Mathematics and Youth Magazine Problems 2012 (Issue 415 - 426)
Issue 415
Let $$A=\dfrac{2011^{2011}}{2012^{2012}},\quad
B=\dfrac{2011^{2011}+2011}{2012^{2012}+2012}.$$ Which number is greater,
$A$ or $B$?
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 $\left[\dfrac{A-B}{A+B}\right]$.
Find all pairs of natural numbers $x,y$ such that \[x^{2}-5x+7=3^{y}.\]
Prove the inequality \[\left(1+\frac{1}{2}\right)\left(1+\frac{1}{2^{2}}\right)\ldots\left(1+\frac{1}{2^{n}}\right)<3.\]
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.
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
\[\min\{b,c\}\leq a\leq\max\{b,c\}.\]
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 $(x+y)z$.
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.
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 $E$.
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 \[b_{n}=a_{1}+\ldots+a_{n}.\]
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 \[\frac{1}{g(\tan
a)}+\frac{1}{g(\tan b)}+\frac{1}{g(\tan c)}<\frac{3\sqrt{6}}{4}.\]
Issue 416
Find all natural numbers $x,y,z$ such that \[2010^{x}+2011^{y}=2012^{z}.\]
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.
Let $a,b,c$ be positive real numbers. Prove the inequality
\[\frac{(a+b)^{2}}{ab}+\frac{(b+c)^{2}}{bc}+\frac{(c+a)^{2}}{ca}\geq9+2\left(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\right).\]
Solve the equation \[4x^{2}+14x+11=4\sqrt{6x+10}.\]
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.
Determine the maximum value of the expression \[A=\frac{mn}{(m+1)(n+1)(m+n+1)}\] where $m,n$ are natural numbers.
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.
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.
Find all polynomials $P(x)$ such that \[P(2)=12,\quad P(x^{2})=x^{2}(x^{2}+1)P(x),\:\forall x\in\mathbb{R}.\]
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 $x$).
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 $x\in\mathbb{R}$.
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.
Issue 417
Which number is bigger, $2^{3100}$ or $3^{2100}$?.
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 $CM\geq CN$.
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 \[|a|+3|b|+|c|\leq7.\]
Solve the equation \[(x-2)(x^{2}+6x-11)^{2}=(5x^{2}-10x+1)^{2}.\]
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
\[\frac{AI}{AJ}=\frac{HP}{HQ}.\]
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 \[P=(x+2)(y+2)(z+2).\]
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
\[m_{a}+m_{b}+m_{c}+l_{a}+l_{b}+l_{c}\leq2\sqrt{3}p.\]
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 $DEF$)
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.
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 ${\displaystyle
e=\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{n}}$.
Find all functions $f:\mathbb{R}^{+}\to\mathbb{R}^{+}$ satisfying \[f(x)f(yf(x))=f(y+f(x)),\quad x,y\in\mathbb{R}^{+}.\]
Given a triangle $ABC$ inscribed in a circle $(O,R)$, with center
$G$ and area $S$. Prove that
\[a^{2}+b^{2}+c^{2}\geq\left(4\sqrt{3}+\frac{OG^{2}}{R^{2}}\right)S+(a-b)^{2}+(b-c)^{2}+(c-a)^{2}.\]
Issue 418
Given \[A=1^{5}+2^{5}+3^{5}+\ldots+2011^{5}.\] Find the last digit of $A$.
Let $ABC$ be an isosceles right triangle with right angle at $A$. On
the half-plane defined by $AB$ containing $C$ draw an isosceles right
triangle $ABD$ with right angle at $B$. Let $E$ be the midpoint of
segment $BD$. Draw $CM$ perpendicular to $AE$ at $M$. Let $N$ be the
midpoint of segment $CM$, $K$ is the intersection of $BM$ and $DN$. Find
the measure of the angle $BKD$.
Find all positive integer solutions of the equation \[3^{x}-32=y^{2}.\]
Find all minimal value of the expression
\[A=\frac{1}{x^{3}+xy+y^{3}}+\frac{4x^{2}y^{2}+2}{xy}\] where $x$ and
$y$ are positive real numbers satisfying $x+y=1$.
Let $ABC$ be an acute triangle with orthocenter $H$. Prove that
$ABC$ is an equilateral triangle if and only if
\[\frac{AH}{BC}=\frac{BH}{CA}=\frac{CH}{AB}.\]
Let $ABC$ be a triangle with circumcenter $O$, and incenter $I$.
$BC$ touches the circle $(I)$ at $D$. The circle whose diameter is $AI$
meets $(O)$ at $M$ ($M\ne A$) and cuts the line passing through $A$
parallel to $BC$ at $N$. Prove that $MO$ passes through the midpoint of
$DN$.
