a^{4}+b^{4}+c^{4}}+\sqrt{a^{2} b^{2}+b^{2} c^{2}+c^{2} a^{2}} \geq \sqrt{a^{3} b+b^{3} c+c^{3} a}+\sqrt{a b^{3}+b c^{3}+c a^{3}}$$ where $a, b, c$ are non negative real numbers.)
5. Let $A B C$ be a triangle right at $A$. For every point $K$ on the side $A C$, construct the circle $(K)$ with center $K$, touching the line $B C$ at $E$. Draw the line $B D$ touching the circle $(K)$ at $D$ (distinct from $E)$. Let $M$, $N$, $P$ and $Q$ be the midpoints of $A B$, $A D$, $B D$ and $M P$ respectively. Let $S$ be the point of intersection of $Q N$ and $B D$. Find the line on which moves the point $S$ when $K$ moves on the side $A C$?.
6. Let $f(x)$ be a polynomial of degree 2003 with $$f(k)=\frac{k^{2}}{k+1},\,\forall k=1,2,3, \ldots 2004.$$ Calculate $f(2005)$.
7. Prove that $$4 x^{2}+4 y^{2} \leq x y+y z+z x+5 z^{2}$$ where $x, y, z$ are positive real numbers satisfying the conditions $x \leq y \leq z$. When does equality occur?
8. Let $r_{a}$, $r_{b}$, $r_{c}$ be the radii of the escribed circles in angles $A, B, C$ of the triangle $A B C$ respectively. Prove that  $$r_{a} \sin (A / 2)+r_{b} \sin (B / 2)+r_{c} \sin (C / 2) \leq \frac{r_{a}^{3}+r_{b}^{3}+r_{c}^{3}}{6}\left(\frac{1}{r_{a}^{2}}+\frac{1}{r_{b}^{2}}+\frac{1}{r_{c}^{2}}\right)$$

### Issue 325

1. Prove that the sum $$A=\frac{2004}{2003^{2}+1}+\frac{2004}{2003^{2}+2}+\ldots+\frac{2004}{2003^{2}+n}+\ldots+\frac{2004}{2003^{2}+2003}$$ ($2003$ terms) is not an integer.
2. Let $A B C D$ be a rectangle with $A B=2 A D$ and let $M$ be the midpoint of the segment $A B$. Let $H$ be the point on side $A B$ such that $\widehat{A D H}=15^{\circ}$. The lines $C H$ and $D M$ intersect at $K$. Compare the lengths of the segments $D H$ and $D K$.
3. Solve the system of equations $$\begin{cases}x^{3}(2+3 y) &=1 \\ x\left(y^{3}-2\right) &=3\end{cases}$$
4. Find the least value of the expression $$A=\frac{1}{x^{3}+y^{3}}+\frac{1}{x y}$$ where $x, y$ are positive real numbers satisfying $x+y=1$.
5. Let be given a convex quadrilateral $A B C D$. $O$ is the midpoint of side $B C$, $E$ is symmetric to $D$ with respect to $O$. A point $M$ moves on the side $A D$. The line $E M$ cuts $O A$ at $I$. The line passing through $I$, parallel to $B C$, cuts $A B$ and $A C$ respectively at $K$ and $H$. Prove that the expression $$\frac{A B}{A K}+\frac{A C}{A H}-\frac{A D}{A M}$$ takes constant value.
6. Let be given $a>1$. Find all triples $(x, y, z)$ such that $|y| \geq 1$ and $$\log _{a}^{2}(x y)+\log _{a}\left(x^{3} y^{3}+x y z\right)^{2}+\frac{8+\sqrt{4 z-y^{2}}}{2}=0$$
7. Find the greatest value of the expression $a c+b d+c d$ where $a, b, c, d$ are real numbers satisfying the conditions $a^{2}+b^{2}=4$ and $c+d=4$.
8. The circles $C_{1}$, $C_{2}$, $C_{3}$ internally touch the circle $C$ respectively at $A_{1}$, $A_{2}$, $A_{3}$ and they externally touch each other. Let $B_{1}$, $B_{2}$, $B_{3}$ be respectively the touching point of $C_{2}$ and $C_{3}$, of $K_{3}$ and $C_{1}$, of $C_{1}$ and $C_{2}$. Prove that the lines $A_{1} B_{1}$, $A_{2} B_{2}$, $A_{3} B_{3}$ are concurrent.

