Found problems: 6530
2020 Caucasus Mathematical Olympiad, 8
Let real $a$, $b$, and $c$ satisfy $$abc+a+b+c=ab+bc+ca+5.$$ Find the least possible value of $a^2+b^2+c^2$.
2004 Gheorghe Vranceanu, 2
[b]a)[/b] Let be an even number $ n\ge 4 $ and $ n $ positive real numbers $ x_1,x_2,\ldots ,x_n. $ Prove that:
$$ \min_{1\le i\le n/2} \frac{x_i}{x_{i+n/2}}\le \frac{x_1+x_2+\cdots +x_{n/2}}{x_{1+n/2}+ x_{2+n/2} +\cdots + x_n}\le \max_{1\le i\le n/2} \frac{x_i}{x_{i+n/2}}$$
[b]b)[/b] Let be $ m\ge 1 $ pairwise distinct natural numbers $ a,b,\ldots ,c. $ Show that:
$$ \frac{ab\cdots c}{a+b+\cdots +c}\ge (m-1)!\cdot\frac{2}{m+1} $$
[i]M. Tetiva[/i]
2020 Brazil Team Selection Test, 8
Let $a_1, a_2,\dots$ be an infinite sequence of positive real numbers such that for each positive integer $n$ we have \[\frac{a_1+a_2+\cdots+a_n}n\geq\sqrt{\frac{a_1^2+a_2^2+\cdots+a_{n+1}^2}{n+1}}.\]
Prove that the sequence $a_1,a_2,\dots$ is constant.
[i]Proposed by Alex Zhai[/i]
2009 Hong Kong TST, 1
Let $ \theta_1, \theta_2,\ldots , \theta_{2008}$ be real numbers. Find the maximum value of
$ \sin\theta_1\cos\theta_2 \plus{} \sin\theta_2\cos\theta_3 \plus{} \ldots \plus{} \sin\theta_{2007}\cos\theta_{2008} \plus{} \sin\theta_{2008}\cos\theta_1$
2005 Serbia Team Selection Test, 4
Let $T$ be the centroid of triangle $ABC$. Prove that \[ \frac 1{\sin \angle TAC} + \frac 1{\sin \angle TBC} \geq 4 \]
2020 Ecuador NMO (OMEC), 5
In triangle $ABC$, $D$ is the middle point of side $BC$ and $M$ is a point on segment $AD$ such that $AM=3MD$.
The barycenter of $ABC$ and $M$ are on the inscribed circumference of $ABC$.
Prove that $AB+AC>3BC$.
1995 Canada National Olympiad, 2
Let $\{a,b,c\}\in \mathbb{R}^{+}$. Prove that $a^a b^b c^c \ge (abc)^{\frac{a+b+c}{3}}$.
1998 Portugal MO, 6
Let $a_0$ be a positive real number and consider the general term sequence $a_n$ defined by $$a_n =a_{n-1} + \frac{1}{a_{n-1}} \,\,\, n=1,2,3,...$$ Prove that $a_{1998} > 63$.
2009 AMC 12/AHSME, 10
In quadrilateral $ ABCD$, $ AB \equal{} 5$, $ BC \equal{} 17$, $ CD \equal{} 5$, $ DA \equal{} 9$, and $ BD$ is an integer. What is $ BD$?
[asy]unitsize(4mm);
defaultpen(linewidth(.8pt)+fontsize(8pt));
dotfactor=4;
pair C=(0,0), B=(17,0);
pair D=intersectionpoints(Circle(C,5),Circle(B,13))[0];
pair A=intersectionpoints(Circle(D,9),Circle(B,5))[0];
pair[] dotted={A,B,C,D};
draw(D--A--B--C--D--B);
dot(dotted);
label("$D$",D,NW);
label("$C$",C,W);
label("$B$",B,E);
label("$A$",A,NE);[/asy]$ \textbf{(A)}\ 11 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 13 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 15$
2021 Taiwan TST Round 1, G
In the plane, there are $n \geqslant 6$ pairwise disjoint disks $D_{1}, D_{2}, \ldots, D_{n}$ with radii $R_{1} \geqslant R_{2} \geqslant \ldots \geqslant R_{n}$. For every $i=1,2, \ldots, n$, a point $P_{i}$ is chosen in disk $D_{i}$. Let $O$ be an arbitrary point in the plane. Prove that \[O P_{1}+O P_{2}+\ldots+O P_{n} \geqslant R_{6}+R_{7}+\ldots+R_{n}.\]
(A disk is assumed to contain its boundary.)
2003 China Girls Math Olympiad, 1
Let $ ABC$ be a triangle. Points $ D$ and $ E$ are on sides $ AB$ and $ AC,$ respectively, and point $ F$ is on line segment $ DE.$ Let $ \frac {AD}{AB} \equal{} x,$ $ \frac {AE}{AC} \equal{} y,$ $ \frac {DF}{DE} \equal{} z.$ Prove that
(1) $ S_{\triangle BDF} \equal{} (1 \minus{} x)y S_{\triangle ABC}$ and $ S_{\triangle CEF} \equal{} x(1 \minus{} y) (1 \minus{} z)S_{\triangle ABC};$
(2) $ \sqrt [3]{S_{\triangle BDF}} \plus{} \sqrt [3]{S_{\triangle CEF}} \leq \sqrt [3]{S_{\triangle ABC}}.$
1961 Polish MO Finals, 2
Prove that if $ a + b = 1 $, then $$
a^5 + b^5 \geq \frac{1}{16}$$
2009 Serbia Team Selection Test, 2
Let $ x,y,z$ be positive real numbers such that $ xy \plus{} yz \plus{} zx \equal{} x \plus{} y \plus{} z$. Prove the inequality
$ \frac1{x^2 \plus{} y \plus{} 1} \plus{} \frac1{y^2 \plus{} z \plus{} 1} \plus{} \frac1{z^2 \plus{} x \plus{} 1}\le1$
When does the equality hold?
