Found problems: 2529
Let $ a_0$, $ a_1$, $ a_2$, $ \ldots$ be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, $ \gcd (a_i, a_{i \plus{} 1}) > a_{i \minus{} 1}$. Prove that $ a_n\ge 2^n$ for all $ n\ge 0$.
[i]Proposed by Morteza Saghafian, Iran[/i]
A deck of $n > 1$ cards is given. A positive integer is written on each card. The deck has the property that the arithmetic mean of the numbers on each pair of cards is also the geometric mean of the numbers on some collection of one or more cards.
For which $n$ does it follow that the numbers on the cards are all equal?
[i]Proposed by Oleg Košik, Estonia[/i]
Find all $(a, b, c, d)\in \mathbb{R}$ satisfying
\[\begin{aligned}
\begin{cases}
a+b+c+d=0,\\
a^2+b^2+c^2+d^2=12,\\
abcd=-3.\\
\end{cases}
\end{aligned}\]
There are some checkers in $n\cdot n$ size chess board.Known that for all numbers $1\le i,j\le n$ if checkwork in the intersection of $i$ th row and $j$ th column is empty,so the number of checkers that are in this row and column is at least $n$.Prove that there are at least $\frac{n^2}{2}$ checkers in chess board.
The sequence $(a_n)$ is defined by $a_1=1,a_2=\frac{1}{2}$,$$n(n+1)
a_{n+1}a_{n}+na_{n}a_{n-1}=(n+1)^2a_{n+1}a_{n-1}(n\ge 2).$$
Prove that $$\frac{2}{n+1}<\sqrt[n]{a_n}<\frac{1}{\sqrt{n}}(n\ge 3).$$
A tree with $n\geq 2$ vertices is given. (A tree is a connected graph without cycles.) The vertices of the tree have real numbers $x_1,x_2,\dots,x_n$ associated with them. Each edge is associated with the product of the two numbers corresponding to the vertices it connects. Let $S$ be a sum of number across all edges. Prove that \[\sqrt{n-1}\left(x_1^2+x_2^2+\dots+x_n^2\right)\geq 2S.\]
(Author: V. Dolnikov)
If $a$, $b$ ,$c$ are three positive real numbers such that $ab+bc+ca = 1$, prove that \[ \sqrt[3]{ \frac{1}{a} + 6b} + \sqrt[3]{\frac{1}{b} + 6c} + \sqrt[3]{\frac{1}{c} + 6a } \leq \frac{1}{abc}. \]
Let $a_1,a_2,\ldots a_n,k$, and $M$ be positive integers such that
$$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=k\quad\text{and}\quad a_1a_2\cdots a_n=M.$$
If $M>1$, prove that the polynomial
$$P(x)=M(x+1)^k-(x+a_1)(x+a_2)\cdots (x+a_n)$$
has no positive roots.
We have $2^m$ sheets of paper, with the number $1$ written on each of them. We perform the following operation. In every step we choose two distinct sheets; if the numbers on the two sheets are $a$ and $b$, then we erase these numbers and write the number $a + b$ on both sheets. Prove that after $m2^{m -1}$ steps, the sum of the numbers on all the sheets is at least $4^m$ .
[i]Proposed by Abbas Mehrabian, Iran[/i]
Let $a,b,c$ be real numbers such that $a+b+c = 4$ and $a,b,c > 1$. Prove that:
\[\frac 1{a-1} + \frac 1{b-1} + \frac 1{c-1} \ge \frac 8{a+b} + \frac 8{b+c} + \frac 8{c+a}\]
Prove that for positive real numbers $a,b,c$ such that $a+b+c=1$,
$$a\sqrt{2b+1}+b\sqrt{2c+1}+c\sqrt{2a+1}\le \sqrt{2-(a^2+b^2+c^2)}.$$
Let $ a,b,c,d,e,f,g$ and $ h$ be distinct elements in the set
\[ \{ \minus{} 7, \minus{} 5, \minus{} 3, \minus{} 2,2,4,6,13\}.
\]What is the minimum possible value of
\[ (a \plus{} b \plus{} c \plus{} d)^2 \plus{} (e \plus{} f \plus{} g \plus{} h)^2
\]$ \textbf{(A)}\ 30\qquad
\textbf{(B)}\ 32\qquad
\textbf{(C)}\ 34\qquad
\textbf{(D)}\ 40\qquad
\textbf{(E)}\ 50$
Let $a,b,c,d \in \mathbb{R}^+$ and suppose that all roots of the equation
\begin{align*} x^5-ax^4+bx^3-cx^2+dx=1 \end{align*}
are real. Prove
\begin{align*} \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} \le \frac{3}{5} \end{align*}
Let $x$ and $y$ be two positive reals. Prove that $xy\le\frac{x^{n+2}+y^{n+2}}{x^n+y^n}$ for all non-negative integers $n$.
