Found problems: 6530
2014 India Regional Mathematical Olympiad, 6
Let $x_1,x_2,x_3 \ldots x_{2014}$ be positive real numbers such that $\sum_{j=1}^{2014} x_j=1$. Determine with proof the smallest constant $K$ such that
\[K\sum_{j=1}^{2014}\frac{x_j^2}{1-x_j} \ge 1\]
1999 Romania National Olympiad, 4
a) Let $a,b\in R$, $a <b$. Prove that $x \in (a,b)$ if and only if there exists $\lambda \in (0,1)$ such that $x=\lambda a +(1-\lambda)b$.
b) If the function $f: R \to R$ has the property:
$$f (\lambda x+(1-\lambda) y) < \lambda f(x) + (1-\lambda)f(y), \forall x,y \in R, x\ne y, \forall \lambda \in (0,1), $$ prove that one cannot find four points on the function’s graph that are the vertices of a parallelogram
2002 JBMO ShortLists, 5
Let $ a,b,c$ be positive real numbers. Prove the inequality:
$ \frac {a^3}{b^2} \plus{} \frac {b^3}{c^2} \plus{} \frac {c^3}{a^2}\ge \frac {a^2}{b} \plus{} \frac {b^2}{c} \plus{} \frac {c^2}{a}$
2016 SDMO (High School), 3
Let $ u, v, w$ be positive real numbers such that $ u\sqrt {vw} \plus{} v\sqrt {wu} \plus{} w\sqrt {uv} \geq 1$. Find the smallest value of $ u \plus{} v \plus{} w$.
1972 Swedish Mathematical Competition, 4
Put $x = \log_{10} 2$, $y = \log_{10} 3$. Then $15 < 16$ implies $1 - x + y < 4x$, so $1 + y < 5x$.
Derive similar inequalities from $80 < 81$ and $243 < 250$. Hence show that \[
0.47 < \log_{10} 3 < 0.482.
\]
2006 Germany Team Selection Test, 2
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]
2019 Saudi Arabia JBMO TST, 4
Let $n$ be positive integer and let $a_1, a_2,...,a_n$ be real numbers. Prove that there exist positive integers $m, k$ $<=n$ , $|$ $(a_1+a_2+...+a_m)$ $-$ $(a_{m+1}+a_{m+2}+...+a_n)$ $|$ $<=$ $|$ $a_k$ $|$
2018 Saint Petersburg Mathematical Olympiad, 4
$$(b+c)x^2+(a+c)x+(a+b)=0$$ has not real roots. Prove that $$4ac-b^2 \leq 3a(a+b+c)$$
2010 Romania National Olympiad, 1
Let $a,b,c$ be integers larger than $1$. Prove that
\[a(a-1)+b(b-1)+c(c-1)\le (a+b+c-4)(a+b+c-5)+4.\]
1997 Brazil Team Selection Test, Problem 5
Let $ABC$ be an acute-angled triangle with incenter $I$. Consider the point $A_1$ on $AI$ different from $A$, such that the midpoint of $AA_1$ lies on the circumscribed circle of $ABC$. Points $B_1$ and $C_1$ are defined similarly.
(a) Prove that $S_{A_1B_1C_1}=(4R+r)p$, where $p$ is the semi-perimeter, $R$ is the circumradius and $r$ is the inradius of $ABC$.
(b) Prove that $S_{A_1B_1C_1}\ge9S_{ABC}$.
2020 Saint Petersburg Mathematical Olympiad, 4.
Let $m$ be a given positive integer. Prove that there exists a positive integer $k$ such that it holds
$$1\leq \frac{1^m+2^m+3^m+\ldots +(k-1)^m}{k^m}<2.$$
1988 China Team Selection Test, 1
Suppose real numbers $A,B,C$ such that for all real numbers $x,y,z$ the following inequality holds:
\[A(x-y)(x-z) + B(y-z)(y-x) + C(z-x)(z-y) \geq 0.\]
Find the necessary and sufficient condition $A,B,C$ must satisfy (expressed by means of an equality or an inequality).
2017 USAMO, 6
Find the minimum possible value of \[\frac{a}{b^3+4}+\frac{b}{c^3+4}+\frac{c}{d^3+4}+\frac{d}{a^3+4}\] given that $a$, $b$, $c$, $d$ are nonnegative real numbers such that $a+b+c+d=4$.
