Mathematics and Youth Magazine Problems 2016 (Issue 463 - 474)
Issue 463
Let $a_{1},a_{2},\ldots,a_{1964}$ be integers such that $$a_{1964}^{2}+a_{1963}^{2}=a_{1962}^{2}-a_{1961}^{2}+a_{1960}^{2}-a_{1959}^{2}+\ldots+a_{2}^{2}-a_{1}^{2}.$$ Prove that $a_{1}\cdot a_{2}\cdots a_{1964}+2015$ can be written as a difference of two perfect squares.
Let $ABC$ be a triangle with its side $BC$ is fixed and its vertex $A$ varies such that the triangle is not isosceles at $A$. Draw the internal angle bisector $AD$ of the triangle. On the ray $CA$ choose $E$ such that $CE=AB$. Let $I$ be the midpoint of $AE$. Prove that the line which passes through a fixed point.
Find all natural numbers $x,y$ such that $$x^{5}=y^{5}+10y^{3}+20y+1.$$
Given a triangle $PQR$ inscribed in a circle $\left(O\right)$. Let $S$ be the midpoint of the arc $QR$ which does not contain $P$. Draw a circle with center $O'$ passing through $P$ and $S$. This circle intersects $PQ$ and $PR$ respectively at $M$ and $N$. Let $H$ and $K$ respectively be the midpoint of $MN$ and $QR$. Prove that $HK$ is perpendicular to $PS$.
Solve the system of equations $$\begin{cases}x^{3}-x^{2}+x\left(y^{2}+1\right) &=y^{2}-y+1 \\ 2y^{3}+12y^{2}+18y-2+z&=0 \\ 3z^{3}-9z+x-7 &=0\end{cases}$$ where $x,y,z\in\mathbb{R}$.
Given positive real numbers $x,y$ and $z$ such that $$\left(x+y\right)\left(y+z\right)\left(z+x\right)=1.$$ Show that $$\dfrac{\sqrt{x^{2}+xy+y^{2}}}{\sqrt{xy}+1}+\dfrac{\sqrt{y^{2}+yz+z^{2}}}{\sqrt{yz}+1}+\dfrac{\sqrt{z^{2}+zx+x^{2}}}{\sqrt{zx}+1}\geq\sqrt{3}.$$
Solve the equation $$\left(\log\left(x^{2}\left(2-x\right)\right)\right)^{3}+\left(\log x\right)^{2}\cdot\log\left(x\left(2-x\right)^{2}\right)=0.$$
Let $d_{a}$, $d_{b}$, $d_{c}$ respectively be the distances from the orthocenter $H$ of an acute triangle $ABC$ to the sides $BC$, $CA$, $AB$. Let $r$ and $R$ respectively be the inradius and circumradius of $ABC$. Prove that $$ d_{a}+d_{b}+d_{c}\leq\dfrac{3}{4}\cdot\dfrac{R^{2}}{r}.$$
Solve the system of equations $$\begin{cases}x\left(x-2\right)+\left(y-2\right)\left(2z+1\right)&=0 \\ x\left(y+1\right)+\left(y-2\right)\left(5z+1\right)&=0 \\ \sqrt{\left(y+1\right)^{2}+\left(5z+1\right)^{2}}&=2\sqrt{\left(x-2\right)^{2}+\left(2z+1\right)^{2}}\end{cases}$$
Find the maximal number $m$ such that the following inequality holds for all non-negative real numbers $a,b,c,d$ $$\left(ab+cd\right)^{2}+\left(ac+bd\right)^{2}+\left(ad+bc\right)^{2}\geq ma\left(b+c\right)\left(b+d\right)\left(c+d\right).$$
Given natural numbers $m,n>2$. Prove that the number $\dfrac{m^{2^{n}-1}-1}{m-1}$ always has a divisor of the form $m^{a}+1$ where $a$ is a natural number.
Given an acute triangle $ABC$ with the circumcenter $O$. Let $D$, $E$ and $F$ respectively be the midpoints of $BC,CA$ and $AB$. Choose a point $M$ on the line through $BC$. Let $N$ be the intersection between $AM$ and $EF$ and $P$ the second intersection between $ON$ and the circumcircle of the triangle $ODM$. Let $Q$ be the reflection point of $M$ through the midpoint of $DP$. Prove that $Q$ belongs to the circumcircle of the triangle $DEF$.
Issue 464
Find all pairs of natural number $\left(m.n\right)$ satisfying $3^{m}-2^{n}=5$.
Given a right triangle $ABC$ with the right angle $A$ and $AB < AC$. Let $AH$ be the altitude from the vertex $A$. On $AC$ choose $D$ such that $AD=AB$. Let $I$ be the midpoint of $BD$. Prove that $\widehat{BIH}=\widehat{ACB}$.
Can we cover a square-shape area of the size $3,5{\rm m}\times3,5{\rm m}$ by rectangle-shape tiles of the size $25{\rm cm}\times100{\rm cm}$ without cutting any tile?.
Construct a triangle $ABC$ given three lines $d_{a}$, $d_{b}$ and $d_{c}$ which contain perpendicular bisector of $ABC$ (assume that they are concurrent at $O$) and given the length of $AH$ where $H$ is the orthocenter of $ABC$.