Solve the system of equations
\[\begin{cases}\sqrt{xy+(x-y)(\sqrt{xy}+2)}+\sqrt{x} &
=y+\sqrt{y}\\(x+1)(y+\sqrt{xy}+x(1-x)) & =4\end{cases}.\]
Let $ABC$ be an acute triangle. Prove the inequaltiy \[\cos^{3}A+\cos^{3}B+\cos^{3}C+\cos A.\cos B.\cos C\geq\frac{1}{2}.\]
For each natural number $n$, let $(S_{n})$ be the sum of all digits
of $n$ (in the decimal system). Put $S_{k}(n)=S(S(\ldots(S(n))\ldots))$
($k$ times). Find all natural numbers $n$ such that
\[S_{1}(n)+S_{2}(n)+\ldots S_{k}(n)+\ldots+S_{223}(n)=n.\]
Does there exist a set $X$ satisfying the following two conditions
$X$ contains $2012$ natural numbers.
The sum of any arbitrary elements in $X$ is the $k$-th power of a positive integer ($k\geq2$).
Find all functions $f:\mathbb{R}\to\mathbb{R}$ satisfing
\[f\left(\frac{xf(y)}{2}\right)+f\left(\frac{yf(x)}{2}\right)=4xy,\:\forall
x,y\in\mathbb{R}.\]
Fix two circles $(K)$ and $(O)$, where $(K)$ is inside $(O)$. Two
circles $(O_{1})$, $(O_{2})$ are moving so that they always externally
touch each other at $M$. Both also internally touch $(O)$, and
externally touch $(K)$. Prove that $M$ belongs to a fixed circle.
Issue 419
Let $$A=\frac{1}{1^{2}}+\frac{1}{2^{2}}+\frac{1}{3^{2}}+\ldots+\frac{1}{50^{2}}$$ and $B=\dfrac{165}{101}$. Compare $A$ and $B$.
Let $A B C$ be a right isosceles triangle with right angle at $A .$
If there exists a point $M$ inside the triangle with $\widehat{M B A}=$
$\widehat{M A C}=\widehat{M C B}$. Find the ratio $M A: M B: M C$.
Find the minimum values of the natural numbers $a, b, c$ satisfying
$$\begin{align} & a+(a+1)+(a+2)+\ldots+(a+6) \\ =&
b+(b+1)+(b+2)+\ldots+(b+8) \\ =
&c+(c+1)+(c+2)+\ldots+(c+10).\end{align}$$
Solve the following equation $$6(x-1) \sqrt{x+1}+\left(x^{2}+2\right)(\sqrt{x-1}-3)=x\left(x^{2}+2\right).$$
Let $M$ be the midpoint of the arc $A B$ of a semicircle with center
$O$ and diameter $A B$. $A C$ meets $M O$ at $D$. Prove that the
circumcenter of triangle $M D C$ always lies on a fixed line when $C$
moves on the semicircle.
Let $a, b, c$ be positive real numbers. Prove that
$$6\left(a^{3}+b^{3}+c^{3}\right) \geq 18 a b
c+\left(\sqrt[3]{a(b-c)^{2}}+\sqrt[3]{b(c-a)^{2}}+\sqrt[3]{c(a-b)^{2}}\right)^{3}.$$
Let $A B C$ be an acute triangle which is not isosceles; and $H$,
$O$ be its orthocenter and circumcenter respectively; let $D$, $E$ be
respectively the foot of the altitude from $A$, $B$. The lines $O D$ and
$B E$ intersect at $K$, $O E$ and $A D$ intersect at $L$. Let $M$ be
the midpoint of edge $A B$. Prove that $K$, $L$, $M$ are collinear if
and only if $C$, $D$, $O$, $H$ lies on the same circle.
Find all pairs of positive integers $(n, k)$ satisfying $C_{3
n}^{n}=3^{n} n^{k},$ where $$C_{p}^{m}=\frac{p !}{m !(p-m) !} ; 0 \leq m
\leq p, p \neq 0, m, p \in \mathbb{N}.$$
Let $a, b, c$ be three positive real numbers satisfying
$$15\left(\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}\right)=10\left(\frac{1}{a
b}+\frac{1}{b c}+\frac{1}{c a}\right)+2012.$$ Find the largest possible
value of the expression $$P=\frac{1}{\sqrt{5 a^{2}+2 a b+2
b^{2}}}+\frac{1}{\sqrt{5 b^{2}+2 b c+2 c^{2}}}+\frac{1}{\sqrt{5 c^{2}+2 c
a+2 a^{2}}}.$$
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
for all $x, y \in \mathbb{R}$ we have $$f(x f(y))+f(f(x)+f(y))=y
f(x)+f(x+f(y)).$$
On the interval $[a ; b],$ pick $k$ distinct points $x_{1}, x_{2},
\ldots, x_{k}$. Let $d_{n}$ be the product of the distances from $x_{n}$
to the $k-1$ remaining points; $n=1,2,3 \ldots, k .$ Find the smallest
value of $\displaystyle \sum_{n=1}^{k} \frac{1}{d_{n}}$.
