Found problems: 6530
1977 IMO Longlists, 46
Let $f$ be a strictly increasing function defined on the set of real numbers. For $x$ real and $t$ positive, set\[g(x,t)=\frac{f(x+t)-f(x)}{f(x) - f(x - t)}.\]
Assume that the inequalities\[2^{-1} < g(x, t) < 2\]
hold for all positive t if $x = 0$, and for all $t \leq |x|$ otherwise.
Show that\[ 14^{-1} < g(x, t) < 14\]
for all real $x$ and positive $t.$
2010 Contests, 4
A $9\times 7$ rectangle is tiled with tiles of the two types: L-shaped tiles composed by three unit squares (can be rotated repeatedly with $90^\circ$) and square tiles composed by four unit squares.
Let $n\ge 0$ be the number of the $2 \times 2 $ tiles which can be used in such a tiling. Find all the values of $n$.
2002 Moldova National Olympiad, 2
For every nonnegative integer $ n$ and every real number $ x$ prove the inequality:
$ |\cos x|\plus{}|\cos 2x|\plus{}\ldots\plus{}|\cos 2^nx|\geq \dfrac{n}{2\sqrt{2}}$
2007 Indonesia TST, 3
Let $a, b, c$ be positive reals such that $a + b + c = 1$ and $P(x) = 3^{2005}x^{2007 }- 3^{2005}x^{2006} - x^2$.
Prove that $P(a) + P(b) + P(c) \le -1$.
2020 Canadian Mathematical Olympiad Qualification, 7
Let $a, b, c$ be positive real numbers with $ab + bc + ac = abc$. Prove that
$$\frac{bc}{a^{a+1}} +\frac{ac}{b^{b+1 }}+\frac{ab}{c^{c+1}} \ge \frac13$$
1994 Cono Sur Olympiad, 3
Let $p$ be a positive real number given. Find the minimun vale of $x^3+y^3$, knowing that $x$ and $y$ are positive real numbers such that $xy(x+y)=p$.
2007 Putnam, 2
Find the least possible area of a convex set in the plane that intersects both branches of the hyperbola $ xy\equal{}1$ and both branches of the hyperbola $ xy\equal{}\minus{}1.$ (A set $ S$ in the plane is called [i]convex[/i] if for any two points in $ S$ the line segment connecting them is contained in $ S.$)
1997 Irish Math Olympiad, 4
Let $ a,b,c$ be nonnegative real numbers. Suppose that $ a\plus{}b\plus{}c\ge abc$. Prove that:
$ a^2\plus{}b^2\plus{}c^2 \ge abc.$
2012 Mathcenter Contest + Longlist, 6
Let $a,b,c>0$ and $abc=1$. Prove that $$\frac{a}{b^2(c+a)(a+b)}+\frac{b}{c^2(a+b)(b+c)}+\frac{c}{a^2(c+a)(a+b)}\ge \frac{3}{4}.$$
[i](Zhuge Liang)[/i]
2005 Canada National Olympiad, 4
Let $ ABC$ be a triangle with circumradius $ R$, perimeter $ P$ and area $ K$. Determine the maximum value of: $ \frac{KP}{R^3}$.
2002 JBMO ShortLists, 10
Let $ ABC$ be a triangle with area $ S$ and points $ D,E,F$ on the sides $ BC,CA,AB$. Perpendiculars at points $ D,E,F$ to the $ BC,CA,AB$ cut circumcircle of the triangle $ ABC$ at points $ (D_1,D_2), (E_1,E_2), (F_1,F_2)$. Prove that:
$ |D_1B\cdot D_1C \minus{} D_2B\cdot D_2C| \plus{} |E_1A\cdot E_1C \minus{} E_2A\cdot E_2C| \plus{} |F_1B\cdot F_1A \minus{} F_2B\cdot F_2A| > 4S$
2006 Cuba MO, 3
Let $a, b, c$ be different real numbers. prove that
$$\left(\frac{2a-b}{a-b}\right)^2+ \left(\frac{2b- c}{b-c}\right)^2+ \left(\frac{2c-a}{c-a}\right)^2 \ge 5. $$
2008 Peru MO (ONEM), 2
Let $a$ and $b$ be real numbers for which the following is true:
$acscx + b cot x \ge 1$, for all $0 <x < \pi$
Find the least value of $a^2 + b$.
2020 Taiwan APMO Preliminary, P6
Let $a,b,c$ be positive reals. Find the minimum value of
$$\dfrac{13a+13b+2c}{2a+2b}+\dfrac{24a-b+13c}{2b+2c}+\dfrac{(-a+24b+13c)}{2c+2a}$$.
(1)What is the minimum value?
(2)If the minimum value occurs when $(a,b,c)=(a_0,b_0,c_0)$,then find $\frac{b_0}{a_0}+\frac{c_0}{b_0}$.