Let $f$ be a real function with a continuous third derivative such that $f(x)$, $f^\prime(x)$, $f^{\prime\prime}(x)$, $f^{\prime\prime\prime}(x)$ are positive for all $x$. Suppose that $f^{\prime\prime\prime}(x)\leq f(x)$ for all $x$. Show that $f^\prime(x)<2f(x)$ for all $x$.
For all positive real numbers $a, b, c$ satisfying $ab+bc+ca \leq 1,$ prove that
\[ a+b+c+\sqrt{3} \geq 8abc \left(\frac{1}{a^2+1}+\frac{1}{b^2+1}+\frac{1}{c^2+1}\right) \]
Positive real numbers $a, b, c$ satisfy $\frac{1}{a} +\frac{1}{b} +\frac{1}{c} = 3.$ Prove the inequality \[\frac{1}{\sqrt{a^3+ b}}+\frac{1}{\sqrt{b^3 + c}}+\frac{1}{\sqrt{c^3 + a}}\leq \frac{3}{\sqrt{2}}.\]
Let $a, b,c$ be positive real numbers.
Prove that $ \sqrt{a^3b+a^3c}+\sqrt{b^3c+b^3a}+\sqrt{c^3a+c^3b}\ge \frac43 (ab+bc+ca)$
Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of
$$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$
[i]Israel[/i]
Let $ P $ be a point in the interior of a triangle $ ABC $, and let $ D, E, F $ be the point of intersection of the line $ AP $ and the side $ BC $ of the triangle, of the line $ BP $ and the side $ CA $, and of the line $ CP $ and the side $ AB $, respectively. Prove that the area of the triangle $ ABC $ must be $ 6 $ if the area of each of the triangles $ PFA, PDB $ and $ PEC $ is $ 1 $.
In a $999 \times 999$ square table some cells are white and the remaining ones are red. Let $T$ be the number of triples $(C_1,C_2,C_3)$ of cells, the first two in the same row and the last two in the same column, with $C_1,C_3$ white and $C_2$ red. Find the maximum value $T$ can attain.
[i]Proposed by Merlijn Staps, The Netherlands[/i]
The $8$ eyelets for the lace of a sneaker all lie on a rectangle, four equally spaced on each of the longer sides. The rectangle has a width of $50$ mm and a length of $80$ mm. There is one eyelet at each vertex of the rectangle. The lace itself must pass between the vertex eyelets along a width side of the rectangle and then crisscross between successive eyelets until it reaches the two eyelets at the other width side of the rectrangle as shown. After passing through these final eyelets, each of the ends of the lace must extend at least $200$ mm farther to allow a knot to be tied. Find the minimum length of the lace in millimeters.
[asy]
size(200);
defaultpen(linewidth(0.7));
path laceL=(-20,-30)..tension 0.75 ..(-90,-135)..(-102,-147)..(-152,-150)..tension 2 ..(-155,-140)..(-135,-40)..(-50,-4)..tension 0.8 ..origin;
path laceR=reflect((75,0),(75,-240))*laceL;
draw(origin--(0,-240)--(150,-240)--(150,0)--cycle,gray);
for(int i=0;i<=3;i=i+1)
{
path circ1=circle((0,-80*i),5),circ2=circle((150,-80*i),5);
unfill(circ1); draw(circ1);
unfill(circ2); draw(circ2);
}
draw(laceL--(150,-80)--(0,-160)--(150,-240)--(0,-240)--(150,-160)--(0,-80)--(150,0)^^laceR,linewidth(1));[/asy]
a，b，c>0，abc=1，prove that(a/b)+(b/c)+(c/a)≥a+b+c.
Let $a$, $b$, $c$ be positive reals such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$. Show that $$a^abc+b^bca+c^cab\ge 27bc+27ca+27ab.$$
[i]Proposed by Milan Haiman[/i]
If $a$, $b$ ,$c$ are three positive real numbers such that $ab+bc+ca = 1$, prove that \[ \sqrt[3]{ \frac{1}{a} + 6b} + \sqrt[3]{\frac{1}{b} + 6c} + \sqrt[3]{\frac{1}{c} + 6a } \leq \frac{1}{abc}. \]