### Issue 326

1. Factorize $2003^{2004}$ in the product of two natural numbers $a$ and $b$. Is the sum $a+b$ divisible by $2004$?
2. The positive integers $a, b, c, d$ satisfy the conditions $a^{2}+c^{2}=1$ and $\dfrac{a^{4}}{b}+\dfrac{c^{4}}{d}=\dfrac{1}{b+d}$. Prove that $$\frac{a^{2004}}{b^{1002}}+\frac{c^{2004}}{d^{1002}}=\frac{2}{(b+d)^{1002}}.$$
3. Find the least prime number $p$ such that $p$ can be written in ten sums of the forms $$p=x_{1}^{2}+y_{1}^{2}=x_{2}^{2}+2 y_{2}^{2}=x_{3}^{2}+3 y_{3}^{2}=\ldots=x_{10}^{2}+10 y_{10}^{2},$$ where $x_{i}, y_{i}$ $(i=1,2, \ldots, 10)$ are positive integers.
4. Solve the equation $$\sqrt{3 x+1}+\sqrt{5-x}+\sqrt{2 x-9}-\sqrt{4 x-3}=0$$
5. Prove that $$\frac{1}{a(b+1)}+\frac{1}{b(c+1)}+\frac{1}{c(a+1)} \geq \frac{3}{a b c+1}$$ for arbitrary positive numbers $a, b, c$.
6. Prove that for every polygon, there exist at least two sides such that the measures $a$, $b$ of which satisfy the conditions $a \leq b \leq 2 a$.
7. From a point $P$ at the outside of a circle with center $O$, draw two tangents $P A$, $P B$ to the circle. Let $M$ and $N$ be respectively the midpoints of $A P$ and $O P$. The line $B M$ cuts again the circle at $K$. Prove that $K N \perp A K$.
8. Find integer-solutions of the equation of two unknowns $$x^{y^{x}}=y^{x^{y}}.$$
9. Prove that $$x y+\max \{x, y\} \leq \frac{3 \sqrt{3}}{4}$$ for arbitrary real nonnegative numbers $x, y$ satisfying the condition $x^{2}+y^{2}=1$.
10. Find all functions $f: \mathbb R^{+} \rightarrow \mathbb R^{+}$ satisfying the condition $$x f(x f(y))=f(f(y)),\,\forall x, y \in \mathbb R^{+}.$$
11. Let $R$ and $r$ bc respectively the circumradius and the inradius of a triangle $A B C$. and let $I$ be its incenter. Prove that $$\frac{1}{I A . I B}+\frac{1}{I B \cdot I C}+\frac{1}{I C . I A} \leq \frac{5 R+2 r}{8 R r^{2}}$$
12. Let $A B C D$ be a regular tetrahedron with side $a$. Let $H$ and $K$ be the midpoints of $A B$ and $C D$ respectively. An arbitrary plane containing the line $H K$ cuts the sides $B C$ and $A D$ at $E$ and $F$ respectively. Prove that $E F \perp$ $H K$. Find the least value of the area of the quadrilateral $H E K F$.