Given a triangle $A B C$ and an arbitrary point $M$. Prove that
$$\frac{M A}{B C}+\frac{M B}{C A}+\frac{M C}{A B} \geq \frac{B C+C A+A
B}{M A+M B+M C}.$$
Issue 420
Find the integer value of the expression $f(x ; y)=\dfrac{x^{2}+x+2}{x y-1}$ where $x, y$ are positive integers.
Let $A B C$ be an acute triangle which is not isosceles at $A$. The
perpendicular bisectors of $A B$, $A C$ cut the median $A M$ at $E$, $F$
respectively. $B E$ and $C F$ meet at $K .$ Prove that $\widehat{A K
B}=\widehat{A K C}$ and $\widehat{M A B}=\widehat{K A C}$.
Find all triples of integers $(x ; y ; z)$ such that $$2 x y+6 y z+3 z x-|x-2 y-z|=x^{2}+4 y^{2}+9 z^{2}-1.$$
For each positive integer $n(n=1,2, \ldots),$ put $a_{n}=\dfrac{4
n}{n^{4}+4} .$ Prove that $$a_{1}+a_{2}+\ldots+a_{n}<\frac{3}{2}$$
Let $A B C$ be an acute triangle. The internal angle-bisector of
angle $B A C$ cuts $B C$ at $D$. $E$, $F$ are the orthogonal projections
of point $D$ on $A B$ and $A C$ respectively, $K$ is the intersection
of $C E$ and $B F, H$ is the intersection of $B F$ with the circumcircle
of triangle $A E K$. Prove that $D H$ is perpendicular to $B F$
Solve the system of equations $$\begin{cases} x+6 \sqrt{x y}-y
&=6 \\ x+\dfrac{6\left(x^{3}+y^{3}\right)}{x^{2}+x
y+y^{2}}-\sqrt{2\left(x^{2}+y^{2}\right)} &=3 \end{cases}.$$
Let $a, b, c$ be non-negative real numbers whose sum equals $1$.
Prove that
$$\left(1+a^{2}\right)\left(1+b^{2}\right)\left(1+c^{2}\right)
\geq\left(\frac{10}{9}\right)^{3}$$
Point $M$ inside the triangle $A B C$ with area $S$. Let $x, y, z$
be distances of $M$ to $A$, $B$, $C$ respectively. Prove that
$$(x+y+z)^{2} \geq 4 \sqrt{3} S.$$ When does the equality hold?
A nonempty set $S \subseteq \mathbb{Z}$ posesses the following properties
There exist $a, b \in S$ such that $(a, b)=(a-2 b-2)=1$,
If $x, y \in S$ then $x^{2}-y \in S$ ($x$, $y$ may be identical).
Prove that $S=\mathbb{Z}$. ($(a, b)$ is the greatest common divisor of two integers $a$ and $b$.)
Find the greatest number $k$ such that the inequality $$\sqrt{a+2
b+3 c}+\sqrt{b+2 c+3 a}+\sqrt{c+2 a+3 b} \geq
k(\sqrt{a}+\sqrt{b}+\sqrt{c})$$ holds for all positive numbers $a, b, c$
Let $\left(x_{n}\right)$ be a sequence defined by
$$x_{1}=\frac{1001}{1003} ,\quad
x_{n+1}=x_{n}-x_{n}^{2}+x_{n}^{3}-x_{n}^{4}+\ldots+x_{n}^{2011}-x_{n}^{2012},\,
\forall n \in \mathbb{N}.$$ Find $\displaystyle \lim _{n
\rightarrow+\infty}\left(n x_{n}\right)$.
Given four distinct points $A$, $B$, $C$, $D$ lying on a circle with
center $O$. Let $I$, $J$ be the feet of the perpendicular to $A B$ and
$A D$ through $C$; $K$, $L$ are the feet of the perpendicular to $B C$
and $B A$ through $D$; $N$ is the midpoint of $C D$; $M$ is the
intersection of $I J$ and $K L$. $I J$ meets $O D$ at $E$ and $K L$
meets $O C$ at $F$. Prove that the five points $M$, $N$, $O$, $E$ and
$F$ lie on the same circle.
Issue 421
Given the sum of $2012$ terms
$$S=\frac{1}{5}+\frac{2}{5^{2}}+\frac{3}{5^{3}}+\frac{4}{5^{4}}+\ldots+\frac{2012}{5^{2012}}$$
Compare $S$ with $\dfrac{1}{3}$.