2015 European Mathematical Cup, 2
Let $m, n, p$ be fixed positive real numbers which satisfy $mnp = 8$. Depending on these constants, find the minimum of $$x^2+y^2+z^2+ mxy + nxz + pyz,$$
where $x, y, z$ are arbitrary positive real numbers satisfying $xyz = 8$. When is the equality attained?
Solve the problem for:
[list=a][*]$m = n = p = 2,$
[*] arbitrary (but fixed) positive real numbers $m, n, p.$[/list]
[i]Stijn Cambie[/i]
2021 Stars of Mathematics, 4
Let $k$ be a positive integer, and let $a,b$ and $c$ be positive real numbers. Show that \[a(1-a^k)+b(1-(a+b)^k)+c(1-(a+b+c)^k)<\frac{k}{k+1}.\]
[i]* * *[/i]
2012 Iran MO (3rd Round), 2
Suppose $S$ is a convex figure in plane with area $10$. Consider a chord of length $3$ in $S$ and let $A$ and $B$ be two points on this chord which divide it into three equal parts. For a variable point $X$ in $S-\{A,B\}$, let $A'$ and $B'$ be the intersection points of rays $AX$ and $BX$ with the boundary of $S$. Let $S'$ be those points $X$ for which $AA'>\frac{1}{3} BB'$. Prove that the area of $S'$ is at least $6$.
[i]Proposed by Ali Khezeli[/i]
2010 Irish Math Olympiad, 3
Suppose $x,y,z$ are positive numbers such that $x+y+z=1$. Prove that
(a) $xy+yz+xz\ge 9xyz$;
(b) $xy+yz+xz<\frac{1}{4}+3xyz$;
2005 Balkan MO, 3
Let $a,b,c$ be positive real numbers. Prove the inequality
\[\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\geq a+b+c+\frac{4(a-b)^2}{a+b+c}.\]
When does equality occur?
2000 USAMO, 6
Let $a_1, b_1, a_2, b_2, \dots , a_n, b_n$ be nonnegative real numbers. Prove that
\[
\sum_{i, j = 1}^{n} \min\{a_ia_j, b_ib_j\} \le \sum_{i, j = 1}^{n} \min\{a_ib_j, a_jb_i\}.
\]
Russian TST 2020, P2
Given a natural number $n{}$ find the smallest $\lambda$ such that\[\gcd(x(x + 1)\cdots(x + n - 1), y(y + 1)\cdots(y + n - 1)) \leqslant (x-y)^\lambda,\] for any positive integers $y{}$ and $x \geqslant y + n$.
2010 Today's Calculation Of Integral, 578
Find the range of $ k$ for which the following inequality holds for $ 0\leq x\leq 1$.
\[ \int_0^x \frac {dt}{\sqrt {(3 \plus{} t^2)^3}}\geq k\int _0^x \frac {dt}{\sqrt {3 \plus{} t^2}}\]
If necessary, you may use $ \ln 3 \equal{} 1.10$.
1982 Bundeswettbewerb Mathematik, 3
Given that $a_1, a_2, . . . , a_n$ are nonnegative real numbers with $a_1 + \cdots + a_n = 1$, prove that the expression
$$ \frac{a_1}{1+a_2 +a_3 +\cdots +a_n }\; +\; \frac{a_2}{1+a_1 +a_3 +\cdots +a_n }\; +\; \cdots \; +\, \frac{a_n }{1+a_1 +a_2+\cdots +a_{n-1} }$$
attains its minimum, and determine this minimum.
2009 Regional Competition For Advanced Students, 1
Find the largest interval $ M \subseteq \mathbb{R^ \plus{} }$, such that for all $ a$, $ b$, $ c$, $ d \in M$ the inequality
\[ \sqrt {ab} \plus{} \sqrt {cd} \ge \sqrt {a \plus{} b} \plus{} \sqrt {c \plus{} d}\]
holds. Does the inequality
\[ \sqrt {ab} \plus{} \sqrt {cd} \ge \sqrt {a \plus{} c} \plus{} \sqrt {b \plus{} d}\]
hold too for all $ a$, $ b$, $ c$, $ d \in M$?
($ \mathbb{R^ \plus{} }$ denotes the set of positive reals.)
2005 MOP Homework, 2
Let $x$, $y$, $z$ be positive real numbers and $x+y+z=1$. Prove that
$\sqrt{xy+z}+\sqrt{yz+x}+\sqrt{zx+y} \ge 1+\sqrt{xy}+\sqrt{yz}+\sqrt{zx}$.
2002 Moldova Team Selection Test, 1
Consider the triangular numbers $T_n = \frac{n(n+1)}{2} , n \in \mathbb N$.
[list][b](a)[/b] If $a_n$ is the last digit of $T_n$, show that the sequence $(a_n)$ is periodic and find its basic period.
[b](b)[/b] If $s_n$ is the sum of the first $n$ terms of the sequence $(T_n)$, prove that for every $n \geq 3$ there is at least one perfect square between $s_{n-1} and $s_n$.[/list]