### Issue 327

1. Can my friend write $7$ distinct $7$-digit numbers so that a) for writing each number, he uses $7$ distinct digits $1,2,3,4,5,6,7$. b) the sum of the $7^{\text {th }}$ powers of some (distinct) numbers among them is equal to the sum of the $7^{\text {h }}$ powers of the others?
2. Prove that $$\frac{1}{65}<\frac{1}{5^{3}}+\frac{1}{6^{3}}+\ldots+\frac{1}{n^{3}}+\ldots+\frac{1}{2004^{3}}<\frac{1}{40}$$ (the sum consists of $2000$ terms).
3. Find all integers $x$ such that $x^{3}-2 x^{2}+7 x-7$ is divisible by $x^{2}+3$.
4. Solve the equation $$4 x^{2}-4 x-10=\sqrt{8 x^{2}-6 x-10}.$$
5. Prove that $$\left(1+\frac{1}{a^{3}}\right)\left(1+\frac{1}{b^{3}}\right)\left(1+\frac{1}{c^{3}}\right) \geq \frac{729}{512}$$ where $a, b, c$ are positive real numbers satisfying $a+b+c=6$.
6. The circle $\left(O_{1}\right)$ with center $O_{1}$, radius $R_{1}$ cuts the circle $\left(O_{2}\right)$ with center $O_{2}$, radius $R_{2}$ at the points $A$ and $B$. The tangent to $\left(O_{1}\right)$ at $A$ cuts $\left(O_{2}\right)$ at $C$. The tangent to $\left(O_{2}\right)$ at $A$ cuts $\left(O_{1}\right)$ at $D .$ Let $M$ be the point of intersection of $A B$ and $C D$, let $N$ be the midpoint of $C D$. Prove that $\widehat{C A M}=\widehat{D A N}$ and $\dfrac{M C}{M D}=\dfrac{R_{2}^{2}}{R_{1}^{2}}$.
7. The quadrilateral $A B C D$ is inscribed in a circle with radius $R$ and circumscribes about a circle with radius $r$. Prove that $R \geq r \sqrt{2}$.
8. The sequences $\left(x_{n}\right)$ and $\left(y_{n}\right)$ $(n=1,$, $2,3, \ldots)$ are defined by $x_{1}=-1$, $y_{1}=1$ and $$x_{n+1}=-3 x_{n}^{2}-2 x_{n} y_{n}+8 y_{n}^{2},\, y_{n+1}=2 x_{n}^{2}+3 x_{n} y_{n}-2 y_{n}^{2},\,\forall n=1,2,3 \ldots$$ Find all prime numbers $p$ such that $x_{p}+y_{p}$ is not divisible by $p$.
9. The positive real numbers $a, b, c, d$ satisfy the conditions $a \leq b \leq c \leq d$ and $b c \leq a d$. Prove that $$a^{b} b^{c} c^{d} d^{a} \geq a^{d} b^{a} c^{b} d^{c}.$$
10. For each positive integer $n$, consider the function $$f_{n}(x)=e^{-x}\sum_{m=0}^{n} \frac{x^{m}}{m !},$$ defined on the set of positive real numbers. a) Prove that for every positive real numbers $k$ with $0<k<1$ and for every positive integer $n$, the equation $f_{n}(x)=k$ has a unique root. b) Let $\alpha_{n}$ be the above mentioned root. Find $\displaystyle\lim_{n \rightarrow+\infty} \frac{1}{\alpha_{n}}$.
11. Let be given a triangle $A B C$ with $B C=a$, $C A=b$, $A B=c$ and with circumradius $R$. Let $l_{a}$, $l_{b}$, $l_{c}$ be respectively the measure of the angled bisector of the angle $A$, $B$, $C$ and let $r_{a}$, $r_{b}$, $r_{c}$ be respectively the radius of the escribed circle in the angle $A$, $B$, $C$. Prove that $$\frac{l_{a}^{2} \cdot l_{b}^{2} \cdot l_{c}^{2}}{a^{2} b^{2} c^{2}} \leq\left(\frac{r_{a}+r_{b}+r_{c}}{6 R}\right)^{3}$$
12. Let $A_{1} A_{2} A_{3} A_{4}$ be a tetrahedron, circumscribing about a sphere with center $O$. Let $B_{i}$ be the touching point of the sphere with the face opposite to the vertex $A_{i}$ $(i=1,2,3,4)$. Prove that among the angles formed by a pair of distinct rays $O B_{1}$, $O B_{2}$, $O B_{3}$, $O B_{4}$ there exists an angle $\alpha$ with $$\sin \alpha \leq \frac{2 \sqrt{2}}{3}.$$