Let $A B C$ be a triangle with $\widehat{A B C}=40^{\circ},
\widehat{A C B}=30^{\circ} .$ Outside this triangle, construct triangle
$A D C$ with $\widehat{A C D}=\widehat{C A D}=50^{\circ} .$ Prove that
the triangle $B A D$ is isosceles.
Find all natural numbers $a, b, c$ such that $c < 20$ and $a^{2}+a b+b^{2}=70 c$.
Find the largest possible value of the expression
$$P=\sqrt{1-\frac{x}{y+z}}+\sqrt{1-\frac{y}{z+x}}+\sqrt{1-\frac{z}{x+y}}$$
where $x, y, z$ are side lengths of a triangle.
Given a circle $(O),$ with a fixed chord $B C$. $A$ is a point
moving on the line $B C$, outside the circle $(O)$. $AM$ and $AN$ are
the tangent lines to circle $(O)$ $(M, N \in (O))$. The line through $B$
and parallel to $A M$ meets $M N$ at $E .$ Prove that the circumcircle
of triangle $B E N$ always passes through two fixed points when point
$A$ moves on the line $B C$.
Given that $\dfrac{1}{3} < x \leq \dfrac{1}{2}$ and $y \geq 1$.
Find the minimum value of $$P=x^{2}+y^{2}+\frac{x^{2} y^{2}}{((4 x-1)
y-x)^{2}}.$$
Let $\left(a_{n}\right)$ be a sequence of positive real numbers, given by
$a_{0}=1$,
$a_{m}<a_{n}$, for all $m, n \in \mathbb{N}$, $m<n$.
$a_{n}=\sqrt{a_{n+1} \cdot a_{n-1}}+1$ and $4 \sqrt{a_{n}}=a_{n+1}-a_{n-1}$ for all $n \in \mathbb{N}^{*}$.
Determine the sum $T=a_{0}+a_{1}+a_{2}+\ldots+a_{2012}$.
The base of a triangular prism $A B C \cdot A^{\prime} B^{\prime}
C^{\prime}$ is an equilateral triangle with side lengths $a$ and the
lengths of its adjacent sides also equal $a$. Let $I$ be the midpoint of
$A B$ and $B^{\prime} I \perp(A B C)$. Find the distance from
$B^{\prime}$ to the plane $\left(A C C^{\prime} A^{\prime}\right)$ in
term of $a$.
Find $\alpha, \beta$ so that the largest value of $$y=|\cos x+\alpha \cos 2 x+\beta \cos 3 x|$$ is smallest possible.
Let $A B C$ be a triangle with side lengths $a$, $b$ and $c$. Let
$S$ and $p$ be respectively the area and the semiperimeter of this
triangle. Prove the inequality
$$\frac{1}{a^{2}(p-a)^{2}}+\frac{1}{b^{2}(p-b)^{2}}+\frac{1}{c^{2}(p-c)^{2}}
\geq \frac{9}{4 S^{2}}$$
Given an acute triangle $A B C$ inscribed the circle $(O)$ with $B
C>C A>A B$. On the circle $(O)$, select six distinct points $M$,
$N$, $P$, $Q$, $R$ and $S$ (which are also distinct from the vertices of
triangle $A B C$) so that $Q B=B C=C R$, $S C=C A=A M$ and $N A=A B=B
P$. Let $I_{A}$, $I_{B}$ and $I_{C}$ be the incenters of triangles $A P
S$, $B N R$ and $C M Q$ respectively. Prove that $\Delta I_{A} I_{B}
I_{C} \sim \Delta A B C$.
Issue 422
Let
$$A=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\ldots+\frac{1}{2011}-\frac{1}{2012},\quad
B=\frac{1}{1007}+\frac{1}{1008}+\ldots+\frac{1}{2012}.$$ Compute the
value of $\left(\dfrac{A}{B}\right)^{2012}$.
Let $f(x)$ be a polynomial with integer coefficients such that $f(3)
\cdot f(4)=5 .$ Prove that $f(x)-6$ does not have any integer solution.
Find all triple of integers $a, b, c$ such that $$2^{a}+8 b^{2}-3^{c}=283.$$
Given a triangle $A B C$, $B C=a$, $C A=b$, $A B=c$, $\widehat{A B
C}=45^{\circ}$ and $\widehat{A C B}=120^{\circ}$. Point $I$ is taken on
the opposite ray of $C B$ such that $\widehat{A I B}=75^{\circ} .$ Find
the length of $A I$ in term of $a$, $b$ and $c$
Point $K$ lies on side $B C$ of a triangle $A B C$. Prove that $$A
K^{2}=A B \cdot A C - K B \cdot K C$$ if and only if $A B=A C$ or
$\widehat{B A K}=\widehat{C A K}$.
