Found problems: 6530
2012 Bogdan Stan, 3
$ \lim_{n\to\infty }\frac{1}{\sqrt[n]{n!}}\left\lfloor \log_5 \sum_{k=2}^{1+5^n} \sqrt[5^n]{k} \right\rfloor $
[i]Taclit Daniela Nadia[/i]
2012 Bogdan Stan, 2
Prove the complex inequality $ |x|+|y|+|z|\le |x+y+z| +|x-z| +|z-y|+|y-z|. $
2000 Belarus Team Selection Test, 7.1
For any positive numbers $a,b,c,x,y, z$, prove the inequality $ \frac{a^3}{x}+ \frac{b^3}{y}+ \frac{c^3}{z} \ge \frac{(a+b+c)^3}{3(x+y+z)}$
2006 Croatia Team Selection Test, 4
Find all natural solutions of $3^{x}= 2^{x}y+1.$
1986 China Team Selection Test, 2
Given a tetrahedron $ABCD$, $E$, $F$, $G$, are on the respectively on the segments $AB$, $AC$ and $AD$. Prove that:
i) area $EFG \leq$ max{area $ABC$,area $ABD$,area $ACD$,area $BCD$}.
ii) The same as above replacing "area" for "perimeter".
1981 Romania Team Selection Tests, 3.
Determine the lengths of the edges of a right tetrahedron of volume $a^3$ so that the sum of its edges' lengths is minumum.
2013 Pan African, 3
Let $x$, $y$, and $z$ be real numbers such that $x<y<z<6$. Solve the system of inequalities:
\[\left\{\begin{array}{cc}
\dfrac{1}{y-x}+\dfrac{1}{z-y}\le 2 \\
\dfrac{1}{6-z}+2\le x \\
\end{array}\right.\]
1995 All-Russian Olympiad Regional Round, 9.1
(Russia 1195) If x, y > 0, prove that
x/(x^4 + y^2) + y/(y^4 + x^2) <= 1/(xy)
Thoughts?
By the way, this was in Kiran Kedlaya's MOP notes and said to be from Russia 1995, but John Scholes' Kalva archive doesn't have this problem under Russia 1995. Odd.
Hint:
[hide]This was in the Power Mean Inequality section in the lecture notes.[/hide]
2000 AMC 10, 10
The sides of a triangle with positive area have lengths $ 4$, $ 6$, and $ x$. The sides of a second triangle with positive area have lengths $ 4$, $ 6$, and $ y$. What is the smallest positive number that is [b]not[/b] a possible value of $ |x \minus{} y|$?
$ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 10$
2006 VJIMC, Problem 1
Given real numbers $0=x_1<x_2<\ldots<x_{2n}<x_{2n+1}=1$ such that $x_{i+1}-x_i\le h$ for $1\le i\le2n$, show that
$$\frac{1-h}2<\sum_{i=1}^nx_{2i}(x_{2i+1}-x_{2i-1})<\frac{1+h}2.$$
1985 Swedish Mathematical Competition, 1
If $a > b > 0$, prove the inequality $$\frac{(a-b)^2}{8a}< \frac{a+b}{2}- \sqrt{ab} < \frac{(a-b)^2}{8b}.$$
2012 South africa National Olympiad, 6
Find all functions $f:\mathbb{N}\to\mathbb{R}$ such that
$f(km)+f(kn)-f(k)f(mn)\ge 1$
for all $k,m,n\in\mathbb{N}$.
2012 Today's Calculation Of Integral, 821
Prove that : $\ln \frac{11}{27}<\int_{\frac 14}^{\frac 34} \frac{1}{\ln (1-x)}\ dx<\ln \frac{7}{15}.$
2012 China National Olympiad, 2
Consider a square-free even integer $n$ and a prime $p$, such that
1) $(n,p)=1$;
2) $p\le 2\sqrt{n}$;
3) There exists an integer $k$ such that $p|n+k^2$.
