Found problems: 6530
2013 Harvard-MIT Mathematics Tournament, 12
For how many integers $1\leq k\leq 2013$ does the decimal representation of $k^k$ end with a $1$?
2019 IFYM, Sozopol, 7
Let $a, b, c$ be positive real numbers such that $abc=8$. Prove that
\[
\frac{a^2}{\sqrt{(1+a^3)(1+b^3)}} +\frac{b^2}{\sqrt{(1+b^3)(1+c^3)}} +\frac{c^2}{\sqrt{(1+c^3)(1+a^3)}} \geq \frac{4}{3}
\]
2005 Germany Team Selection Test, 1
Find the smallest positive integer $n$ with the following property:
For any integer $m$ with $0 < m < 2004$, there exists an integer $k$ such that
\[\frac{m}{2004}<\frac{k}{n}<\frac{m+1}{2005}.\]
2011 Cuba MO, 1
Let $P(x) = x^3 + (t - 1)x^2 - (t + 3)x + 1$. For what values of real $t$ the sum of the squares and the reciprocals of the roots of $ P(x)$ is minimum?
2001 Switzerland Team Selection Test, 5
Let $a_1 < a_2 < ... < a_n$ be a sequence of natural numbers such that for $i < j$ the decimal representation of $a_i$ does not occur as the leftmost digits of the decimal representation of $a_j$ . (For example, $137$ and $13729$ cannot both occur in the sequence.) Prove that $\sum_{i=1}^n \frac{1}{a_i} \le 1+\frac12 +\frac13 +...+\frac19$
.
2012 China Team Selection Test, 2
Given an integer $k\ge 2$. Prove that there exist $k$ pairwise distinct positive integers $a_1,a_2,\ldots,a_k$ such that for any non-negative integers $b_1,b_2,\ldots,b_k,c_1,c_2,\ldots,c_k$ satisfying $a_1\le b_i\le 2a_i, i=1,2,\ldots,k$ and $\prod_{i=1}^{k}b_i^{c_i}<\prod_{i=1}^{k}b_i$, we have
\[k\prod_{i=1}^{k}b_i^{c_i}<\prod_{i=1}^{k}b_i.\]
1999 Israel Grosman Mathematical Olympiad, 2
Find the smallest positive integer $n$ for which $0 <\sqrt[4]{n}- [\sqrt[4]{n}]< 10^{-5}$
.
1992 Kurschak Competition, 1
Define for $n$ given positive reals the [i]strange mean[/i] as the sum of the squares of these numbers divided by their sum. Decide which of the following statements hold for $n=2$:
a) The strange mean is never smaller than the third power mean.
b) The strange mean is never larger than the third power mean.
c) The strange mean, depending on the given numbers, can be larger or smaller than the third power mean.
Which statement is valid for $n=3$?
2009 Hong Kong TST, 5
Let $ a,b,c$ be the three sides of a triangle. Determine all possible values of $ \frac {a^2 \plus{} b^2 \plus{} c^2}{ab \plus{} bc \plus{} ca}$
2012 ELMO Shortlist, 6
Let $a,b,c\ge0$. Show that $(a^2+2bc)^{2012}+(b^2+2ca)^{2012}+(c^2+2ab)^{2012}\le (a^2+b^2+c^2)^{2012}+2(ab+bc+ca)^{2012}$.
[i]Calvin Deng.[/i]
2006 Federal Math Competition of S&M, Problem 1
Let $x,y,z$ be positive numbers with the sum $1$. Prove that
$$\frac x{y^2+z}+\frac y{z^2+x}+\frac z{x^2+y}\ge\frac94.$$
1974 Bulgaria National Olympiad, Problem 6
In triangle pyramid $MABC$ at least two of the plane angles next to the edge $M$ are not equal to each other. Prove that if the bisectors of these angles form the same angle with the angle bisector of the third plane angle, the following inequality is true
$$8a_1b_1c_1\le a^2a_1+b^2b_1+c^2c_1$$
where $a,b,c$ are sides of triangle $ABC$ and $a_1,b_1,c_1$ are edges crossed respectively with $a,b,c$.
