Mathematical Contests Collection

Mathematics and Youth Magazine Problems - Oct 2017, Issue 484

1. Compute $A=3+4+6+9+13+18+\ldots+4953$ where the terms are determined by the formular $a_{n+1}=a_{n}+n$, $n\in\mathbb{N}^{*}$.
2. Find all natural numbers $x,y$ such that $(1+x!)(1+y!)=(x+y)!$ where $n!=1.2...n$.
3. In each square in a $8\times8$ chess board we place some small stones such that the sum of the stones in an row or any column is even. Prove that the sum of the stones in the black squares is even.
4. Let $ABCD$ be a rectangle with $AB=BC\sqrt{2}$. Choose some point $M$ on the line segment $CD$ such that $M$ is different from $D$. Draw $BI\perp AM$ ($I\in AM$). Assume that $CI$ and $DI$ intersect $AB$ at $E$ and $F$ respectively. Prove that $AE$, $BF$ and $AB$ can be the lengths of the three sides of a right triangle.
5. Solve the equation $2x^{4}-8x^{3}+60x^{2}-104x-240.$
6. Find the real roots of the following equation $$3\sqrt{x^{2}+y^{2}-2x-4y+5}+2\sqrt{5x^{2}+5y^{2}+10x+50y+130}+ \\ +\sqrt{5x^{2}+5y^{2}-30x+45} = \sqrt{102x^{2}+102y^{2}-204x+204y+1360}.$$
7. Prove that the following inequalities hold for every positive integer $n$ $\ln\frac{n+1}{2}<\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n}<\log_{2}\frac{n+1}{2}.$ And hence deduce that $\lim_{n\to+\infty}\left(\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n}\right)=+\infty.$
8. Assume that the incircle $I$ of the triangle $ABC$ is tangent to $BC$, $CA$ and $AB$ respectively at $D$, $E$ and $F$. Draw $DG$ perpendicular to $EF$ ($G$ belongs to $EF$). Let $J$ be the midpoint of $DG$. The line $EJ$ intersects the circle $(I)$ at $H$. Let $K$ be the circumcenter of the triangle $FGH$. Prove that $IK||BH$.
9. Given positivenumbers $x,y,z$ satisfying $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{16}{x+y+z}.$ Find the minimum value of the expression $P=\frac{x}{y}+\frac{y}{z}+\frac{z}{x}.$
10. For each positive integer $n$, let $f(n)$ be the sum of the squares of the positive divisors of $n$. Find all positive integers $n$ such that $\sum_{k=1}^{n}f(k)\geq\frac{10n^{3}+15n^{2}+2n}{24}.$
11. Given the sequence $(x_{n})$ as follows $x_{1}=1,\,x_{2}=\frac{1}{2},\quad x_{n+2}x_{n}=x_{n+1}^{2}+4^{-n},\,n\in\mathbb{N}^{*}.$ Find ${\displaystyle \lim_{n\to\infty}(3-\sqrt{5})^{n}x^{n}}$.
12. Given a triangle $ABC$. Let $(O)$ and $I$ respectively be the circumcircle and the incenter of $ABC$. Assume that $AI$ intersects $BC$ at $A_{1}$ and intersects $(O)$ at another point $A_{2}$. Similarly we get the points $B_{1},B_{2}$ and $C_{1},C_{2}$. Suppose that $B_{1}C_{1}$ intersects $B_{2}C_{2}$at $A_{3}$, $A_{1}C_{1}$ intersects $A_{2}C_{2}$ at $B_{3}$, $A_{1}B_{1}$ intersects $A_{2}B_{2}$ at $C_{3}$. Prove that $A_{3},B_{3},C_{3}$ both belong to a line which is perpendicular to $OI$.
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Mathematics and Youth Magazine Problems - 2004, 40th Anniversary Contest

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
i) $1\leq a_{i}\leq n$ for all $i=0,1,\ldots,p-1$.
ii) $[a_{0},a_{1},\ldots,a_{p-1}]=p\min\{a_{0},a_{1},\ldots,a_{p-1}\}$,
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 and Youth Magazine Problems - Mar 2012, Issue 417