Prove that there exists pairwise distinct positive integers $a,b,c$ such that $n=ab+bc+ca$.
[i]Proposed by Hongbing Yu[/i]
2007 All-Russian Olympiad, 5
Given a set of $n>2$ planar vectors. A vector from this set is called [i]long[/i], if its length is not less than the length of the sum of other vectors in this set. Prove that if each vector is long, then the sum of all vectors equals to zero.
[i]N. Agakhanov[/i]
2013 Kyiv Mathematical Festival, 2
For every positive $a, b,c, d$ such that $a + c \le ac$ and $b + d \le bd$
prove that $\frac{ab}{a + b} +\frac{bc}{b + c} +\frac{cd}{c + d} +\frac{da}{d + a} \ge 4$
1984 Canada National Olympiad, 5
Given any $7$ real numbers, prove that there are two of them $x,y$ such that $0\le\frac{x-y}{1+xy}\le\frac{1}{\sqrt{3}}$.
1986 IMO Longlists, 1
Let $k$ be one of the integers $2, 3,4$ and let $n = 2^k -1$. Prove the inequality
\[1+ b^k + b^{2k} + \cdots+ b^{nk} \geq (1 + b^n)^k\]
for all real $b \geq 0.$
2011 Uzbekistan National Olympiad, 1
Let a,b,c Postive real numbers such that $a+b+c\geq 6$. Find the minimum value $A=\sum_{cyc}{a^2}$+$\sum_{cyc}{\frac{a}{b^2+c+1}}$
2002 Vietnam National Olympiad, 2
Determine for which $ n$ positive integer the equation: $ a \plus{} b \plus{} c \plus{} d \equal{} n \sqrt {abcd}$ has positive integer solutions.
2008 Moldova MO 11-12, 1
Consider the equation $ x^4 \minus{} 4x^3 \plus{} 4x^2 \plus{} ax \plus{} b \equal{} 0$, where $ a,b\in\mathbb{R}$. Determine the largest value $ a \plus{} b$ can take, so that the given equation has two distinct positive roots $ x_1,x_2$ so that $ x_1 \plus{} x_2 \equal{} 2x_1x_2$.
2015 Turkey MO (2nd round), 2
$x$, $y$ and $z$ are real numbers where the sum of any two among them is not $1$. Show that, \[ \dfrac{(x^2+y)(x+y^2)}{(x+y-1)^2}+\dfrac{(y^2+z)(y+z^2)}{(y+z-1)^2} + \dfrac{(z^2+x)(z+x^2)}{(z+x-1)^2} \ge 2(x+y+z) - \dfrac{3}{4}\]Find all triples $(x,y,z)$ of real numbers satisfying the equality case.
1981 IMO Shortlist, 11
On a semicircle with unit radius four consecutive chords $AB,BC, CD,DE$ with lengths $a, b, c, d$, respectively, are given. Prove that
\[a^2 + b^2 + c^2 + d^2 + abc + bcd < 4.\]
2023 SAFEST Olympiad, 2
There are $n!$ empty baskets in a row, labelled $1, 2, . . . , n!$. Caesar
first puts a stone in every basket. Caesar then puts 2 stones in every second basket.
Caesar continues similarly until he has put $n$ stones into every nth basket. In
other words, for each $i = 1, 2, . . . , n,$ Caesar puts $i$ stones into the baskets labelled
$i, 2i, 3i, . . . , n!.$
Let $x_i$ be the number of stones in basket $i$ after all these steps. Show that
$n! \cdot n^2 \leq \sum_{i=1}^{n!} x_i^2 \leq n! \cdot n^2 \cdot \sum_{i=1}^{n} \frac{1}{i} $
1980 Bulgaria National Olympiad, Problem 4
Let $a $, $b $, and $c $ be non-negative reals. Prove that $a^3+b^3+c^3+6abc\ge \frac{(a+b+c)^3}{4} $.