Found problems: 6530
2024 Singapore MO Open, Q2
Let $n$ be a fixed positive integer. Find the minimum value of $$\frac{x_1^3+\dots+x_n^3}{x_1+\dots+x_n}$$ where $x_1,x_2,\dots,x_n$ are distinct positive integers.
2007 Irish Math Olympiad, 3
Let $ ABC$ be a triangle the lengths of whose sides $ BC,CA,AB,$ respectively, are denoted by $ a,b,$ and $ c$. Let the internal bisectors of the angles $ \angle BAC, \angle ABC, \angle BCA,$ respectively, meet the sides $ BC,CA,$ and $ AB$ at $ D,E,$ and $ F$. Denote the lengths of the line segments $ AD,BE,CF$ by $ d,e,$ and $ f$, respectively. Prove that:
$ def\equal{}\frac{4abc(a\plus{}b\plus{}c) \Delta}{(a\plus{}b)(b\plus{}c)(c\plus{}a)}$ where $ \Delta$ stands for the area of the triangle $ ABC$.
2003 Vietnam Team Selection Test, 2
Let $A$ be the set of all permutations $a = (a_1, a_2, \ldots, a_{2003})$ of the 2003 first positive integers such that each permutation satisfies the condition: there is no proper subset $S$ of the set $\{1, 2, \ldots, 2003\}$ such that $\{a_k | k \in S\} = S.$
For each $a = (a_1, a_2, \ldots, a_{2003}) \in A$, let $d(a) = \sum^{2003}_{k=1} \left(a_k - k \right)^2.$
[b]I.[/b] Find the least value of $d(a)$. Denote this least value by $d_0$.
[b]II.[/b] Find all permutations $a \in A$ such that $d(a) = d_0$.
2005 IMO Shortlist, 7
In an acute triangle $ABC$, let $D$, $E$, $F$ be the feet of the perpendiculars from the points $A$, $B$, $C$ to the lines $BC$, $CA$, $AB$, respectively, and let $P$, $Q$, $R$ be the feet of the perpendiculars from the points $A$, $B$, $C$ to the lines $EF$, $FD$, $DE$, respectively.
Prove that $p\left(ABC\right)p\left(PQR\right) \ge \left(p\left(DEF\right)\right)^{2}$, where $p\left(T\right)$ denotes the perimeter of triangle $T$ .
[i]Proposed by Hojoo Lee, Korea[/i]
1999 Miklós Schweitzer, 5
Let $\alpha>-2$ , $n\in \mathbb{N}$ and $y_1,\cdots,y_n$ be the solutions to the system of equations:
$\sum_{j=1}^n \frac{y_j}{j+k+\alpha}= \frac{1}{n+1+k+\alpha}$ , $k=1,\cdots,n$
Prove that $y_{j-1}y_{j+1}\leq y_j^2 \,\forall 1<j<n$
2006 Kazakhstan National Olympiad, 5
Prove that for every $ x $ such that $ \sin x \neq 0 $, exists natural $ n $ such that $ | \sin nx | \geq \frac {\sqrt {3}} {2} $.
2021 Thailand TST, 1
[i]Version 1[/i]. Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all real $x$,
\[
\sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant a_{n}(x-1)^{2}+x .
\]
[i]Version 2[/i]. For every positive integer $N$, determine the smallest real number $b_{N}$ such that, for all real $x$,
\[
\sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant b_{N}(x-1)^{2}+x .
\]
2006 AMC 12/AHSME, 20
Let $ x$ be chosen at random from the interval $ (0,1)$. What is the probability that
\[ \lfloor\log_{10}4x\rfloor \minus{} \lfloor\log_{10}x\rfloor \equal{} 0?
\]Here $ \lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $ x$.
$ \textbf{(A) } \frac 18 \qquad \textbf{(B) } \frac 3{20} \qquad \textbf{(C) } \frac 16 \qquad \textbf{(D) } \frac 15 \qquad \textbf{(E) } \frac 14$
2013 Princeton University Math Competition, 15
Prove: \[|\sin a_1|+|\sin a_2|+|\sin a_3|+\ldots+|\sin a_n|+|\cos(a_1+a_2+a_3+\ldots+a_n)|\geq 1.\]
2001 Polish MO Finals, 1
Prove the following inequality:
$x_1 + 2x_2 + 3x_3 + ... + nx_n \leq \frac{n(n-1)}{2} + x_1 + x_2 ^2 + x_3 ^3 + ... + x_n ^n$
where $\forall _{x_i} x_i > 0$
2002 Iran Team Selection Test, 3
A "[i]2-line[/i]" is the area between two parallel lines. Length of "2-line" is distance of two parallel lines. We have covered unit circle with some "2-lines". Prove sum of lengths of "2-lines" is at least 2.
