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2004 Anniversary 40th

Junior

  1. Write $2004$ natural numbers from $1$ to $2004$ in an abritrary order to get a sequence $a_{1},a_{2},\ldots a_{2014}$ and caculate the sum \[P=\sqrt{a_{1}+a_{2}}+\sqrt{a_{3}+a_{4}}+\ldots+\sqrt{a_{2013}+a_{2014}}.\] Find the greatest value of $P$ for all these sequences.
  2. Let $ABCD$ be a convex quadrilateral such that $AC=BD$ and $AC$ is perpecdicular to $BD$. Construct at the outside of the quadrilateral the equilateral triangles $ABX$, $BCY$, $CDZ$, $DAT$. Prove that $XZ=YT$ and $XZ$ is perpendicular to $YT$.
  3. Let $x,y$ be two integers distinct from $-1$ such that $\dfrac{x^{3}+1}{y+1}+\dfrac{y^{3}+1}{x+1}$ is an integer. Prove that $x^{2014}-1$ is devisible by $y+1$.
  4. Let $H$ be the orthocenter of a non right triangle $ABC$. Let $D$ and $E$ be the midpoints of $BC$ and $AH$ respectively. Let $F$ be the orthogonal projection of $H$ on the angle bisector of $\widehat{BAC}$. Prove that $D,R,F$ are collinear.
  5. Solve the equation \[x+\sqrt{5+\sqrt{x-1}}=6.\]
  6. Find the least value of the expression \[P=\frac{a^{3}}{b^{2}}+\frac{b^{3}}{c^{2}}+\frac{c^{3}}{a^{2}}+27\left(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}\right)\] where $a,b,c$ are positive numbers satisfying the condition $a+b+c\leq3$.
  7. Let $a,b,c$ be given positive real numbers satisfying the conditon \[6\left(\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}\right)\leq1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}.\] Prove that \[\frac{1}{10a+b+c}+\frac{1}{a+10b+c}+\frac{1}{a+b+10c}\leq\frac{1}{12}.\]
  8. Let $ABC$ be a triangle, right at $A$ and $\widehat{ABC}=60^{0}$. A line passing through $B$ cuts the line $AC$ at $D$ and cuts the circle with center $A$ and radius $AC$ at $E$ and $F$. Prove that \[\left|\frac{1}{BE}-\frac{1}{BF}\right|=\frac{1}{BD}.\]
  9. Find all positive integers $x,y$ such that $A=x^{2}y^{4}-y^{3}+1$ is a perfect square.
  10. The sequence of numbers $(x_{n})$ ($n=1,2,3\ldots$) is defined by the formulas \[x_{n}=\begin{cases}0 & \text{when }[(n+1)\sqrt{2004}]-[n\sqrt{2004}]\text{ is odd}\\1 & \text{when }[(n+1)\sqrt{2004}]-[n\sqrt{2004}]\text{ is even}\end{cases}\] for every $n=1,2,3,\ldots$ where $[x]$ denotes the greatest integer not exceeding $x$. Find the sum ($41$ terms) \[S=x_{1964}+x_{1965}+\ldots+x_{2004}.\]

Senior

  1. The sequence of numbers $(a_{n})$ ($n=1,2,3,\ldots$) is defined by \[a_{1}=\frac{1}{2^{1965}},\quad a_{n}=\frac{-1}{2^{1964+n}},\:n=2,3,\ldots,40.\] Prove that ${\displaystyle \sum_{i,j=1}^{40}a_{i}a_{j}|b_{i}-b_{j}|\leq0}$ for arbitrary given real numbers $b_{1},b_{2},\ldots,b_{40}$.
  2. Find all funtion $f:\mathbb{R}^{+}\to\mathbb{R}^{+}$ satisfying the condition \[f\left(\frac{f(x)}{y}\right)=yf(y)f(f(x))\] for all positive numbers $x,y$.
  3. Find all positive integers $a,m,n$ satisfying the condition \[(a-1)^{m}=a^{n}-1.\]
  4. Let $A,B,C,D$ be four points lying on a circle. Prove that the three four points lying on a circle. Prove that the three radical axes of three pairs of circles respectively with diameters $AB$ and $CD$, $BC$ and $DA$, $AC$ and $BD$ are concurrent. 
  5. The sequence of numbers $(u_{n})$ ($n=1,2,3,\ldots$) satisfies the following conditons for every $n=1,2,3,\ldots$ \[u_{n}=u_{n+2004},\:\sum_{i=1}^{2n}u_{i}\leq0,\:\sum_{i=1}^{2n-1}u_{i}\geq0.\] Prove that $|u_{2003}|\geq|u_{2004}|$.
  6. Let $A_{1}A_{2}\ldots A_{n}$ be a regular $n$-polygon inscribed in the unit circle. Prove that for every point $M$, \[\sum_{i=1}^{n}MA_{i}\cdot MA_{i+1}\geq n\] where $A_{n+1}=A_{1}$.
  7. Find the greatest real number $c$ satisfying the condition: for arbitrary given positive integers $m,n$, there exists a real number $x$ such that \[\sin(mx)+\sin(nx)\geq c.\]
  8. Let $ABCDA'B'C'D'$ be a cube. A plane touches the sphere inscribed in the cube at $Q$ and cuts the sides $AB,AD,A'B',A'D'$ of cube at $M,N,M',N'$ respectively. Prove that \[\widehat{MQN}+\widehat{M'QN'}=90^{0}.\]
  9. Let $n$ be a given number and a prime number $p$. Determine the number of sets of $p$ distinct natural numbers $\{a_{0},a_{1},\ldots,a_{p-1}\}$ satisfying the conditions
    • $1\leq a_{i}\leq n$ for all $i=0,1,\ldots,p-1$.
    • $[a_{0},a_{1},\ldots,a_{p-1}]=p\min\{a_{0},a_{1},\ldots,a_{p-1}\}$, where $[a_{0},a_{1},\ldots,a_{p-1}]$ denotes the lease common multiple of the numbers $a_{0},a_{1},\ldots,a_{p-1}$.
  10. Consider a convex hexagon inscribed in a circle such that the opposite sides are parallel. Prove that the sums of the lengths of opposite sides are the same if and only if the distances of the opposite sides are the same if and only if the distances of the oppostite sides are the same.

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Mathematics & Youth: 2004 Anniversary 40th
2004 Anniversary 40th
Mathematics & Youth
https://www.molympiad.org/2017/11/mathematics-and-youth-magazine-problems_28.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2017/11/mathematics-and-youth-magazine-problems_28.html
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