Solve the equation $$2010-\sqrt[3]{6+\sqrt[3]{6+\sqrt[3]{2016-x}}}=x.$$
Solve the system of equations $$\begin{cases} \dfrac{12y}{x} & =3+x-2\sqrt{4y-x}\\ \sqrt{y+3}+y & =x^{2}-x-3 \end{cases}.$$
Let $x,y,z$ be real numbers such that $x^{2}+y^{2}+z^{2}=8$. Find the maximum and minimum values of the expression $$H=\left|x^{3}-y^{3}\right|+\left|y^{3}-z^{3}\right|+\left|z^{3}-x^{3}\right|.$$
Given an acute triangle $ABC$ with the circumcenter $O$. Let $P$ be an arbitrary point on the circumcircle of the triangle $ABC$ and $P$ os different from $B$ and $C$. The bisector of the angles $\widehat{CPA}$ and $\widehat{APB}$ respectively intersects $CA$ and $AB$ at $E$ and $F$. Let $I$, $L$ and $K$ respectively be the incenters of the incircles of the triangles $PEF$, $PCA$ and $PAB$. Prove that $I$, $K$ and $L$ are colinear.
Find positive integers $x,y$ such that $x^{3}+y^{3}=x^{2}+12xy+y^{2}$.
Given $n$ real numbers $a_{1},a_{2},\ldots,a_{n}$ $\left(n\geq3\right)$ satisfying $$a_{1}+a_{2}+\ldots+a_{n}\geq n,\quad a_{1}^{2}+a_{2}^{2}+\ldots+a_{n}^{2}\geq n^{2}.$$ Prove that $\max\left\{ a_{1},a_{2},\ldots a_{n}\right\} \geq2$.
Consider the following sequence of real numbers $\left(a_{n}\right)$ $$a_{1} \geq 0,\quad a_{n+1} =10^{n}a_{n}^{2},\\ n\geq 1.$$ Find all possible values for $a_{i}$ so that ${\displaystyle \lim_{n\to\infty}a_{n}=0}$.
Given an acute triangle $ABC$. Let $E$ and $F$ respectively be the perpendicular projecions of $B$ and $C$ on $AC$ and $AB$. Let $I$ and $J$ respectively be the excenters of the excircles relative to the vertices $F$ and $E$ of the triangles $AFC$ and $AEB$. Assume that $BJ$ intersects $CI$ at $K$. Choose $Q$ on the circumcircle of the triangle $BKC$ such that circumcircle of the $BKC$ such that $\widehat{AQK}=90^{\circ}$. Prove that $AQ$, $BI$ and $CJ$ are concurrent.
Issue 465
Let $$S=1\cdot2^{0}+2\cdot2^{1}+3\cdot2^{2}+\ldots+2016\cdot2^{2015}$$ where $2^{0}=1$. Compare $S$ and $2015\cdot2^{2016}$.
Given a right triangle $ABC$ with the right angle $A$ and $AB<AC$. On the opposite ray of the ray $AB$ choose $D$ so that $BD=AC$ and so the opposite ray of the ray $CA$ choose $E$ so that $CE=AD$. The ray $DC$ intersects $BE$ at $F$. Find the angle $\widehat{CFB}$.
Let $a,b,c$ be positive integers such that $$\dfrac{a^{2}+1}{bc}+\dfrac{b^{2}+1}{ca}+\dfrac{c^{2}+1}{ab}$$ is also a posotive integer. Prove that $$\left(a,b,c\right)\leq\left[\sqrt[3]{a+b+c}\right],$$ where $\left(a,b,c\right)$ is the greatest common divisor of $a,b,c$ and $\left[x\right]$ is the integer part of $x$.
Given a right triangle $ABC$ with the right angle $A$ with $AB<AC$, $BC=2+2\sqrt{3}$, and the inradius is equal to $1$. Find the angles $B$ and $C$.
Solve the equation $$\dfrac{1}{\sqrt{x+1}}+\dfrac{1}{\sqrt{2x+1}}+\dfrac{1}{\sqrt{1-2x}}=\dfrac{4\sqrt{10}}{5}.$$
Solve the system of equations $$\begin{cases}e^{x}&=y+\sqrt{z^{2}+1} \\ e^{y}& =z+\sqrt{x^{2}+1} \\ e^{z}&=x+\sqrt{y^{2}+1}\end{cases}$$
Which number is bigger $$\sin\left(\cos x\right)\text{ or }\cos\left(\sin x\right)?$$
Let $A$, $B$ and $C$ be the angles of a triangle. Prove that $$\dfrac{1}{\sin A}+\dfrac{1}{\sin B}+\dfrac{1}{\sin C}\leq\dfrac{2}{3}\left(\cot\dfrac{A}{2}+\cot\dfrac{B}{2}+\cot\dfrac{C}{2}\right).$$
Find the maximal positive value of $T$ such that the following inequality $$\dfrac{a+b}{b\left(a+1\right)}+\dfrac{b+c}{c\left(b+1\right)}+\dfrac{c+a}{a\left(c+1\right)}\geq T$$ always holds for all positive numbers $a,b,c$ satisfying $abc=1$.
Find all pairs of positive integers $x,y$ so that $x^{2}$ is divisible by $2xy^{2}-y^{2}+1$.
Find all monotonic funtions $f:\left(0,+\infty\right)\to\mathbb{R}$ such that $$f\left(x+y\right)=x^{2016}f\left(\dfrac{1}{x^{2015}}\right)+y^{2016}f\left(\dfrac{1}{y^{2015}}\right)$$ for all positive numbers $x$ and $y$.