1. Which number is bigger, $2^{3100}$ or $3^{2100}$?.
2. Let $ABC$ be an isosceles triangle with $AB=AC$. $BM$ is the median from $B$. $N$ is a point on $BC$ such that $\widehat{CAN}=\widehat{ABM}$. Prove that $CM\geq CN$.
3. Let $a,b,c$ be positive numbers such that $|a+b+c|\leq1,\,|a-b+c|\leq1,\,|4a+2b+c|\leq8,\,|4a-2b+c|\leq8.$ Prove the inequality $|a|+3|b|+|c|\leq7.$
4. Solve the equation $(x-2)(x^{2}+6x-11)^{2}=(5x^{2}-10x+1)^{2}.$
5. Let $ABC$ be a right triangle, with right angle at$A$, $AH$ is the altitude from $A$ and $I,J$ ae the incenters of triangles $HAB$ and $HAC$, respectively. $IJ$ cuts $AB$ at $M$ and meets $AC$ at $N$. Let $X$ and $Y$ be the intersections of $HI$ with $AB$ and $HJ$ with $AC$; $BY$, $CX$ cuts $MN$ at $P$ and $Q$ respectively. Prove that $\frac{AI}{AJ}=\frac{HP}{HQ}.$
6. Let $x,y,z$ be real numbers such that $x^{2}+y^{2}+z^{2}=3$. Find the minimum and maximum value of the expression $P=(x+2)(y+2)(z+2).$
7. In a triangle $ABC$, let $m_{a},m_{b},m_{c}$ be its median lengths, and $l_{a},l_{b},l_{c}$ be the lengths of its inner bisectors, $p$ is half of its perimeter. Prove the inequality $m_{a}+m_{b}+m_{c}+l_{a}+l_{b}+l_{c}\leq2\sqrt{3}p.$
8. Let $S.ABC$ be a pyramid where surface $SAB$ is a isosceles triangle at $S$ and $\widehat{BSA}=120^{0}$, the plane $(SAB)$ is perpendicular to $(ABC)$. Prove that $\dfrac{S_{ABC}}{S_{SAC}}\leq\sqrt{3}$, when does the equality occur?. (Denote by $S_{DEF}$ the area of triangle $DEF$)
9. A natural number $n$ is a good number if it is possible to partition any square into $n$ smaller squares such that at least two of them are not equal.
a) Prove that if $n$ is a good number, then $n\geq4$.
b) Prove that both $4$ and $5$ are not good.
c) Find all good numbers.
10. A sequence $a_{0},a_{1},\ldots,a_{n}$ ($n\geq2$) is defined by $a_{0}=0,\quad a_{k}=\frac{1}{n+1}+\frac{1}{n+2}+\ldots+\frac{1}{n+k},\,k=1,2,\ldots,n.$ Prove the inequality $\sum_{k=0}^{n-1}\frac{e^{a_{k}}}{n+k+1}+(\ln2-a_{n})e^{a_{n}}<1$ where ${\displaystyle e=\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{n}}$.
11. Find all functions $f:\mathbb{R}^{+}\to\mathbb{R}^{+}$ satisfying $f(x)f(yf(x))=f(y+f(x)),\quad x,y\in\mathbb{R}^{+}.$
12. Given a triangle $ABC$ inscribed in a circle $(O,R)$, with center $G$ and area $S$. Prove that $a^{2}+b^{2}+c^{2}\geq\left(4\sqrt{3}+\frac{OG^{2}}{R^{2}}\right)S+(a-b)^{2}+(b-c)^{2}+(c-a)^{2}.$
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Mathematics and Youth Magazine Problems - Feb 2012, Issue 416

1. Find all natural numbers $x,y,z$ such that $2010^{x}+2011^{y}=2012^{z}.$
2. The natural numbers $a_{1},a_{2},\ldots,a_{100}$ satisfy the equation $\frac{1}{a_{1}}+\frac{1}{a_{2}}+\ldots+\frac{1}{a_{100}}=\frac{101}{2}.$Prove that there are at least two equal numbers.
3. Let $a,b,c$ be positive real numbers. Prove the inequality $\frac{(a+b)^{2}}{ab}+\frac{(b+c)^{2}}{bc}+\frac{(c+a)^{2}}{ca}\geq9+2\left(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\right).$
4. Solve the equation $4x^{2}+14x+11=4\sqrt{6x+10}.$
5. In a triange $ABC$, te incircle $(I)$ meets $BC$, $CA$ at $D$, $E$ respectively. Let $K$ be the point of reglection of $D$ through the midpoint of $BC$, the line through $K$ and perpendicular to $BC$ meets $DE$ at $L$, $N$ is the midpoint of $KL$. Prove that $BN$ and $AK$ are orthogonal.
6. Determine the maximum value of the expression $A=\frac{mn}{(m+1)(n+1)(m+n+1)}$ where $m,n$ are natural numbers.
7. Triangle $ABC$ ($AB>AC$) is inscribed in circle $(O)$. The exterior angle bisector of $BAC$ meets $(O)$ at another point $E$; $M,N$ are the midpoints of $BC$, $CA$ respectively; $F$ os the perpendicular foot of $E$ on $AB$, $K$ is the intersection of $MN$ and $AE$. Prove that $KF$ and $BC$ are parallel.
8. Solve the equation $\sin^{2n+1}x+\sin^{n}2x+(\sin^{n}x-\cos^{n}x)^{2}-2=0$ where $n$ is a given positive integer.
9. Find all polynomials $P(x)$ such that $P(2)=12,\quad P(x^{2})=x^{2}(x^{2}+1)P(x),\:\forall x\in\mathbb{R}.$
10. Let $r_{1},r_{2},\ldots,r_{n}$ be $n$ rational numbers such that $0<r_{i}\leq\dfrac{1}{2}$, ${\displaystyle \sum_{i=1}^{n}r_{i}=1}$ ($n>1$), and let $f(x)=[x]+\left[x+\dfrac{1}{2}\right]$. Find the greatest value of the expression ${\displaystyle P(k)=2k-\sum_{i=1}^{n}f(kr_{i})}$ where $k$ runs over the integers $\mathbb{Z}$ (the notation $[x]$ means the greatest integer not exceeding $x$).
11. Suppose that $f:\mathbb{R}\to\mathbb{R}$ is a continuous funtion such that $f(x)+f(x+1006)$ is a rational number if and only if $x\in\mathbb{R}$, $f(x+20)+f(x+12)+f(x+2012)$ is itrational. Prove that $f(x)=f(x+2012)$ for all $x\in\mathbb{R}$.
12. Prove the following inequality $\frac{m_{a}}{h_{a}}+\frac{m_{b}}{h_{b}}=\frac{m_{c}}{h_{c}}\leq1+\frac{R}{r},$ where $m_{a},b_{b},m_{c}$ are medians; $h_{a},h_{b},h_{c}$ are the altitudes from $A,B,C$ and $R,r$ are the circumradius and inradius, respectively.
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Mathematics and Youth Magazine Problems - Jan 2012, Issue 415

