Found problems: 6530
2019 Ramnicean Hope, 3
For this exercise, $ \{\} $ denotes the fractional part.
[b]a)[/b] Let be a natural number $ n. $ Compare $ \left\{ \sqrt{n+1} -\sqrt{n} \right\} $ with $ \left\{ \sqrt{n} -\sqrt{n-1} \right\} . $
[b]b)[/b] Show that there are two distinct natural numbers $ a,b, $ such that $ \left\{ \sqrt{a} -\sqrt{b} \right\} =\left\{ \sqrt{b} -\sqrt{a} \right\} . $
[i]Traian Preda[/i]
2021 Bundeswettbewerb Mathematik, 1
A cube with side length $10$ is divided into two cuboids with integral side lengths by a straight cut. Afterwards, one of these two cuboids is divided into two cuboids with integral side lengths by another straight cut.
What is the smallest possible volume of the largest of the three cuboids?
1994 Abels Math Contest (Norwegian MO), 3a
Let $x_1,x_2,...,x_{1994}$ be positive real numbers. Prove that
$$\left(\frac{x_1}{x_2}\right)^{\frac{x_1}{x_2}}\left(\frac{x_2}{x_3}\right)^{\frac{x_2}{x_3}}...\left(\frac{x_{1993}}{x_{1994}}\right)^{\frac{x_{1993}}{x_{1994}}} \ge \left(\frac{x_1}{x_2}\right)^{\frac{x_2}{x_1}}\left(\frac{x_2}{x_3}\right)^{\frac{x_3}{x_2}}...\left(\frac{x_{1993}}{x_{1994}}\right)^{\frac{x_{1994}}{x_{1993}}}$$
2014 China Northern MO, 2
Define a positive number sequence sequence $\{a_n\}$ by \[a_{1}=1,(n^2+1)a^2_{n-1}=(n-1)^2a^2_{n}.\]Prove that\[\frac{1}{a^2_1}+\frac{1}{a^2_2}+\cdots +\frac{1}{a^2_n}\le 1+\sqrt{1-\frac{1}{a^2_n}}
.\]
2018 Bulgaria JBMO TST, 1
For real numbers $a$ and $b$, define
$$f(a,b) = \sqrt{a^2+b^2+26a+86b+2018}.$$
Find the smallest possible value of the expression $$f(a, b) + f (a,-b) + f(-a, b) + f (-a, -b).$$
2010 Indonesia TST, 2
Consider a polynomial with coefficients of real numbers $ \phi(x)\equal{}ax^3\plus{}bx^2\plus{}cx\plus{}d$ with three positive real roots. Assume that $ \phi(0)<0$, prove that \[ 2b^3\plus{}9a^2d\minus{}7abc \le 0.\]
[i]Hery Susanto, Malang[/i]
2010 Romania Team Selection Test, 3
Let $n$ be a positive integer number. If $S$ is a finite set of vectors in the plane, let $N(S)$ denote the number of two-element subsets $\{\mathbf{v}, \mathbf{v'}\}$ of $S$ such that
\[4\,(\mathbf{v} \cdot \mathbf{v'}) + (|\mathbf{v}|^2 - 1)(|\mathbf{v'}|^2 - 1) < 0. \]
Determine the maximum of $N(S)$ when $S$ runs through all $n$-element sets of vectors in the plane.
[i]***[/i]
2022 Bulgarian Autumn Math Competition, Problem 12.3
The sequence $a_{n}$ is defined by $a_{1}\geq 2$ and the recurrence formula
\[a_{n+1}=a_{n}\sqrt{\frac{a_{n}^3+2}{2(a_{n}^3+1)}}\]
for $n\geq 1$. Prove that for every integer $n$, the inequality $a_{n}>\sqrt{\frac{3}{n}}$ holds.
2006 Croatia Team Selection Test, 4
Find all natural solutions of $3^{x}= 2^{x}y+1.$
2010 China Team Selection Test, 3
Let $n_1,n_2, \cdots, n_{26}$ be pairwise distinct positive integers satisfying
(1) for each $n_i$, its digits belong to the set $\{1,2\}$;
(2) for each $i,j$, $n_i$ can't be obtained from $n_j$ by adding some digits on the right.
Find the smallest possible value of $\sum_{i=1}^{26} S(n_i)$, where $S(m)$ denotes the sum of all digits of a positive integer $m$.
1984 Austrian-Polish Competition, 8
The functions $f_0,f_1 : (1,\infty) \to (1,\infty)$ are given by $ f_0(x) = 2x$ and$ f_1(x) =\frac{x}{x-1}$. Show that for any real numbers $a, b$ with $1 \le a < b$ there exist a positive integer $k$ and indices $i_1,i_2,...,i_k \in \{0,1\}$ such that $a <f_{i_k}(f_{i_{k-1}}(...(f_{i_j}(2))...))< b$.
1996 Greece Junior Math Olympiad, 3
Determine the minimum value of the expression $2x^4 - 2x^2y^2 + y^4 - 8x^2 + 18$ where $x, y \in R$.