Given an isosceles triangle $ABC$ with the vertex angle $\hat{A}<90^{0}$. Let $CD$ be the altitude from $C$. Let $E$ be the midpoint of $BD$ and let $M$ be the midpoint of $CE$. The angle bisector of $\widehat{BDC}$ intersects $CE$ at $P$. The circle with center at $C$ and with radius $\left|CD\right|$ intersects $AC$ at $Q$. Let $K=PQ\cap AM$. Prove that $CKD$ is a right triangle.
Issue 466
Let $x,y$ and $z$ be integers satisfying $x\ne y$ and $\dfrac{x}{y}=\dfrac{x^{2}+z^{2}}{y^{2}+z^{2}}$. Prove that $x^{2}+y^{2}+z^{2}$ is not a prime number.
Given a triangle $ABC$ with $AB<AC$. Let $AD$ be the angle bisector. Let $E$ and $F$ respectivly be points on the sides $AB$ and $AC$ such that $BE=CF$. If $G$ and $H$ respectively are the midpoints of $BF$ and $CE$. Prove that $GH\perp AD$.
Given $3$ different positive integers $a,b$ and $c$ satisfying $ab+bc+ca\geq674$. Find the minimum values of the expression $$ P=\dfrac{a^{3}+b^{3}+c^{3}}{3}-abc.$$
Given an acute triangle $ABC$ with the altitude $AH$. Let $E\in AB$ and $F\in AC$ such that $HE$ is perpendicular to $AB$ and $HF$ is perpendicular to $AC$. Assume that $HE$ and $BF$ intersect at $K$ and $HF$ and $CE$ intersect at $G$. Choose $M$ and $N$ respectively on the line segments $HB$ and $HC$ such that $EMHK$ and $FNHK$ are inscribed quadrilaterals. Prove that $KN=GM$.
Solve the system of equations $$\begin{cases}x^{5}+y^{3}&=2z\\ y^{5}+z^{3}&=2x \\z^{5}+x^{3}&=2y\end{cases}$$
Let $a$, $b$ and $c$ be nonnegative numbers such that $a+b+c\leq\pi$. Prove that $$ 0\leq a-\sin a-\sin b-\sin c+\sin\left(a+b\right)+\sin\left(a+c\right)\leq\pi.$$
Let $a,b$ and $c$ be three sides of a triangle. Prove that $$\dfrac{3}{2}<\sqrt{\dfrac{a}{b+c}}+\sqrt{\dfrac{b}{c+a}}+\sqrt{\dfrac{c}{a+b}}<\dfrac{4\pi}{5}.$$
Given a triangle $ABC$ with the altitude $AH$ ($H\in BC$). Let $\left(I\right)$ be the circle with the diameter $AH$ and the center $I$. The circle $\left(I\right)$ intersects $AB$ and $AC$ respectively at $T$ and $S$. The tangent of $\left(I\right)$ at $T$ and $S$ intersect at $M$. Let $N$, $K$, $J$ respectively be the intersections of $BC$ and $MA$, $BS$ and $CT$, $TS$ and $IM$. Prove that $NJ$ goes through the midpoint of $AK$.
Given $a_{1},a_{2},\ldots a_{n}\in\mathbb{R}$, $n\in\mathbb{N}$, $n\geq3$. Find $$\max\left(\min\left\{ \dfrac{a_{1}+a_{2}}{1+a_{3}^{2}};\dfrac{a_{2}+a_{3}}{1+a_{3}^{2}};\ldots;\dfrac{a_{n-2}+a_{n-1}}{1+a_{n}^{2}}; \dfrac{a_{n-1}+a_{n}}{1+a_{1}^{2}};\dfrac{a_{n}+a_{1}}{1+a_{2}^{2}}\right\} \right).$$
Let $n\geq3$ be a natural number. Prove that $\dfrac{(3n)!}{n!(n+1)!(n+2)!}$ is also a natural number.
Given two sequences $\left(x_{n}\right)$ and $\left(y_{n}\right)$ such that $$\begin{cases} \sqrt{y_{n}-x_{n+1}^{2}+1} & =x_{n}\\ \sqrt{3+x_{n+1}^{2}-y_{n}}+\sqrt{10+y_{n-1}-x_{n-1}^{2}} & =\dfrac{5n-1}{n} \end{cases}\quad\forall n\in\mathbb{N}^{*}.$$ Prove that $\left(x_{n}\right)$ and $\left(y_{n}\right)$ converge. Find the limits.
Given an acute triangle $ABC$ which is not isosceles and is inscribed in a circle $\left(O\right)$. Let $P$ be a point inside the triangle such that $AP\perp BC$. Let $E$ and $F$ respectivly be the circumcircle of the triangle $AEF$ intersects $\left(O\right)$ at another point $G$ besides $A$. Prove that $GP$, $BE$ and $CF$ are concurrent.
Issue 467
Given the following increasing sequence $1,3,5,7,9,\ldots$ where all the digits of all the terms are odd. Find the $2016^{{\rm th}}$ term of the above sequence.
Given an acute triangle $ABC$ and the median $AM$. Draw $BH\perp AC$. The line which goes through $A$ and is perpendicular to $AM$ intersects $BH$ at $E$. On the opposite ray of the ray $AE$ choose $F$ so that $AE=AF$. Prove that $CF\perp AB$.