1. Let $$A=\dfrac{2011^{2011}}{2012^{2012}},\quad B=\dfrac{2011^{2011}+2011}{2012^{2012}+2012}.$$ Which number is greater, $A$ or $B$?.
2. Given $A=\sqrt{6+\sqrt{6+\ldots+\sqrt{6}}},\:B=\sqrt[3]{6+\sqrt[3]{6+\ldots+\sqrt[3]{6}}},$ where there are exactly $n$ square roots in $A$ and $n$ cube roots in $B$. Write $[x]$ for the greatest integer not exceeding $x$. Determine the value of $\left[\dfrac{A-B}{A+B}\right]$.
3. Find all pairs of natural numbers $x,y$ such that $x^{2}-5x+7=3^{y}.$
4. Prove the inequality $\left(1+\frac{1}{2}\right)\left(1+\frac{1}{2^{2}}\right)\ldots\left(1+\frac{1}{2^{n}}\right)<3.$
5. Let $ABCD$ be aparallelogram. Points $H$ and $K$ are chosen on lines $AB$ and $BC$ such that triangles $KAB$ and $HCB$ are isosceles ($KA=AB$, $HC=CB$). Prove that
a) Triangle $KDH$ is also isosceles.
b) Triangle $KAB$, $BCH$ and $KDH$ are similar.
6. In a triangle $ABC$ with $a=BC$, $b=CA$, $c=AB$, $A_{1}$ is the midpoint of $BC$; $O$ and $I$ are its circumcenter and incenter respectively. Prove that if $AA_{1}$ isperpendicular to $OI$ then $\min\{b,c\}\leq a\leq\max\{b,c\}.$
7. The real numbers $x,y$ and $z$ are such that $\begin{cases}\sqrt{x}\sin\alpha+\sqrt{y}\cos\alpha-\sqrt{z} & =-\sqrt{2(x+y+z)}\\ 2x+2y-13\sqrt{z} & =7 \end{cases},\quad\pi\leq\alpha\leq\frac{3\pi}{2}.$Determine the value of $(x+y)z$.
8. Solve the following system of equations in two variables $\begin{cases}\log_{2}x & =2^{y+2}\\ 2\sqrt{1+x}+xy\sqrt{4+y^{2}} & =0 \end{cases}.$
9. A collection of prime numbers (each prime can be repeated) is said to be beautiful if their product is exactly ten times their sum. Find all beautiful collections.
10. Points $A,B,C,D,E$ in clockwise order, lie on the same circle. $M,N,P,Q$ are the feet of perpendicular lines from $E$ onto $AB$, $BC$, $CD$, $DA$. Prove that $MN$, $NP$, $PQ$, $QM$ are tangent lines to a certain parabole whose focus point if $E$.
11. The sequence $(a_{n})$ is defined recursively by the following rules $a_{1}=1,\quad a_{n+1}=\frac{1}{a_{1}+\ldots+a_{n}}-\sqrt{2},\:n=1,2,\ldots.$ Find the limit of the sequence $(b_{n})$ where $b_{n}=a_{1}+\ldots+a_{n}.$
12. Let $\alpha$ and $\beta$ be two real roots of the equation $4x^{2}-4tx-1=0$ where $t$ is a parameter. Let $f(x)=\dfrac{2x-t}{x^{2}+1}$ be a funtion defined on the interval $[\alpha;\beta]$, and let $g(t)=\max_{x\in[\alpha;\beta]}f(x)-\min_{x\in[\alpha;\beta]}f(x).$ Prove that if a triple $a,b,c\in\left(0;\frac{\pi}{2}\right)$ are such that $\sin a+\sin b+\sin c=1$, then $\frac{1}{g(\tan a)}+\frac{1}{g(\tan b)}+\frac{1}{g(\tan c)}<\frac{3\sqrt{6}}{4}.$
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