[i]Proposed by Titu Andreescu[/i]
1998 Abels Math Contest (Norwegian MO), 1
Let $a_0,a_1,a_2,...$ be an infinite sequence of positive integers such that $a_0 = 1$ and $a_i^2 > a_{i-1}a_{i+1}$ for all $i > 0$.
(a) Prove that $a_i < a_1^i$ for all $i > 1$.
(b) Prove that $a_i > i$ for all $i$.
1985 IMO Longlists, 23
Let $\mathbb N = {1, 2, 3, . . .}$. For real $x, y$, set $S(x, y) = \{s | s = [nx+y], n \in \mathbb N\}$. Prove that if $r > 1$ is a rational number, there exist real numbers $u$ and $v$ such that
\[S(r, 0) \cap S(u, v) = \emptyset, S(r, 0) \cup S(u, v) = \mathbb N.\]
2010 China Western Mathematical Olympiad, 5
Let $k$ be an integer and $k > 1$. Define a sequence $\{a_n\}$ as follows:
$a_0 = 0$,
$a_1 = 1$, and
$a_{n+1} = ka_n + a_{n-1}$ for $n = 1,2,...$.
Determine, with proof, all possible $k$ for which there exist non-negative integers $l,m (l \not= m)$ and positive integers $p,q$ such that $a_l + ka_p = a_m + ka_q$.
2007 ITest, 9
Suppose that $m$ and $n$ are positive integers such that $m<n$, the geometric mean of $m$ and $n$ is greater than $2007$, and the arithmetic mean of $m$ and $n$ is less than $2007$. How many pairs $(m,n)$ satisfy these conditions?
$\textbf{(A) }0\hspace{14em}\textbf{(B) }1\hspace{14em}\textbf{(C) }2$
$\textbf{(D) }3\hspace{14em}\textbf{(E) }4\hspace{14em}\textbf{(F) }5$
$\textbf{(G) }6\hspace{14em}\textbf{(H) }7\hspace{14em}\textbf{(I) }2007$
2013 Online Math Open Problems, 25
Let $ABCD$ be a quadrilateral with $AD = 20$ and $BC = 13$. The area of $\triangle ABC$ is $338$ and the area of $\triangle DBC$ is $212$. Compute the smallest possible perimeter of $ABCD$.
[i]Proposed by Evan Chen[/i]
2025 Bulgarian Winter Tournament, 12.1
Let $a,b,c$ be positive real numbers with $a+b>c$. Prove that $ax + \sin(bx) + \cos(cx) > 1$ for all $x\in \left(0, \frac{\pi}{a+b+c}\right)$.
2011 Estonia Team Selection Test, 4
Let $a,b,c$ be positive real numbers such that $2a^2 +b^2=9c^2$.Prove that $\displaystyle \frac{2c}{a}+\frac cb \ge\sqrt 3$.
2009 Tuymaada Olympiad, 4
The sum of several non-negative numbers is not greater than 200, while the sum of their squares is not less than 2500. Prove that among them there are four numbers whose sum is not less than 50.
[i]Proposed by A. Khabrov[/i]
2017 Azerbaijan JBMO TST, 1
Let $x,y,z,t$ be positive numbers.Prove that
$\frac{xyzt}{(x+y)(z+t)}\leq\frac{(x+z)^2(y+t)^2}{4(x+y+z+t)^2}.$
2009 Hungary-Israel Binational, 2
Let $ x$, $ y$ and $ z$ be non negative numbers. Prove that \[ \frac{x^2\plus{}y^2\plus{}z^2\plus{}xy\plus{}yz\plus{}zx}{6}\le \frac{x\plus{}y\plus{}z}{3}\cdot\sqrt{\frac{x^2\plus{}y^2\plus{}z^2}{3}}\]
2009 Jozsef Wildt International Math Competition, W. 26
If $a_i >0$ ($i=1, 2, \cdots , n$) and $\sum \limits_{i=1}^n a_i^k=1$, where $1\leq k\leq n+1$, then $$\sum \limits_{i=1}^n a_i + \frac{1}{\prod \limits_{i=1}^n a_i} \geq n^{1-\frac{1}{k}}+n^{\frac{n}{k}}$$
2016 Swedish Mathematical Competition, 2
Determine whether the inequality $$ \left|\sqrt{x^2+2x+5}-\sqrt{x^2-4x+8}\right|<3$$ is valid for all real numbers $x$.