[i]V. Petkov[/i]
2016 Estonia Team Selection Test, 4
Prove that for any positive integer $n\ge $, $2 \cdot \sqrt3 \cdot \sqrt[3]{4} ...\sqrt[n-1]{n} > n$
2013 Princeton University Math Competition, 16
Is $\cos 1^\circ$ rational? Prove.
MathLinks Contest 7th, 4.3
Let $ a,b,c$ be positive real numbers such that $ ab\plus{}bc\plus{}ca\equal{}3$. Prove that
\[ \frac 1{1\plus{}a^2(b\plus{}c)} \plus{} \frac 1{1\plus{}b^2(c\plus{}a)} \plus{} \frac 1 {1\plus{}c^2(a\plus{}b) } \leq \frac 3 {1\plus{}2abc} .\]
2011 India Regional Mathematical Olympiad, 6
Find the largest real constant $\lambda$ such that
\[\frac{\lambda abc}{a+b+c}\leq (a+b)^2+(a+b+4c)^2\]
For all positive real numbers $a,b,c.$
2004 Baltic Way, 18
A ray emanating from the vertex $A$ of the triangle $ABC$ intersects the side $BC$ at $X$ and the circumcircle of triangle $ABC$ at $Y$. Prove that $\frac{1}{AX}+\frac{1}{XY}\geq \frac{4}{BC}$.
2024 Balkan MO, 3
Let $a$ and $b$ be distinct positive integers such that $3^a + 2$ is divisible by $3^b + 2$. Prove that $a > b^2$.
[i]Proposed by Tynyshbek Anuarbekov, Kazakhstan[/i]
2001 Singapore Senior Math Olympiad, 2
Let $n$ be a positive integer, and let $f(n) =1^n + 2^{n-1} + 3^{n-2}+ 4^{n-3}+... + (n-1)^2 + n^1$
Find the smallest possible value of $\frac{f(n+2)}{f(n)}$ .Justify your answer.
2005 USA Team Selection Test, 2
Let $A_{1}A_{2}A_{3}$ be an acute triangle, and let $O$ and $H$ be its circumcenter and orthocenter, respectively. For $1\leq i \leq 3$, points $P_{i}$ and $Q_{i}$ lie on lines $OA_{i}$ and $A_{i+1}A_{i+2}$ (where $A_{i+3}=A_{i}$), respectively, such that $OP_{i}HQ_{i}$ is a parallelogram. Prove that
\[\frac{OQ_{1}}{OP_{1}}+\frac{OQ_{2}}{OP_{2}}+\frac{OQ_{3}}{OP_{3}}\geq 3.\]
2018 Belarusian National Olympiad, 9.2
For every integer $n\geqslant2$ prove the inequality
$$
\frac{1}{2!}+\frac{2}{3!}+\ldots+\frac{2^{n-2}}{n!}\leqslant\frac{3}{2},
$$
where $k!=1\cdot2\cdot\ldots\cdot k$.
2004 India IMO Training Camp, 3
For $a,b,c$ positive reals find the minimum value of \[ \frac{a^2+b^2}{c^2+ab}+\frac{b^2+c^2}{a^2+bc}+\frac{c^2+a^2}{b^2+ca}. \]
2015 District Olympiad, 1
For any $ n\ge 2 $ natural, show that the following inequality holds:
$$ \sum_{i=2}^n\frac{1}{\sqrt[i]{(2i)!}}\ge\frac{n-1}{2n+2} . $$
MathLinks Contest 7th, 5.3
If $ a\geq b\geq c\geq d > 0$ such that $ abcd\equal{}1$, then prove that \[ \frac 1{1\plus{}a} \plus{} \frac 1{1\plus{}b} \plus{} \frac 1{1\plus{}c} \geq \frac {3}{1\plus{}\sqrt[3]{abc}}.\]
2024 Assara - South Russian Girl's MO, 5
Prove that $(100!)^{99} > (99!)^{100} > (100!)^{98}$.
[i]K.A.Sukhov[/i]