Let $a,b$ and $c$ be three different nonzero rational numbers such that $$\left(\dfrac{a}{b-c}\right)^{2}+\left(\dfrac{b}{c-a}\right)^{2}+\left(\dfrac{c}{a-b}\right)^{2}\leq2.$$ Prove that $$\sqrt{\left(\dfrac{b-c}{a}\right)^{2}+\left(\dfrac{c-a}{b}\right)^{2}+\left(\dfrac{c-a}{b}\right)^{2}}$$ is a rational number.
Given a pentagon $ABCDE$ with $\widehat{ACB}=\widehat{ADE}=90^{0}$ and $\widehat{CAB}=\widehat{DAE}$. Let $O$ be the intersection of $BD$ and $CE$. Prove that $AO\perp DC$.
Solve the equation $$x^{4}-2x^{3}+\sqrt{2x^{3}+x^{2}+2}-2=0.$$
Given three real numbers $x,y$ and $z$ such that $x+y+z=3$ and $xy+yz+zx=-9$. Find the maximum and minimum values of $xyz$.
Let $x,y$ and $x$ be real numbers satisfying $x^{2}+y^{2}+z^{2}=1$. Find the maximum value of the expression $$P=\left(x^{2}-yz\right)\left(y^{2}-zx\right)\left(z^{2}-xy\right).$$
Let $p$ and $R$ respectively be the semiperimeter and the circuradius of the circumcircle of a triangle $ABC$. Prove that $$\left(\dfrac{p}{3R}\right)^{2}+\left(\dfrac{3R}{p}\right)^{2}\geq\dfrac{25}{12}.$$ When does the equality happen?.
Find all real numbers $k>-2$ such that the inequality $$\dfrac{x}{x+y+kz}+\dfrac{y}{y+z+kx}+\dfrac{z}{z+x+ky}\geq\dfrac{3}{k+2}$$ holds for all nonnegative $x,y$ and $z$.
Find all natural numbers $x$ and $y$ such that $$\left(\sqrt{x}-1\right)\left(\sqrt{y}-1\right)=2017.$$
Determine the sequence $\left(x_{n}\right)$ as follows $$ x_{1}=1;\quad x_{n+1}=\dfrac{n+1}{n+2}x_{n}+n^{2},\quad n=1,2,\ldots$$ Find ${\displaystyle \lim_{n\to\infty}\left(\dfrac{\sqrt[3]{x_{n}}}{n+1}\right)}$.
Assume that $ABC$ is a triangle inscribed in a circle $\left(O\right)$ and $AD,BE,CF$ are its altitudes. The line through $AO$ intersects $BC$ at $A'$. Let $M$ be the midpoint of $BC$ and let $S$ be the intersection between two tangent lines of $\left(O\right)$ at $B$ and $C$, The line through $EF$ intersects $SD$ and $SA'$ respectively at $I$ and $J$. Show that $BC$ is tangent to the circumcircle of the triangle $IJM$ at $M$.
Issue 468
Find positive integers $x,y,z$ such that $$2^{x}+3^{y}+5^{z}=136.$$
Given a right triangle $ABC$ with the right angle $A$. Let $AH$ be the altitude. Choose a point $D$ on the opposite ray of the ray $AH$ such that $AD=BC$ and choose $E$ on the opposite ray of the ray $CA$ such that $AB=CE$. The line which goes through $A$ and is perpendicular to $BD$ intersects $BD$ and $DE$ repectively at $I$ and $K$. Find the angle $\widehat{CKE}$.
Find all pairs of primes $\left(p,q\right)$ such that $p^{2}-pq-q^{3}=27$.
Given a rhombus $ABCD$ with $\widehat{BAD}=120^{0}$. The points $M$ and $N$ respectively vary on the sides $BC$ and $CD$ such that $\widehat{MAN}=30^{0}$. Find the locus of the circumcenter $O$ of the triangle $AMN$.
Solve the system of equations $$\begin{cases}\left(x+y\right)^{2}+\sqrt{3\left(x+y\right)} & =\sqrt{2\left(x+y+1\right)}+4\\ \left(x^{2}+y-2\right)\sqrt{2x+1} & =x^{3}+2y-5.\end{cases}$$
Solve the equation $$-3x^{2}+x+3+\left(\sqrt{3x+2}-4\right)\sqrt{3x-2x^{2}}+\left(x-1\right)\sqrt{3x+2}=0.$$
Let $ABC$ be an acute triangle. Prove that $$\tan^{2}A+\tan^{2}B+\tan^{2}C>4\left(\cot^{2}A+\cot^{2}B+\cot^{2}C\right). $$
Let $ABCD$be a tetrahedron inscribed in a unit sphere $O$. Assume futher more that $O$ lies inside the $ABCD$. Prove that at least one face of $ABCD$ has perimeter greater than $2\sqrt{3}$.
Find the maximum and minimum values of the expression $$A=\dfrac{x^{4}+y^{4}+2xy+9}{x^{4}+y^{4}+3},\quad\forall x,y\in\mathbb{R}^{3}.$$
A worker has to tile floors of sizes $2^{n}\times2^{n}$ units square either by domino shaped tiles or by set square shaped tiles which are both 3 unites square. For every floor, he has to cover all except one $1\times1$ unit square spot which is finised later for the decoration purpose. Since ser square shaped tiles are much more expensive than the other tiles are much more expensive than the other ones, he wants to use them the fewer the better. He observes that no matter where the special spot is, he never needs more than $n$ set square shaped tiles. Show that his observation is mathematically correct for any positive integer $n$.