### Issue 328

1. Compare the numbers $2^{3^{2^{3}}}$ and $3^{2^{3^{2}}}$.
2. Calculate the following sum of 2004 numbers $$f\left(\frac{1}{2005}\right)+f\left(\frac{2}{2005}\right)+\ldots+f\left(\frac{2004}{2005}\right)$$ where $f(x)=\dfrac{100^{x}}{100^{x}+10}$.
3. Find positive integer solutions of the equation $$(n+1)(2 n+1)=10 m^{2}$$
4. Find all positive integers $n$ such that the polynomial with $n+1$ terms $$P(x)=x^{4 n}+x^{4(n-1)}+\ldots+x^{8}+x^{4}+1$$ is divisible by the polynomial with $n+1$ terms $$Q(x)=x^{2 n}+x^{2(n-1)}+\ldots+x^{4}+x^{2}+1.$$
5. Find the greatest value of the expression $$T=\frac{a^{2}+1}{b^{2}+1}+\frac{b^{2}+1}{c^{2}+1}+\frac{c^{2}+1}{a^{2}+1}$$ where $a, b, c$ are non negative real numbers satisfying $a+b+c=1$.
6. Let $A B C$ be a triangle with acute angle $A$ and $A C=2 A B$. The angle bisector $A D$ cuts the altitude $B H$ at $K$ ($D$ lies on $B C$, $H$ on $A C)$. The line $C K$ cuts $A B$ at $E$. Prove that $\triangle A B C$ is right at $B$ when and only when the areas of the triangles $B D E$ and $H D E$ are equal.
7. On the side $A B$ of an equilateral triangle $A B C$ take a point $N$, on the side $A C$ take a point $M$ so that $A N>N B$ and $A M>M C$. The line $B M$ cuts $C N$ at $H$. Let $P$ and $Q$ be respectively the orthocenters of $\triangle A B M$ and $\triangle A C N$. Prove that $B N=C M$ when and only when $H P=H Q$.
8. Find the least prime number $p$ such that $\left[(3+\sqrt{p})^{2 n}\right]+1$ is divisible by $2^{n+1}$ for every natural number $n$, where $[x]$ denotes the greatest integer not exceeding $x$.
9. Prove that $$\left(\frac{a}{b+c}\right)^{k}+\left(\frac{b}{c+a}\right)^{k}+\left(\frac{c}{a+b}\right)^{k} \geq \frac{3}{2^{k}}$$ where $a, b, c, k$ are positive real numbers and $k \geq \dfrac{2}{3}$.
10. Find all positive real numbers $a$ such that there exist a positive real number $k$ and a function $f: \mathbb R \rightarrow \mathbb R$ satisfying the condition $$\frac{f(x)+f(y)}{2} \geq f\left(\frac{x+y}{2}\right)+k \cdot|x-y|^{a}$$ for all real numbers $x, y$.
11. The altitudes $A D$, $B E$, $C F$ of an acute triangle $A B C$ intersect at $H$ so that $A H>H D$, $B H>H E$, $C H>H F$. Prove that $$\tan^{2} A+\tan^{2} B+\tan^{2} C>6$$
12. Let be given $n$ dinstinct points $A_{1}$, $A_{2}, \ldots, A_{n}$. Prove that $$\sum_{i=1}^{n} \widehat{A_{i}A_{i+1}A_{i+2}} \geq \pi \quad \text{and} \quad \sum_{i=1}^{n} \widehat{A_{i} Q A_{i+1}} \leq(n-1) \pi$$ where $A_{n+1}$ is considered as $A_{1}, A_{n+2}$ is considered as $A_{2}$ and $Q$ is an arbitrary point distinct from $A_{1}, A_{2}, \ldots, A_{n}$.

