Found problems: 6530
2015 BMT Spring, 5
Let $x$ and $y$ be real numbers satisfying the equation $x^2-4x+y^2+3=0$. If the maximum and minimum values of $x^2+y^2$ are $M$ and $m$ respectively, compute the numerical value of $M-m$.
2016 Austria Beginners' Competition, 2
Prove that all real numbers $x \ne -1$, $y \ne -1$ with $xy = 1$ satisfy the following inequality:
$$\left(\frac{2+x}{1+x}\right)^2 + \left(\frac{2+y}{1+y}\right)^2 \ge \frac92$$
(Karl Czakler)
2015 Switzerland - Final Round, 10
Find the largest natural number $n$ such that for all real numbers $a, b, c, d$ the following holds:
$$(n + 2)\sqrt{a^2 + b^2} + (n + 1)\sqrt{a^2 + c^2} + (n + 1)\sqrt{a^2 + d^2} \ge n(a + b + c + d)$$
MathLinks Contest 7th, 3.3
Find the greatest positive real number $ k$ such that the inequality below holds for any positive real numbers $ a,b,c$:
\[ \frac ab \plus{} \frac bc \plus{} \frac ca \minus{} 3 \geq k \left( \frac a{b \plus{} c} \plus{} \frac b{c \plus{} a} \plus{} \frac c{a \plus{} b} \minus{} \frac 32 \right).
\]
2009 Junior Balkan MO, 3
Let $ x$, $ y$, $ z$ be real numbers such that $ 0 < x,y,z < 1$ and $ xyz \equal{} (1 \minus{} x)(1 \minus{} y)(1 \minus{} z)$. Show that at least one of the numbers $ (1 \minus{} x)y,(1 \minus{} y)z,(1 \minus{} z)x$ is greater than or equal to $ \frac {1}{4}$
2004 CHKMO, 1
Find the greatest real number $K$ such that for all positive real number $u,v,w$ with $u^{2}>4vw$ we have $(u^{2}-4vw)^{2}>K(2v^{2}-uw)(2w^{2}-uv)$
1967 Putnam, A1
Let $f(x)= a_1 \sin x + a_2 \sin 2x+\cdots +a_{n} \sin nx $, where $a_1 ,a_2 ,\ldots,a_n $ are real numbers and where $n$ is a positive integer. Given that $|f(x)| \leq | \sin x |$ for all real $x,$ prove that
$$|a_1 +2a_2 +\cdots +na_{n}|\leq 1.$$
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 $1$?
1985 Kurschak Competition, 3
We reflected each vertex of a triangle on the opposite side. Prove that the area of the triangle formed by these three reflection points is smaller than the area of the initial triangle multiplied by five.
1991 AIME Problems, 13
A drawer contains a mixture of red socks and blue socks, at most 1991 in all. It so happens that, when two socks are selected randomly without replacement, there is a probability of exactly $1/2$ that both are red or both are blue. What is the largest possible number of red socks in the drawer that is consistent with this data?
1975 Chisinau City MO, 113
Prove that any integer $n$ satisfying the inequality $n <(44 + \sqrt{1975})^100 <n + 1$ is odd.
1989 Tournament Of Towns, (240) 4
The set of natural numbers is represented as a union of pairwise disjoint subsets, whose elements form infinite arithmetic progressions with positive differences $d_1,d_2,d_3,...$. Is it possible that the sum $\frac{1}{d_1}+\frac{1}{d_1}+\frac{1}{d_3}+... $ does not exceed $0.9$? Consider the cases where
(a) the total number of progressions is finite, and
(b) the number of progressions is infinite.
(In this case the condition that $\frac{1}{d_1}+\frac{1}{d_1}+\frac{1}{d_3}+... $ does not exceed $0.9$ should be taken to mean that the sum of any finite number of terms does not exceed 0.9.)
(A. Tolpugo, Kiev)
1998 China National Olympiad, 2
Given a positive integer $n>1$, determine with proof if there exist $2n$ pairwise different positive integers $a_1,\ldots ,a_n,b_1,\ldots b_n$ such that $a_1+\ldots +a_n=b_1+\ldots +b_n$ and
\[n-1>\sum_{i=1}^{n}\frac{a_i-b_i}{a_i+b_i}>n-1-\frac{1}{1998}.\]
1994 India National Olympiad, 2
If $x^5 - x ^3 + x = a,$ prove that $x^6 \geq 2a - 1$.