A non-isosceles triangle $A B C$ has $B C=a$, $C A=b$, $A B=c$. Let
$\left(A A_{1}, A A_{2}\right)$, $\left(B B_{1}, B B_{2}\right)$,
$\left(C C_{1}, C C_{2}\right)$ be the median and the altitude from
vertices $A$, $B$ and respectively. Prove that
$$\frac{a^{2}}{b^{2}-c^{2}} \overline{A_{1}
A_{2}}+\frac{b^{2}}{c^{2}-a^{2}} \overrightarrow{B_{1}
B_{2}}+\frac{c^{2}}{a^{2}-b^{2}} \overrightarrow{C_{1}
C_{2}}=\overrightarrow{0}$$
Let $a, b, c \in(0 ; 1)$ and $$a b+b c+c a+a+b+c=1+a b c.$$ Prove
that $$\frac{1+a}{1+a^{2}}+\frac{1+b}{1+b^{2}}+\frac{1+c}{1+c^{2}} \leq
\frac{3}{4}(3+\sqrt{3})$$
Let $A B C$ be an acute triangle with all angles greater than
$45^{\circ}$. Prove that $$\frac{2}{1+\tan A}+\frac{2}{1+\tan
B}+\frac{2}{1+\tan C} \leq 3(\sqrt{3}-1).$$ When does equality occur?
Two sequences $\left(a_{n}\right)$ and $\left(b_{n}\right)$ are
defined inductively as follows $$a_{0}=3, b_{0}=-3,\quad a_{n}=3
a_{n-1}+2 b_{n-1},\,b_{n}=4_{n-1}+3 b_{n-1},\,\forall n \geq 1.$$ Find
all natural numbers $n$ such that
$\displaystyle\prod_{k=0}^{n}\left(b_{k}^{2}+9\right)$ is a perfect
square.
Let $n$ be a positive integer. How many strings of length $n: a_{1}
a_{2} \ldots a_{n}$ where $a_{i}$ is chosen from $\{0,1,2, \ldots,
9\}(i=1,2, \ldots, n)$ are there such that the number of occurrences of 0
is even?
Let $\left(u_{n}\right)$ be a sequence defined by $u_{0}=a \in[0 ;
2), u_{n}=\dfrac{u_{n-1}^{2}-1}{n}$ for all $n=1,2,$ $3, \ldots$ Find
$\displaystyle\lim _{n \rightarrow+\infty}\left(u_{n} \sqrt{n}\right)$.
Let $A B C$ be a triangle, inscribed in the circle $(O)$ with
altitudes $A D$, $B E$ and $C F$. $A A^{\prime}$ is a diameter of $(O)$.
$A^{\prime} B$, $A^{\prime} C$ intersect $A C$, $A B$ at $M$, $N$
respectively. Points $P$, $Q$ are in $E F$, such that $P B$, $Q C$ are
perpendicular to $B C$. The line passing through $A$ and orthogonal to
$Q N$, $P M$ cuts $(O)$ at $X$, $Y$ respectively. The tangents to circle
$(O)$ at $X$ and $Y$ meet at $J$. Prove that $J A^{\prime}$ is
perpendicular to $B C$.
Issue 423
Find all numbers abcde, where all five digits are distinct and $\overline{a b c d}=(5 e+1)^{2}$
Find all positive integers $x, y, z$ such that $x+3=2^{y}$ và $3 x+1=4^{z}$
Find the last digit of the sum $$S=1^{2}+2^{2}+3^{3}+\ldots+n^{n}+\ldots+2012^{2012}.$$
Given a function $f$ such that
$$f\left(1+\frac{\sqrt{2}}{x}\right)=\frac{(1+2011) x^{2}+2 \sqrt{2
x}+2}{x^{2}}$$ for all nonzero $x$. Determine
$f(\sqrt{2012-\sqrt{2011}})$
Let $A B C$ be a triangle inscribed in the circle $(O)$. The
tangents of $(O)$ at $B$ and $C$ meet at $T$. The line passing through
$T$ and parallel to $B C$ cuts $A B$ and $A C$ respectively at $B_{1}$
and $C_{1}$ Prove that $\widehat{B_{1} O C_{1}}$ is an acute angle.
On the outside of triangle $A B C$, construct equilateral triangles
$A B C_{1}$, $B C A_{1}$, $CAB_{1}$ and inside of $A B C$ construct
equilateral triangles $A B C_{2}$, $B C A_{2}$, $C A B_{2}$. Let
$G_{1}$, $G_{2}$, $G_{3}$ be respectively the centroids of $A B C_{1}$,
$B C A_{1}$, $C A B_{1}$ and let $G_{4}$, $G_{5}$, $G_{6}$ be
respectively the centroids of triangles $A B C_{2}$, $BCA_{2}$ and
$CAB_{2}$. Prove that the centroids of triangle $G_{1} G_{2} G_{3}$ and
of triangle $G_{4} G_{5} G_{6}$ coincide.