1964 IMO Shortlist, 2
Suppose $a,b,c$ are the sides of a triangle. Prove that \[ a^2(b+c-a)+b^2(a+c-b)+c^2(a+b-c) \leq 3abc \]
1953 Putnam, A3
$a, b, c$ are real, and the sum of any two is greater than the third.
Show that $\frac{2(a + b + c)(a^2 + b^2 + c^2)}{3} > a^3 + b^3 + c^3 + abc$.
2023 Korea Summer Program Practice Test, P7
Determine the smallest value of $M$ for which for any choice of positive integer $n$ and positive real numbers $x_1<x_2<\ldots<x_n \le 2023$ the inequality
$$\sum_{1\le i < j \le n , x_j-x_i \ge 1} 2^{i-j}\le M$$
holds.
2010 Putnam, B2
Given that $A,B,$ and $C$ are noncollinear points in the plane with integer coordinates such that the distances $AB,AC,$ and $BC$ are integers, what is the smallest possible value of $AB?$
2002 IMC, 10
Let $OABC$ be a tetrahedon with $\angle BOC=\alpha,\angle COA =\beta$ and $\angle AOB =\gamma$. The angle between the faces $OAB$ and $OAC$ is $\sigma$ and the angle between the faces $OAB$ and $OBC$ is $\rho$.
Show that $\gamma > \beta \cos\sigma + \alpha \cos\rho$.
KoMaL A Problems 2021/2022, A. 809
Let the lengths of the sides of triangle $ABC$ be denoted by $a,b,$ and $c,$ using the standard notations. Let $G$ denote the centroid of triangle $ABC.$ Prove that for an arbitrary point $P$ in the plane of the triangle the following inequality is true: \[a\cdot PA^3+b\cdot PB^3+c\cdot PC^3\geq 3abc\cdot PG.\][i]Proposed by János Schultz, Szeged[/i]
1976 Dutch Mathematical Olympiad, 5
$f(k) = k + \left[ \frac{n}{k}\right ] $,$k \in \{1,2,..., n\}$, $k_0 =\left[ \sqrt{n} \right] + 1$.
Prove that $f(k_0) < f(k)$ if $k \in \{1,2,..., n\}$
2003 China Team Selection Test, 1
Let $S$ be the set of points inside and on the boarder of a regular haxagon with side length 1. Find the least constant $r$, such that there exists one way to colour all the points in $S$ with three colous so that the distance between any two points with same colour is less than $r$.
2001 Romania National Olympiad, 2
In the tetrahedron $OABC$ we denote by $\alpha,\beta,\gamma$ the measures of the angles $\angle BOC,\angle COA,$ and $\angle AOB$, respectively. Prove the inequality
\[\cos^2\alpha+\cos^2\beta+\cos^2\gamma<1+2\cos\alpha\cos\beta\cos\gamma \]
1970 AMC 12/AHSME, 17
If $r\ge 0$, then for all $p$ and $q$ such that $pq\neq 0$ and $pr>qr$, we have
$\textbf{(A) }-p>-q\qquad\textbf{(B) }-p>q\qquad\textbf{(C) }1>-q/p\qquad$
$\textbf{(D) }1<q/p\qquad \textbf{(E) }\text{None of These}$
MathLinks Contest 7th, 4.3
Let $ a,b,c$ be positive real numbers such that $ ab\plus{}bc\plus{}ca\equal{}3$. Prove that
\[ \frac 1{1\plus{}a^2(b\plus{}c)} \plus{} \frac 1{1\plus{}b^2(c\plus{}a)} \plus{} \frac 1 {1\plus{}c^2(a\plus{}b) } \leq \frac 3 {1\plus{}2abc} .\]
2023 Kazakhstan National Olympiad, 3
$a,b,c$ are positive real numbers such that $\max\{\frac{a(b+c)}{a^2+bc},\frac{b(c+a)}{b^2+ca},\frac{c(a+b)}{c^2+ab}\}\le \frac{5}{2}$. Prove inequality $$\frac{a(b+c)}{a^2+bc}+\frac{b(c+a)}{b^2+ca}+\frac{c(a+b)}{c^2+ab}\le 3$$
1992 All Soviet Union Mathematical Olympiad, 566
Show that for any real numbers $x, y > 1$, we have $$\frac{x^2}{y - 1}+ \frac{y^2}{x - 1} \ge 8$$
2004 Abels Math Contest (Norwegian MO), 2
(a) Prove that $(x+y+z)^2 \le 3(x^2 +y^2 +z^2)$ for any real numbers $x,y,z$.
(b) If positive numbers $a,b,c$ satisfy $a+b+c \ge abc$, prove that $a^2 +b^2 +c^2 \ge \sqrt3 abc$