2012 Today's Calculation Of Integral, 805
Prove the following inequalities:
(1) For $0\leq x\leq 1$,
\[1-\frac 13x\leq \frac{1}{\sqrt{1+x^2}}\leq 1.\]
(2) $\frac{\pi}{3}-\frac 16\leq \int_0^{\frac{\sqrt{3}}{2}} \frac{1}{\sqrt{1-x^4}}dx\leq \frac{\pi}{3}.$
2017 China Team Selection Test, 3
Suppose $S=\{1,2,3,...,2017\}$,for every subset $A$ of $S$,define a real number $f(A)\geq 0$ such that:
$(1)$ For any $A,B\subset S$,$f(A\cup B)+f(A\cap B)\leq f(A)+f(B)$;
$(2)$ For any $A\subset B\subset S$, $f(A)\leq f(B)$;
$(3)$ For any $k,j\in S$,$$f(\{1,2,\ldots,k+1\})\geq f(\{1,2,\ldots,k\}\cup \{j\});$$
$(4)$ For the empty set $\varnothing$, $f(\varnothing)=0$.
Confirm that for any three-element subset $T$ of $S$,the inequality $$f(T)\leq \frac{27}{19}f(\{1,2,3\})$$ holds.
1997 Hungary-Israel Binational, 2
Find all the real numbers $ \alpha$ satisfy the following property: for any positive integer $ n$ there exists an integer $ m$ such that $ \left |\alpha\minus{}\frac{m}{n}\right|<\frac{1}{3n}$.
2021 Estonia Team Selection Test, 3
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.)
2008 Pre-Preparation Course Examination, 4
Sarah and Darah play the following game. Sarah puts $ n$ coins numbered with $ 1,\dots,n$ on a table (Each coin is in HEAD or TAIL position.) At each step Darah gives a coin to Sarah and she (Sarah) let him (Dara) to change the position of all coins with number multiple of a desired number $ k$. At the end, all of the coins that are in TAIL position will be given to Sarah and all of the coins with HEAD position will be given to Darah. Prove that Sarah can put the coins in a position at the beginning of the game such that she gains at least $ \Omega(n)$ coins.
[hide="Hint:"]Chernov inequality![/hide]
2023 Miklós Schweitzer, 3
Let $X =\{x_0, x_1,\ldots , x_n\}$ be the basis set of a finite metric space, where the points are inductively listed such that $x_k$ maximizes the product of the distances from the points $\{x_0, x_1,\ldots , x_{k-1}\}$ for each $1\leqslant k\leqslant n.$ Prove that if for each $x\in X$ we let $\Pi_x$ be the product of the distances from $x{}$ to every other point, then $\Pi_{x_n}\leqslant 2^{n-1}\Pi_x$ for any $x\in X.$
1953 Moscow Mathematical Olympiad, 253
Given the equations
(1) $ax^2 + bx + c = 0$
(2)$ -ax^2 + bx + c = 0$
prove that if $x_1$ and $x_2$ are some roots of equations (1) and (2), respectively, then there is a root $x_3$ of the equation $$\frac{a}{2}x^2 + bx + c = 0$$ such that either $x_1 \le x_3 \le x_2$ or $x_1 \ge x_3 \ge x_2$.
2019 Jozsef Wildt International Math Competition, W. 2
If $0<a\leq c\leq b$ then $$\frac{(b^{30}-a^{30})(b^{30}-c^{30})}{36b^{10}}\leq \frac{(b^{25}-a^{25})(b^{25}-c^{25})}{25}\leq \frac{(b^{30}-a^{30})(b^{30}-c^{30})}{36(ac)^{10}}$$
2001 India IMO Training Camp, 3
In a triangle $ABC$ with incircle $\omega$ and incenter $I$ , the segments $AI$ , $BI$ , $CI$ cut $\omega$ at $D$ , $E$ , $F$ , respectively. Rays $AI$ , $BI$ , $CI$ meet the sides $BC$ , $CA$ , $AB$ at $L$ , $M$ , $N$ respectively. Prove that:
\[AL+BM+CN \leq 3(AD+BE+CF)\]
When does equality occur?
2014 Contests, 3
Let $D, E, F$ be points on the sides $BC, CA, AB$ of a triangle $ABC$, respectively such that the lines $AD, BE, CF$ are concurrent at the point $P$. Let a line $\ell$ through $A$ intersect the rays $[DE$ and $[DF$ at the points $Q$ and $R$, respectively. Let $M$ and $N$ be points on the rays $[DB$ and $[DC$, respectively such that the equation
\[ \frac{QN^2}{DN}+\frac{RM^2}{DM}=\frac{(DQ+DR)^2-2\cdot RQ^2+2\cdot DM\cdot DN}{MN} \]
holds. Show that the lines $AD$ and $BC$ are perpendicular to each other.
2022 Brazil Team Selection Test, 1
Let $a, b, c$ be positive real numbers. Show that $$a^5+b^5+c^5 \geq 5abc(b^2-ac)$$ and determine when the equality occurs.