Solve the equation $$3^{3 x}+3^{x}=\log _{3}\left(2^{x}+x\right)+2^{x}+3^{2^{x}+x}.$$
Let $A$, $B$, $C$ be the three angles of an acute triangle. Prove
the inequality $$\sqrt{\frac{\cos A \cos B}{\cos C}}+\sqrt{\frac{\cos B
\cos C}{\cos A}}+\sqrt{\frac{\cos C \cos A}{\cos B}}>2.$$
Find the largest positive integer $n$ $(n \geq 3)$ such that there
exists a sequence of positive integers $a_{1}, a_{2}, \ldots, a_{n}$
satisfying the condition $$a_{k+1}+1=\frac{a_{k}^{2}+1}{a_{k-1}+1},\, k
\in\{2,3, \ldots, n-1\}.$$
Let $p$ be an odd prime number, $n$ is a positive integer so that
$p-1$, $p$, $n$ and $n+1$ are pairwise coprime. Find all positive
integers $x$, $y$ satisfying $$x^{p-1}+x^{p-2}+\ldots+x+2=y^{n+1}.$$
Solve the system of equations $$\begin{cases}\sqrt{5 x^{2}+2 x y+2
y^{2}}+\sqrt{2 x^{2}+2 x y+5 y^{2}} &=3(x+y) \\ \sqrt{2 x+y+1}+2
\sqrt[3]{7 x+12 y+8} &=2 x y+y+5\end{cases}.$$
Let $A B C$ be a triangle inscribed in the circle $(O)$ and let $I$
be its incenter. $A I$, $B I$, $Cl$ cut the circle $(O)$ at
$A^{\prime}$, $B^{\prime}$ and $C^{\prime}$ respectively; $A^{\prime}
C^{\prime}$, $A^{\prime} B^{\prime}$ cut $B C$ at $M$, $N$; $B^{\prime}
A^{\prime}$; $B^{\prime} C^{\prime}$ cut $C A$ at $P$, $Q$; $C^{\prime}
B^{\prime}$, $C^{\prime} A$ cut $A B$ at $R$, $S$. Prove that
$$\frac{2}{3} S_{A B C} \leq S_{M N P Q R S} \leq \frac{2}{3}
S_{A^{\prime} B^{\prime} C^{\prime}}.$$
Issue 424
Find all $2$-digit numbers such that when multiplied by $2,3,4,$
$5,6,7,8,9,$ the sum of the digits of the resulting numbers are equal.
Let
$$S=\frac{2}{2013+1}+\frac{2^{2}}{2012^{2}+1}+\frac{2^{3}}{2013^{2^{2}}+1}+\ldots+\frac{2^{2014}}{2013^{2^{2013}}+1}.$$
Which number is greater? $S$ or $\dfrac{1}{1006}$?.
Find all integer solutions of the equation $$(y-2) x^{2}+\left(y^{2}-6 y+8\right) x=y^{2}-5 y+62$$
Let $x$, $y$ be two rational numbers such that
$$x^{2}+y^{2}+\left(\frac{x y+1}{x+y}\right)^{2}=2 .$$ Prove that
$\sqrt{1+x y}$. is also a rational number.
Let $O$ denote the point of intersection of the two diagonals $A C$
and $B D$ of a convex quadrilateral $A B C D$. Let $E$, $F$, $H$ be the
feet of the altitudes from $B$, $C$ and $O$ respectively onto $A D$.
Prove that $$ A D \cdot B E \cdot C F \leq A C \cdot B D \cdot O H.$$
When does equality holds?
$a, b, c$ are positive real numbers satisfying $a b c=1$. Prove that
$$\frac{a^{3}+5}{a^{3}(b+c)}+\frac{b^{3}+5}{b^{3}(c+a)}+\frac{c^{3}+5}{c^{3}(a+b)}
\geq 9$$
Solve the equation $$\left(x^{3}+\frac{1}{x^{3}}+1\right)^{4}=3\left(x^{4}+\frac{1}{x^{4}}+1\right)^{3}$$
Let $A B C$ be a triangle with acute angle $A$. Point $P$ inside the
triangle $A B C$ such that $\widehat{B A P}=\widehat{A C P}$ and
$\widehat{C A P}=\widehat{A B P}$. Let $M$ and $N$ be the incenters of
triangles $A B P$ and $A C P$ respectively, $R$ is the circumradius of
triangle $A M N$. Prove that $$\frac{1}{R}=\frac{1}{A B}+\frac{1}{A
C}+\frac{1}{A P}.$$
Solve the equation $$[x]^{3}+2 x^{2}=x^{3}+2[x]^{2}$$ where $[t]$ denotes the largest integer not exceeding $t$.