### Issue 329

1. Let $p$ and $q$ be two primes satisfying $p>q>3$ and $p-q=2$. Prove that $p+q$ is divisible by $12$.
2. Find the greatest value of the expression $$P=(a-b)^{4}+(b-c)^{4}+(c-a)^{4}$$ where $a, b, c$ are real numbers not less than $1$ and not greater than $2$.
3. Prove that the following sum (of $1999$ terms) $$s=1^{100}-2^{100}+3^{100}-4^{100}+\ldots+n^{100}-(n+1)^{100}+\ldots-1998^{100}+1999^{100}$$ is divisible by $201899$.
4. Solve the equation $$x=(2004+\sqrt{x})(1-\sqrt{1-\sqrt{x}})^{2}.$$
5. Prove that $$\frac{a}{a+\sqrt{(a+b)(a+c)}}+\frac{b}{b+\sqrt{(b+c)(b+a)}}+\frac{c}{c+\sqrt{(c+a)(c+b)}} \leq 1$$ where $a, b, c$ are positive real numbers.
6. Let $M N P Q$ be a quadrilateral inscribed in a circle and let $E$ be the point of intersection of $M P$ and $N Q$. Let $K$ be a point on the segment $M E$ ($K$ distinct from $M$, $E$). The tangent at $E$ to the circumcircle of triangle $N E K$ cuts the lines $Q M$ and $Q P$ respectively at $F$ and $G$. Prove that $$\dfrac{E G}{E F}=\dfrac{K P}{K M}$$
7. Consider the triangles $A B C$ with given perimeter $a+b+c=k$ (const), $a=B C$, $b=C A$, $c=A B$. Find the greatest value of the expression $$T=\frac{a b}{a+b+2 c}+\frac{b c}{2 a+b+c}+\frac{a c}{a+2 b+c}$$
8. Let $a$, $b$ be two real numbers distinct from $0$. Consider the sequence of numbers $\left(u_{n}\right)(n=0,1,2, \ldots)$ defined by $$u_{0}=0,\, u_{1}=1,\quad u_{n+2}=a u_{n+1}-b u_{n},\,\forall n=2,3, \ldots$$ Prove that if there exist four consecutive terms of the sequence that are integers then all terms of the sequence are intergers.
9. Find all values of the parameter $p$ so that the roots $x_{1}, x_{2}, x_{3}$ of the equation $$x^{3}-3 x^{2}-p x-1=0$$ satisfy the conditions $$\frac{1}{2005}<\frac{1}{\left(x_{1}-1\right)^{3}}+\frac{1}{\left(x_{2}-1\right)^{3}}+\frac{1}{\left(x_{3}-1\right)^{3}}<\frac{1}{2004}$$
10. Given positive numbers $a_{i}$, $b_{i}$ $(i=1,2, \ldots, n)$. Prove that $$\frac{a_{1}^{r}}{b_{1}^{s}}+\frac{a_{2}^{r}}{b_{2}^{s}}+\ldots+\frac{a_{n}^{r}}{b_{n}^{s}} \geq \frac{\left(a_{1}+a_{2}+\ldots+a_{n}\right)^{r}}{n^{r-s-1}\left(b_{1}+b_{2}+\ldots+b_{n i}\right)^{s}}$$ where $r, s$ are positive rational numbers and $r \geq s+1$.
11. Suppose that the quadrilateral $A B C D$ is inscribed in a circle with center $O$ with radius $R$ and the opposite rays of the rays $B A$, $D A$, $C B$, $C D$ touch a circle with center $I$ and radius $r$. Prove that by putting $d=O I$, we have $$\frac{1}{(d+R)^{2}}+\frac{1}{(d-R)^{2}}=\frac{1}{r^{2}}$$
12. For a tetrahedron $A B C D$ with $A B=C D$, $A C=B D$, $A D=B C$, let $\varphi_{1}$, $\varphi_{2}$, $\varphi_{3}$ be respectively the measures of the dihedral angles with sides $B C$, $C A$, $A B$. Prove that $$\cos \frac{\varphi_{1}}{2} \cdot \cos \frac{\varphi_{2}}{2} \cdot \cos \frac{\varphi_{3}}{2}=\frac{\sqrt{\cos A \cdot \cos B \cdot \cos C}}{\sin A \cdot \sin B \cdot \sin C}$$ where $A$, $B$, $C$ denote the angles of triangle $A B C$.