Let $a,,b,c$ and $S$ respectively the sides and the area of a given triangle. Let $x,y$ and $z$ be positive real numbers. Prove that a) $xa^{2}+yb^{2}+zc^{2}\geq4\sqrt{xy+yz+zx}S$ b) $\left(y+z\right)bc+\left(z+x\right)ca+\left(x+y\right)ab-xa^{2}-yb^{2}-zc^{2}\geq4\sqrt{xy+yz+zx}S.$ When do the equalities happen?.
Let $ABCD$ be a quadrilateral circumscribed about a circle $\left(I\right)$. The rays $AB$ and $CD$ intersect at $E$, the rays $DA$ and $CB$ intersect at $F$. Let $\left(I_{1}\right),\left(I_{2}\right)$ respectively be the incircles of the triangles $EFB$ and $EFD$. Prove that $\widehat{I_{1}IB}=\widehat{I_{2}ID}$.
Issue 469
Compare $A$ and $B$ where $$A=\dfrac{1}{2!}+\dfrac{2}{3!}+\dfrac{3}{4!}+\ldots+\dfrac{2015}{2016!}$$ (with $n!$ is the usual notation for the product $1,2\ldots,n$) and $B=1,02015$.
Given 5 line segments where any 3 of them can form a triangle. Prove that there is at least one acute triangle among the ones which can be formed bt any 3 line segments.
Suppose that $a,b$ and $c$ are positive numbers and $n$ is an integer which is greater or equal to 2. Prove that $$\sqrt[n]{\dfrac{a}{a+nb}}+\sqrt[n]{\dfrac{b}{b+nc}}+\sqrt[n]{\dfrac{c}{c+na}}>1.$$
Given a rhombus $ABCD$ and let the length of the side $AB$ be $2a$. Let $R_{1},R_{2}$ respectively be the circumradius of the triangles $ABC$ and $ABD$. Prove that $R_{1}R_{2}\geq2a^{2}$. When do we have the equality?.
Find integral solutions of the following equation $$\dfrac{11x}{5}-\sqrt{2x+1}=3y-\sqrt{4y-1}+2.$$
Given a system of equations $$\begin{cases} x\sqrt{y-m}+y\sqrt{z-m}+z\sqrt{x-m} & =6m\sqrt{m},\\ x^{2}+y^{2}+z^{2} & =12m^{2}, \end{cases}$$ where $m$ is a positive number. Find $x,y$ and $z$ in terms $m$.
Find the maximum and minimum of the function $$y=f\left(x\right)=\cos x+4\cos\dfrac{x}{2}+7\cos\dfrac{x}{4}+6\cos\dfrac{x}{8}.$$
Given a triangle $ABC$. Let $M$ be the midpoint of $BC$. Prove that if $\widehat{BAC}\leq\dfrac{\pi}{2}$ then $$2AM\leq\sqrt{2\left(AB^{2}+AC^{2}\right)}\cos\dfrac{\widehat{BAC}}{2}.$$ When do we have the equality?.
Let $x,y$ and $z$ be positive numbers such that $xyz=1$. Prove that $$x+y+z+xy+yz+zx\leq 3+\left(\dfrac{x}{y}\right)^{n}+\left(\dfrac{y}{z}\right)^{n}+\left(\dfrac{z}{x}\right)^{n}$$ for every $n\in\mathbb{N}^{*}$.
Find the largest $n$ for which there exist $n$ different positive numbers $x_{1},x_{2},\ldots,x_{n}$ such that $$\dfrac{x_{i}}{x_{j}}+\dfrac{x_{j}}{x_{i}}+8\left(\sqrt{3}-2\right)\geq\left(7-4\sqrt{3}\right)\left(\dfrac{1}{x_{i}x_{i}}+x_{i}x_{j}\right) $$ for any two different indices $i,j$.
Find all functions $f:\mathbb{R}^{+}\to\mathbb{R}$ satisfying $$f\left(\dfrac{x+y}{2016}\right)=\dfrac{f\left(x\right)+f\left(y\right)}{2015}$$ for all $x,y\in\mathbb{R}^{+}$.
Two circles $\left(O\right)$ and $\left(O_{1}\right)$ intersect at $B$ and $C$, Let $M$ be the midpoint of $BC$. Let $A$ be a point which varies on $\left(O\right)$ but is different from $B$ and $C$. The lines $AB$ and $CA$ respectively intersect $\left(O_{1}\right)$ at $F$ and $E$. Assume that $P$ and $Q$ respectively be the perpendicular projections of $M$ on $BE$ and $CF$. Construct the parallelogram $MPKQ$. Prove that $AK$ always goes through a fixed point.
Issue 470
Find all integers $x,y,z$ and $t$ such that $$38\left(xyzt+xy+xt+zt+1\right)=49\left(yzt+y+t\right).$$
Given an isosceles triangle $ABC$ with the vertex angle $A$. Let $M$ be a point inside the triangle such that $\widehat{AMB}>\widehat{AMC}$, and $N$ is a point between $A$ and $M$. Prove that $\widehat{ANB}>\widehat{ANC}$.
Find the maximal $k$ such that there exist $k$ positive integers which do not exceed 100 and have the property that any number among them cannot be a power of any remain one.