In the interior of a unit square, there are $n\left(n \in
\mathbb{N}^{*}\right)$ circles whose sum of areas is greater than $n-1$.
Prove that the circles has at least a common point of intersection.
Given that the following equation $$a_{0} x^{n}+a_{1}
x^{n-1}+\ldots+a_{n-1} x+a_{n}=0 $$ has $n$ distinct roots. Prove that
$$\frac{n-1}{n}>\frac{2 a_{0} a_{2}}{a_{1}^{2}}.$$
Let $O$, $I$ and $I_{a}$ denote the circumcenter, incenter and
excenter in the angle $A$ of a triangle $A B C$. $A I$ meets $B C$ at
$D$. BI meets $C A$ at $E$. The line through $I$ and perpendicular to $O
I_{a}$ intersects $A C$ at $M$. Prove that $D E$ passes through the
midpoint of line segment $I M$.
Issue 425
Find all natural numbers $N$ such that $N$ decreases by a factor of $1997$ after truncating the last several digits.
Let $A B C$ be a right triangle with right angle at $A$ and
$\widehat{A C B}=15^{\circ}$, Point $D$ on edge $A C$ such that the line
passing through $D$ and perpendicular to $B D$ cuts $B C$ at $E$ and $D
E=2 D A$. Find the measure of angle $A D B$.
Find all positive integers $n$ such that $[A]=4951$ where $A$ is the
sum of $n$ terms
$$A=\left(1+\frac{1}{2}\right)+\left(2+\frac{2}{2^{2}}\right)+\left(3+\frac{3}{2^{3}}\right)+\ldots+\left(n+\frac{n}{2^{n}}\right).$$
Here $[x]$ denotes the largest integer not exceeding $x$
Find the minimum value of the expression
$$P=\frac{1+\sqrt[3]{x}+\sqrt[3]{y}+\sqrt[3]{z}}{x y+y z+z x},$$ where
$x, y, z$ are positive numbers satisfying $x+y+z=3$
Solve the equation $$x^{2}-2 x+7+\sqrt{x+3}=2 \sqrt{1+8 x}+\sqrt{1+\sqrt{1+8 x}}.$$
Let $A B C$ be a non-isosceles triangle with medians $A A^{\prime}$,
$B B^{\prime}$ and $C C^{\prime}$; and altitudes $A H$, $B F$ and CK.
Given that $C K=B B^{\prime}$, $B F=A A^{\prime}$. Determine the ratio
$\dfrac{C C^{\prime}}{A H}$.
$a_{1}, a_{2}, \ldots, a_{n}$ $(n \geq 3)$ are positive numbers that
$$\left(a_{1}+a_{2}+\ldots+a_{n}\right)^{2}>\frac{3
n-1}{3}\left(a_{1}^{2}+a_{2}^{2}+\ldots+a_{n}^{2}\right).$$ Prove that
for any triple $a_{i}, a_{j}, a_{k}$ are three edge lengths of some
triangle, where natural numbers $i, j,$ $k$ satisfying $0<i<j<k
\leq n$.
The volume of a given parallelogrambased pyramid $S.ABCD$ is $V$.
Assume that plane $(P)$ cuts$S A$, $S B$, $S C$, $S D$ at $A^{\prime}$,
$B^{\prime}$, $C^{\prime}$, $D^{\prime}$ respectively such that
$$\frac{S A}{S A^{\prime}}+\frac{S B}{S B^{\prime}}+\frac{S C}{S
C^{\prime}}+\frac{S D}{S D^{\prime}}=8.$$ Denote the volume of the
pyramid $S . A^{\prime} B^{\prime} C^{\prime}$ by $V_{1}$ and that of $S
. A^{\prime} C^{\prime} D^{\prime}$ by $V_{2}$. Prove the inequality
$$\frac{1}{\sqrt[3]{V_{1}}}+\frac{1}{\sqrt[3]{V_{2}}} \leq \frac{4
\sqrt[3]{2}}{\sqrt[3]{V}}.$$
Write $2012^{2013}$ as a sum of $2013$ positive interger $a_{1},
a_{2}, a_{3}, \ldots, a_{2013} ;$ and let
$$T=a_{1}^{13}+a_{2}^{13}+a_{3}^{13}+\ldots+a_{2013}^{13}.$$ Prove that
$T+2012^{2013}$ is not a perfect square.
The incircle $(I)$ of a triangle $A B C$ touches the edges $B C$, $C
A$, $A B$ at $D$, $E$, $F$, respectively. $M$ is the intersection of $B
C$ and the internal angle bisector of angle $B I C$, $N$ is the
intersection of $E F$ and the internal angle bisector of angle $E D F$.