### Issue 330

1. Find the integers $x$, $y$, $z$ satisfying the equalities $$\frac{x}{y}+\frac{y}{z}+\frac{z}{x}=\frac{y}{x}+\frac{z}{y}+\frac{x}{z}=x+y+z=3$$
2. Let $A B C$ be a triangle with $\widehat{A C B}=50^{\circ}, \widehat{B A C}=100^{\circ}$, let $M$ be the point on the side $A B$ such that $A M=A C$. Compare $C M$ with $A B$.
3. Find all integer roots of the equation $$x^{y}+y^{z}+z^{x}=2(x+y+z).$$
4. Solve the equation $$\sqrt{\sqrt{3}-x}=x \sqrt{\sqrt{3}+x}$$
5. Prove the inequality $$\frac{a^{3}+b^{3}+c^{3}}{2 a b c}+\frac{a^{2}+b^{2}}{c^{2}+a b}+\frac{b^{2}+c^{2}}{a^{2}+b c}+\frac{c^{2}+a^{2}}{b^{2}+a c} \geq \frac{9}{2}$$ where $a, b, c$ are positive real numbers.
6. Let be given a triangle $A B C$ with $A B=A C$. From every point $M$ on the side $B C$, draw $M P \perp A B$ and $M Q \perp A C$ ($P$, $Q$ lie respectively on the lines $A B$, $A C)$. Prove that the perpendicular bisector of $P Q$ passes through a fixed point when $M$ moves on the side $B C$.
7. Let $A B C$ be a triangle with the altitude $A H$ ($H$ distinct from $B$, $C$). Draw $H E \| A C$, $H M \perp A B$ ($E$ and $M$ lie on the line $A B$), draw $H F \| A B$, $H N \perp A C$ ($F$ and $N$ lie on the line $A C$). Prove that the lines $E F$, $M N$ and $B C$ are concurrent.
8. Find the greatest and the least values of the expression $P=x^{y^{z}}$ where $x, y, z$ are integers greater than $2$ and satisfy $x+y+z=20$.
9. Let $M$ and $m$ be respectively the greatest value and the least value of the function $$f(x)=\cos (2002 x)+k \cos (x+\alpha)$$ where $k$, $\alpha$ are real parameters. Prove that $$M^{2}+m^{2} \geq 2.$$
10. Let be given a postive integer $n$. Consider a continuous function $f(x):[0 ; n] \rightarrow \mathbb R$ satisfying $f(0)=f(n)$. Prove that there exist $n$ couples of numbers $a_{i}, b_{i}$ $(i=1,2, \ldots, n)$ belonging to $[0 ; n]$ such that $b_{i}-a_{i}$ are positive integers and $f\left(a_{i}\right)=f\left(b_{i}\right)$ for all $i=1,2, \ldots, n$
11. In plane let be given a line $x y$, a segment $A B$ perpendicular to $x y$ at $A$, a point $C$ on the ray $A x$, a point $D$ on the ray $A y$ ($C$, $D$ distinct from $A$). Draw $A E \perp B C$ ($E$ lies on $B C$), $A F \perp B D$ ($F$ lies on $B D$). A line passing through the midpoint $Q$ of $A B$ cuts the lines $x y$, $B C$, $B D$ respectively at $P$, $M$, $N$. Prove that $P$, $E$, $F$ are collinear when and only when $Q$ is the midpoint of $M N$.
12. Given a regular tetrahedron $A_{1} A_{2} A_{3} A_{4}$. Let $d_{i}(i=1,2,3,4)$ be the distance from a point $M$ in space to the face opposite to vertex $A_{i}$ of the tetrahedron $A_{1} A_{2} A_{3} A_{4}$. Prove that $$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2} \leq 9\left(d_{1}^{2}+d_{2}^{2}+d_{3}^{2}+d_{4}^{2}\right)$$ where $x_{i}=M A_{i}$ $(i=1,2,3,4)$.
 MOlympiad.NET là dự án thu thập và phát hành các đề thi tuyển sinh và học sinh giỏi toán. Quý bạn đọc muốn giúp chúng tôi chỉnh sửa đề thi này, xin hãy để lại bình luận facebook (có thể đính kèm hình ảnh) hoặc google (có thể sử dụng $\LaTeX$) bên dưới. BBT rất mong bạn đọc ủng hộ UPLOAD đề thi và đáp án mới hoặc liên hệbbt.molympiad@gmail.comChúng tôi nhận tất cả các định dạng của tài liệu: $\TeX$, PDF, WORD, IMG,... 