1998 Poland - First Round, 2
Show that for all real numbers $ a,b,c,d,$ the following inequality holds:
\[ (a\plus{}b\plus{}c\plus{}d)^2 \leq 3 (a^2 \plus{} b^2 \plus{} c^2 \plus{} d^2) \plus{} 6ab\]
2004 Czech and Slovak Olympiad III A, 1
Find all triples $(x,y,z)$ of real numbers such that
\[x^2+y^2+z^2\le 6+\min (x^2-\frac{8}{x^4},y^2-\frac{8}{y^4},z^2-\frac{8}{z^4}).\]
2012 Turkey MO (2nd round), 4
For all positive real numbers $x, y, z$, show that $ \frac{x(2x-y)}{y(2z+x)}+\frac{y(2y-z)}{z(2x+y)}+\frac{z(2z-x)}{x(2y+z)} \geq 1$ is true.
2002 China Team Selection Test, 2
$ A_1$, $ B_1$ and $ C_1$ are the projections of the vertices $ A$, $ B$ and $ C$ of triangle $ ABC$ on the respective sides. If $ AB \equal{} c$, $ AC \equal{} b$, $ BC \equal{} a$ and $ AC_1 \equal{} 2t AB$, $ BA_1 \equal{} 2rBC$, $ CB_1 \equal{} 2 \mu AC$. Prove that:
\[ \frac {a^2}{b^2} \cdot \left( \frac {t}{1 \minus{} 2t} \right)^2 \plus{} \frac {b^2}{c^2} \cdot \left( \frac {r}{1 \minus{} 2r} \right)^2 \plus{} \frac {c^2}{a^2} \cdot \left( \frac {\mu}{1 \minus{} 2\mu} \right)^2 \plus{} 16tr \mu \geq 1
\]
2010 Contests, 1
Show that $\frac{(x - y)^7 + (y - z)^7 + (z - x)^7 - (x - y)(y - z)(z - x) ((x - y)^4 + (y - z)^4 + (z - x)^4)}
{(x - y)^5 + (y - z)^5 + (z - x)^5} \ge 3$
holds for all triples of distinct integers $x, y, z$. When does equality hold?
2010 Iran MO (3rd Round), 1
[b]two variable ploynomial[/b]
$P(x,y)$ is a two variable polynomial with real coefficients. degree of a monomial means sum of the powers of $x$ and $y$ in it. we denote by $Q(x,y)$ sum of monomials with the most degree in $P(x,y)$.
(for example if $P(x,y)=3x^4y-2x^2y^3+5xy^2+x-5$ then $Q(x,y)=3x^4y-2x^2y^3$.)
suppose that there are real numbers $x_1$,$y_1$,$x_2$ and $y_2$ such that
$Q(x_1,y_1)>0$ , $Q(x_2,y_2)<0$
prove that the set $\{(x,y)|P(x,y)=0\}$ is not bounded.
(we call a set $S$ of plane bounded if there exist positive number $M$ such that the distance of elements of $S$ from the origin is less than $M$.)
time allowed for this question was 1 hour.
PEN N Problems, 9
Let $ q_{0}, q_{1}, \cdots$ be a sequence of integers such that
a) for any $ m > n$, $ m \minus{} n$ is a factor of $ q_{m} \minus{} q_{n}$,
b) item $ |q_n| \le n^{10}$ for all integers $ n \ge 0$.
Show that there exists a polynomial $ Q(x)$ satisfying $ q_{n} \equal{} Q(n)$ for all $ n$.
1999 Moldova Team Selection Test, 6
Let $n\in\mathbb{N}, x_0=0$ and $x_1,x_2,\ldots,x_n$ be postive real numbers such that $x_1+x_2+\ldots+x_n=1$. Show that $$1\leq\sum_{i=1}^{n}\frac{x_i}{\sqrt{1+x_0+x_1+\ldots+x_{i-1}}\cdot\sqrt{x_i+x_{i+1}+\ldots+x_n}}<\frac{\pi}{2}.$$
2016 India Regional Mathematical Olympiad, 4
Let $a,b,c$ be positive real numbers such that $a+b+c=3$. Determine, with certainty, the largest possible value of the expression $$ \frac{a}{a^3+b^2+c}+\frac{b}{b^3+c^2+a}+\frac{c}{c^3+a^2+b}$$
1992 Iran MO (2nd round), 2
In the sequence $\{a_n\}_{n=0}^{\infty}$ we have $a_0=1$, $a_1=2$ and
\[a_{n+1}=a_n+\dfrac{a_{n-1}}{1+a_{n-1}^2} \qquad \forall n \geq 1\]
Prove that
\[52 < a_{1371} < 65\]
2010 IMAC Arhimede, 6
Consider real numbers $a, b ,c \ge0$ with $a+b+c=2$. Prove that:
$\frac{bc}{\sqrt[4]{3a^2+4}}+\frac{ca}{\sqrt[4]{3b^2+4}}+\frac{ab}{\sqrt[4]{3c^2+4}} \le \frac{2*\sqrt[4] {3}}{3}$
1994 China National Olympiad, 3
Find all functions $f:[1,\infty )\rightarrow [1,\infty)$ satisfying the following conditions:
(1) $f(x)\le 2(x+1)$;
(2) $f(x+1)=\dfrac{1}{x}[(f(x))^2-1]$ .
1968 IMO Shortlist, 12
If $a$ and $b$ are arbitrary positive real numbers and $m$ an integer, prove that
\[\Bigr( 1+\frac ab \Bigl)^m +\Bigr( 1+\frac ba \Bigl)^m \geq 2^{m+1}.\]