Prove that $A$, $M$, $N$ are collinear.
If $p(x)$ and $q(x)$ are polynomials with integer coefficients,
write $p(x) \equiv q(x) \pmod 2$ if the coefficients of $p(x)-q(x)$ are
all even. A sequence of polynomials $p_{n}(x)$ is such that
$p_{1}(x)=p_{2}(x)=1$ and $$p_{n+2}(x)=p_{n+1}(x)+x p_{n}(x),\,\forall n
\geq 1.$$ Prove that $p_{2^{n}}(x) \equiv 1\pmod 2, \forall n \in
\mathbb{N}$.
Let $A B C$ be an acute triangle. Prove the inequality $$\frac{\cos B
\cos C}{\cos \frac{B-C}{2}}+\frac{\cos C \cos A}{\cos
\frac{C-A}{2}}+\frac{\cos A \cos B}{\cos \frac{A-B}{2}} \leq
\frac{3}{4}$$
Issue 426
Prove that for any natural number $n>4$ there exists a pair of
natural numbers $x, y$ with $\dfrac{n}{2} \leq x<n$ and $\dfrac{n}{2}
\leq y<n,$ such that $x^{2}-y$ is divisible by $n$
Let $A B C$ be an isosceles right triangle with right angle at $A$.
On the ray $A C$ choose two points $E$ and $F$ such that $\widehat{A B
E}=15^{\circ}$ and $C E=C F$. What is the measure of the angle $C B F$?.
Find all positive integer solutions of the equation $$65\left(x^{3} y^{3}+x^{2}+y^{2}\right)=81\left(x y^{3}+1\right).$$
Solve the system of equations $$\begin{cases} 9 x^{2}+9 x y+5 x-4
y+9 \sqrt{y} &=7 \\ \sqrt{x-y+2}+1 &=9(x-y)^{2}+\sqrt{7 x-7 y}
\end{cases}.$$
Let $A B C$ be an isosceles triangle where the angle $B A C$ is
obtuse. Suppose $D$ is a point on edge $A B$ such that $B C=C D
\sqrt{2}$. The line through $D$ and perpendicular to $A B$ meets $C A$
at $E$. Prove that $C D$ passes through the midpoint of $B E$.
Let
$$S=\sqrt{2}+\sqrt[3]{\frac{3}{2}}+\sqrt[4]{\frac{4}{3}}+\sqrt[5]{\frac{5}{4}}+\ldots+2013
\sqrt[2]{\frac{2013}{2012}}.$$ Find the integer part of $S$.
Given that $a, b, c$ are edge lengths of a triangle. Prove the
following inequality $$\sqrt{(a+b-c)(b+c-a)(c+a-b)} \leq \frac{3
\sqrt{3} a b c}{(a+b+c) \sqrt{a+b+c}}.$$
Let $A B C D$ be a trirectangular tetrahedron where the edges $A B$,
$A C$, $A D$ are pairwise perpendicular. $M$ is an arbitrary point in
the space. Given that $A B=4$, $A C=8$, $A D=12$. Find the minimum value
of the expression $$P=\sqrt{7} M A+\sqrt{11} M B+\sqrt{23} M
C+\sqrt{43} M D.$$
Let $\left(a_{n}\right)$ be a sequence of positive integers where
$$a_{1}=1,\, a_{2}=2,\quad a_{n+2}=4 a_{n+1}+a_{n},\,\forall n \geq 1
.$$ Prove that
a) $a_{n} a_{n+2}+(-1)^{n} .5$ is a perfect square for all $n \geq 1$.
b) The equation $x^{2}-4 x y-y^{2}=5$ has infinitely many positive integer solutions.
Let $a$ be a real number from $(0,1)$ and $b$ is a complex number,
$|b|<1$. Prove that $$|b|+\left|\frac{a-b}{1-a b}\right| \geq a.$$
Let $0<\alpha<\dfrac{\pi}{2}$. Prove that $$(\cot \alpha)^{\cos 2 \alpha} \geq \frac{1}{\sin 2 \alpha}$$
A tetrahedron $A B C D$ is inscribed in a sphere centered at $O$.
Point $G$ does not belong to planes $(B C D)$, $(C D A)$, $(D A B)$, $(A
B C)$ and the sphere $(O)$. $X$, $Y$, $Z$, $T$ are the centers of the
circumscribed spheres of the tetrahedron $G B C D$, $G C D A$, $G D A
B$, $G A B C$ respectively. Prove that $G$ is the centroid of
tetrahedron $A B C D$ if and only if $O$ is the centroid of tetrahedron
$X Y Z T$.