Assume that $ABC$ be a triangle inscribed in the semicircle with the center $O$ and the diameter $BC$. Two tangent lines to the semicircle $\left(O\right)$ at $A$ and $B$ intersect at $D$. Prove that $DC$ goes through the midpoint $I$ of the altitude $AH$ of $ABC$.
Find positive integer solutions of the equation $$xyz=x+2y+3z-5.$$
Let $a,b$ and $c$ are real numbers such that $\left|a+b\right|+\left|b+c\right|+\left|c+a\right|=8$. Find the maximum and minimum values of the expression $P=a^{2}+b^{2}+c^{2}$.
Solve the system of equations $$\begin{cases} x & =2^{1-y},\\ y & =2^{1-x}. \end{cases}$$
Let $ABC$ be a triangle without any obtuse angle. Prove that $$\cos A\cos B+\cos B\cos C+\cos C\cos A\geq2\sqrt{\cos A\cos B\cos C}.$$
Given $n$ non-negative real numbers $x_{i}$, $i=1,2,\ldots,n$ $\left(n\geq2\right)$ satisfying $x_{1}+x_{2}+\ldots+x_{n}=1$. Show that $$\dfrac{1}{n}\left(\dfrac{x_{1}}{1+x_{1}}+\dfrac{x_{2}}{1+x_{2}}+\ldots\dfrac{x_{n}}{1+x_{n}}\right)<\dfrac{x_{1}^{2}}{1+x_{1}^{2}}+\dfrac{x_{2}^{2}}{1+x_{2}^{2}}+\ldots\dfrac{x_{n}^{2}}{1+x_{n}^{2}}.$$
A natural number is called $k$-\emph{success} if the sum of its digits equals to $k$. Let $a_{ik}$ be the number of all $k$-success numbers which have $i$ digits. Find the sum $\sum_{k=1}^{m}\sum_{i=1}^{n}a_{ik}$ where $k,i,m,n$ are positive integers.
Find all triples $\left(x,y,p\right)$ of two non-negative integers $x,y$ and a prime number $p$ such that $p^{x}-y^{p}=1$.
Given a triangle ABC and $H$ is a point inside the triangle such that $\widehat{ABH}=\widehat{ACH}$. A circle which goes through $B$ and $C$ intersects the circle with the diameter $AH$ at two different points $X$ and $Y$, $AH$ intersects $BC$ at $D$. Let $K$ be the perpendicular projection of $D$ on $XY$. Prove that $\widehat{BKD}=\widehat{CKD}$.
Issue 471
Let $a,b$ and $c$ be three integers such that $abc=2015^{2016}$. Find the remainder when $19a^{2}+5b^{2}+1890c^{2}$ is divided by $24$.
Let $a,b\in\mathbb{Z}$. Prove that if $10\mid\left(a^{2}+ab+b^{2}\right)$ then $1000\mid\left(a^{3}-b^{3}\right)$.
Solve the equation $$\sqrt{2x-2}+\sqrt[3]{x-2}=\dfrac{9-x}{\sqrt[3]{8x-16}}.$$
Given a triangle $ABC$. Outside the triangle, we draw two equilateral triangles $ABD$ and $ACE$. Let $O$ be the centroid of $ACE$. On the opposite ray of the ray $OE$ choose $F$ such that $OF=OE$. Prove that $DF=BO$.
Consider the quadratic function $f\left(x\right)=20x^{2}-11x+2016$. Show that there exists an integer $\alpha$ such that $2^{20^{11^{1960}}}\mid f\left(\alpha\right)$
Solve the equation $$\dfrac{\log_{2}x}{x^{4}-10x^{2}+26}=\dfrac{5-x^{2}}{\log_{2}^{2}x+1}.$$
Let $x$ and $y$ be positive numbers such that $\left(\sqrt{x}+1\right)\left(2\sqrt{y}+4\right)+y\geq13$. Find the minimum value of the expression $$P=\dfrac{x^{4}}{y}+\dfrac{y^{3}}{x}+y.$$
Given an acute triangle $ABC$. Prove that $$\sqrt[3]{\cot A+\cot B}\geq\dfrac{16}{3}\cot A\cot B\cot C.$$
Let $a,b$ and $c$ be three positive numbers such that $a+b+c=abc$. Prove that the following inequality $$\begin{align*} & \sqrt{\left(1+a^{2}\right)\left(1+b^{2}\right)}+\sqrt{\left(1+b^{2}\right)\left(1+c^{2}\right)}+\sqrt{\left(1+c^{2}\right)\left(1+c^{2}\right)}\\ \geq & \sqrt{\left(1+a^{2}\right)\left(1+b^{2}\right)\left(1+c^{2}\right)}+4. \end{align*}$$
A school organizes 7 summer classes. Each student in the school attend at lease one class and each class has exactly 40 students. Besides, for any two classes, there are no more 9 students attending both of them. Prove that the number of students in that school is at least 120.
Find all continuous functions $f:\mathbb{R}\to\mathbb{R}$ such that $$f\left(xy\right)=f\left(\dfrac{x^{2}+y^{2}}{2}\right)+\left(x-y\right)^{2}$$ for all $x,y\in\mathbb{R}$.
Given a triangle $ABC$ and let $I$ be its incenter. Assume that $\Delta$ is the line which goes through $I$ and is perpendicular to $AI$. The points $E$ and $F$ belong to $\Delta$ such that $\widehat{EBA}=\widehat{FCA}=90^{0}$. The points $M$ and $N$ belong to $BC$ such that $ME\parallel NF\parallel AI$. Prove that the circumcircles of the triangles $ABC$ and $AMN$ are tangent to each other.