2016 Bulgaria JBMO TST, 2
a, b, c are positive real numbers and a+b+c=k. Find the minimum value of $ b^2/(ka+bc)^1/2+c^2/(kb+ac)^1/2+a^2/(kc+ab)^1/2 $
2007 Switzerland - Final Round, 1
Determine all positive real solutions of the following system of equations:
$$a =\ max \{ \frac{1}{b} , \frac{1}{c}\} \,\,\,\,\,\, b = \max \{ \frac{1}{c} , \frac{1}{d}\} \,\,\,\,\,\, c = \max \{ \frac{1}{d}, \frac{1}{e}\} $$
$$d = \max \{ \frac{1}{e} , \frac{1}{f }\} \,\,\,\,\,\, e = \max \{ \frac{1}{f} , \frac{1}{a}\} \,\,\,\,\,\, f = \max \{ \frac{1}{a} , \frac{1}{b}\}$$
2009 Today's Calculation Of Integral, 485
In the $x$-$y$ plane, for the origin $ O$, given an isosceles triangle $ OAB$ with $ AO \equal{} AB$ such that $ A$ is on the first quadrant and $ B$ is on the $ x$ axis.
Denote the area by $ s$. Find the area of the common part of the traingle and the region expressed by the inequality $ xy\leq 1$ to give the area as the function of $ s$.
2016 Middle European Mathematical Olympiad, 1
Let $n \ge 2$ be an integer, and let $x_1, x_2, \ldots, x_n$ be reals for which:
(a) $x_j > -1$ for $j = 1, 2, \ldots, n$ and
(b) $x_1 + x_2 + \ldots + x_n = n.$
Prove that $$ \sum_{j = 1}^{n} \frac{1}{1 + x_j} \ge \sum_{j = 1}^{n} \frac{x_j}{1 + x_j^2} $$
and determine when does the equality occur.
2011 Kosovo Team Selection Test, 1
Let $a,b,c$ be real positive numbers. Prove that the following inequality holds:
\[{
\sum_{\rm cyc}\sqrt{5a^2+5c^2+8b^2\over 4ac}\ge 3\cdot \root 9 \of{8(a+b)^2(b+c)^2(c+a)^2\over (abc)^2}
}\]
2004 China Team Selection Test, 2
Find the largest positive real $ k$, such that for any positive reals $ a,b,c,d$, there is always:
\[ (a\plus{}b\plus{}c) \left[ 3^4(a\plus{}b\plus{}c\plus{}d)^5 \plus{} 2^4(a\plus{}b\plus{}c\plus{}2d)^5 \right] \geq kabcd^3\]
1996 AMC 12/AHSME, 27
Consider two solid spherical balls, one centered at $(0, 0, \frac{21}{2} )$ with radius $6$, and the other centered at $(0, 0, 1)$ with radius $\frac 92$ . How many points $(x, y, z)$ with only integer coordinates (lattice points) are there in the intersection of the
balls?
$\text{(A)}\ 7 \qquad \text{(B)}\ 9 \qquad \text{(C)}\ 11 \qquad \text{(D)}\ 13 \qquad \text{(E)}\ 15$
2021 BMT, 24
Suppose that $a, b, c$, and p are positive integers such that $p$ is a prime number and $$a^2 + b^2 + c^2 = ab + bc + ca + 2021p$$.
Compute the least possible value of $\max \,(a, b, c)$.
2020 China Northern MO, BP1
For all positive real numbers $a,b,c$, prove that
$$\frac{a^3+b^3}{ \sqrt{a^2-ab+b^2} } + \frac{b^3+c^3}{ \sqrt{b^2-bc+c^2} } + \frac{c^3+a^3}{ \sqrt{c^2-ca+a^2} } \geq 2(a^2+b^2+c^2)$$
2009 Iran MO (2nd Round), 3
Let $ ABC $ be a triangle and the point $ D $ is on the segment $ BC $ such that $ AD $ is the interior bisector of $ \angle A $. We stretch $ AD $ such that it meets the circumcircle of $ \Delta ABC $ at $ M $. We draw a line from $ D $ such that it meets the lines $ MB,MC $ at $ P,Q $, respectively ($ M $ is not between $ B,P $ and also is not between $ C,Q $).
Prove that $ \angle PAQ\geq\angle BAC $.
2013 USAMTS Problems, 3
An infinite sequence of positive real numbers $a_1,a_2,a_3,\dots$ is called [i]territorial[/i] if for all positive integers $i,j$ with $i<j$, we have $|a_i-a_j|\ge\tfrac1j$. Can we find a territorial sequence $a_1,a_2,a_3,\dots$ for which there exists a real number $c$ with $a_i<c$ for all $i$?