Issue 472
Find all $3$ digit perfecy squares such that if we pick any of those numbers, say $n^{2}$, and then interchange the tens digit and the units digit of that one, we will get $\left(n+1\right)^{2}$.
Consider the following sequence $1$; $3+5$; $7+9+11$; $13+15+17+19$; $\ldots$. Prove that each term of the sequence is a cube of some positive integer.
Find integers $x$ and $y$ such that $x^{2}-2x=27y^{3}$.
Given a triangle $ABC$ with $AB^{2}-AC^{2}=\dfrac{BC^{2}}{2}$. Suppose futhermore that the angle $BAC$ is obtuse. Let $D$ be the point on the side $AB$ such that $BC=2CD$. The line which goes through $D$ and is perpendicular to $AB$ intersects the line through $AC$ at $E$. Let $K$ be the intersection between the line through $CD$ the line through $BE$. Prove that $K$ is the midpoint of $BE$.
Given positive numbers $a$ and $b$. Prove that $$\dfrac{a+b}{1+ab}+\left(\dfrac{1}{1+a}+\dfrac{1}{1+b}\right)ab+\dfrac{a+b+2ab}{\left(1+a\right)\left(1+b\right)ab}\geq3.$$ When does the equality happen?.
Solve the equation $$\sqrt[3]{2x^{3}+6}=x+\sqrt{x^{2}-3x+3}.$$
Find the integral solutions of the inequality $$x^{6}-2x^{3}-6x^{2}-6x-17<0.$$
Given a scalene triangle $ABC$ with the orthocenter $H$. The incircle $\left(I\right)$ of the triangle if tangent to the sides $BC$, $CA$, $AB$ respectively at $A_{1}$, $B_{1}$, $C_{1}$. The line $d_{1}$ which goes through $I$ and is paralled to $BC$ intersects the line through $\left(B_{1}C_{1}\right)$ at $A_{2}$. Similarly, we construct the points $B_{2}$, $C_{2}$. Prove that $A_{2}$, $B_{2}$, $C_{2}$ are colinear and the line through these points is perpendicular to $IH$.
Given the equation $$x^{n}+a_{1}x^{n-1}+\ldots+a_{n-1}x+a_{n}=0$$ where the coefficients satisfy $$\begin{align*} \sum_{i=1}^{n}a_{i} & =0,\\ \,\sum_{i=1}^{n-1}\left(n-1\right)a_{i} & =2-n,\\ \sum_{i=1}^{n-2}\left(n-i\right)\left(n-i-1\right)a_{i} & =3-n\left(n-1\right). \end{align*}$$ Suppose that $x_{1},x_{2},\ldots,x_{n}$ are $n$ solutions of the above equation. Compute the value of the following expressions $$\begin{align*} E & =\sum_{1\leq i<j\leq n}\dfrac{1}{\left(1-x_{i}\right)\left(1-x_{i}\right)},\\ F & =\sum_{i=1}^{n}\dfrac{1}{\left(1-x_{i}\right)^{2}}. \end{align*}$$
There are $20$ stones and we divide them into $3$ piles. Then we start to moving stones from one pile to another with the follow rule. Each time we can move half of a pile with even number of stones to any pile. Prove that nomatter how we divide the stones into $3$ piles, after finite of moves, we will get a pile with $10$ stones.
For each positive integer $t$, let $\phi\left(t\right)$ denote the numbers of positive integers which are not greater than $t$ and are relatively prime to $t$ (Euler phi function). Now assume that $n,k$ are positive integers and $p$ is an odd prime number. Prove that there exists a positive integer $a$ such that $p^{k}$ is a divisor of all numbers $\phi\left(a\right),\phi\left(a+1\right),\ldots$ and $\phi\left(a+n\right)$.
Given an acute triangle $ABC$ with the altitudes $AD$, $BE$ and $CF$. The circles with the diameters $AB$ and $AC$ respectively intersect the rays $DF$ and $DE$ at $Q$ and $P$. Let $N$ be the cicumcenter of the triangle $DEF$. Prove that a) $AN\perp PQ$. b) $AN$, $BP$ and $CQ$ are concurrent.
Issue 473
Find the last two digits of the difference $2^{9^{2016}}-2^{9^{1945}}$.
Find natural numbers $n$ such that $$n^{6}+8n^{3}+19n^{2}-33n-90=0.$$
Find positive integers $x$ and $y$ so that $$A=x^{2}+y^{2}+\dfrac{x^{2}y^{2}}{\left(x+y\right)^{2}}$$ is a perfect square.
Given an acute triangle $ABC$ with two altitudes $BE$ and $CK$. Let $R$ and $S$ respectively be the perpendicular projections of $K$ on $BC$ and $BE$, and $P$ and $Q$ respectively the perpendicular projections of $E$ on $BC$ and $CK$. Prove that the lines $RS,PQ$ and $EK$ are concurrent.
Solve the system of equations $$\begin{cases} 5x^{2}+2y^{2}+z^{2} & =2\\ xy+yz+zx & =1 \end{cases}$$
Find integers $m$ so that the equation $$x^{3}+\left(m+1\right)x^{2}-\left(2m-1\right)x-\left(2m^{2}+m+4\right)=0$$ has an integer solution.
Let $x,y,x$ be real numbers satisfying $$\begin{cases} x+z-yz & =1\\ y-3z+xz & =1 \end{cases}.$$ Find the maximum and minimum values of the expression $T=x^{2}+y^{2}$.
Given a triangle $ABC$. Let $\left(O\right)$ be its circumcircle. The excircle $\left(I\right)$ relative to the vertex $A$ is tangent to $AB$ and $AC$ respectively at $D$ and $E$. Let $A'$ be the intersection between $OI$ and $DE$. Similarly, we construct the points $B'$ and $C'$. Prove that $AA',BB'$ and $CC'$ are concurrent.
Let $a,b,c$ be positive numbers such that $a+b+c=1$. Prove that $$a^{b}b^{c}c^{a}\leq ab+bc+ca.$$
Show that there exist infinitely many positive integers $n$ so that $n^{2}+1$ has a prime divisor $p$ which is greater than $2n+\sqrt{10n}$.
Given two arbitrary positive integers $n$ and $p$. Find the number of functions $$f:\left\{ 1,2,3,\ldots,n\right\} \to\left\{ -p,-p+1,-p+2,\ldots,p\right\}$$ which satisfy the property $\left|f\left(i\right)-f\left(j\right)\right|\leq p$ for any $i,j\in\left\{ 1,2,3,\ldots,n\right\} $.
Given a non-right triangle $ABC$. Let $O$ and $H$ respectively be the circumcenter and the orthocenter of $ABC$. Choose an arbitrary point $M$ on $\left(O\right)$ so that $M$ is different from $A,B$ and $C$. Let $N$ denote the symmetric point of $M$ through $BC$. Let $P$ be the second intersection between $AM$ and the circumcircle of $OMN$. Prove that $HN$ goes through the orthocenter of $AOP$.
Issue 474
Prove that there do not exist two coprime integers $a$ and $b$ which are greater than $1$ and satisfy the fact that $a^{2007}+b^{2007}$ is divisible by $a^{2006}+b^{2006}$.
Find prime numbers $a,b,c,d,e$ such that $$a^{4}+b^{4}+c^{4}+d^{4}+e^{4}=abcde.$$
Given $f\left(x\right)=a^{2016}x^{2}+bx+a^{2016}c-1$ where $a,b,c\in\mathbb{Z}$. Suppose that the equation $f\left(x\right)=-2$ has two positive integral solutions. Prove that $$A=\dfrac{f^{2}\left(1\right)+f^{2}\left(-1\right)}{2}$$ is a composite number.
Given a triangle $ABC$ and $\left(I\right)$ is its incircle. Let $D$ be the point of tangency between $\left(I\right)$ and $BC$. The line which goes through $D$ and is perpendicular to $AI$ intersects $IB$ and $IC$ respectively at $P$ and $Q$. Prove that all $B$, $C$, $P$ and $Q$ lie on a circle whose center is on $AD$.
Find all the real numbers $k$ such that for all non-negative numbers $a,b,c$ we always have the following inequality $$\left[a+k\left(b-c\right)\right]\left[b+k\left(c-a\right)\right]\left[c+k\left(a-b\right)\right]\leq abc.$$
Find the minimum value of the expression $$f= \sqrt{x^{2}-2x+2}+\sqrt{x^{2}-8x+32}+\sqrt{x^{2}-6x+25}+\sqrt{x^{2}-4x+20}+\sqrt{x^{2}-10x+26}.$$
How many solutions does the following equations have $$\log_{\dfrac{5\pi}{2}}x=\cos x.$$
Given a tetrahedron $SABC$ such that $SA$, $SB$ and $SC$ are mutually perpendicular. A moving plane $\left(P\right)$ which contains the centroid of the given tetrahedron intersects $SA,SB$ and $SC$ respectively at $A_{1},B_{1}$, and $C_{1}$. Prove that $$\dfrac{1}{SA_{1}^{2}}+\dfrac{1}{SB_{1}^{2}}+\dfrac{1}{SC_{1}^{2}}\geq\dfrac{4}{R^{2}},$$ where $R$ is the radius of the circumscribed sphere of the tetrahedron $SABC$.
Find the integral part of the following number $$T=\dfrac{2}{1}\cdot\dfrac{4}{3}\cdot\dfrac{6}{5}\ldots\dfrac{2016}{2015}.$$
A positive interger is called a "HV2015'' number if in its decima representation there are $2015$ consecutive digits $9$. Let a stricly increasing sequence of positive integers such that the sequence $\left(\dfrac{a_{n}}{n}\right)$, $n=1,2,3,\ldots$ is bounded. Prove that there are infinitely many "HV2015'' numbers in the given sequence $\left(a_{n}\right),n=1,2,3,\ldots$.
Given a grid of the size $1\times n$ $\left(n>2\right)$. Now we fill in each square with $0$ or $1$. A way to fill the grid is called "good'' if any three consecutive squares do not contain the same number. Let $a_{n}$ be the number of "good'' ways to fill the grid. Compute $a_{n}$.
Given a triangle $ABC$. Prove that $$\sin\dfrac{A}{2}+\sin\dfrac{B}{2}+\sin\dfrac{C}{2}\geq\dfrac{1}{\sqrt{2}}\left(\sqrt{\sin\dfrac{A}{2}}+\sqrt{\sin\dfrac{B}{2}}+\sqrt{\sin\dfrac{C}{2}}\right).$$.
