4
votes
2answers
80 views

Finding the value of $(bc-ad)(ac-bd)(ab-cd)$

Let $a,b,c,d$ be $4$ distinct non-zero integers such that $a+b+c+d = 0$. It is know that the number $$M = (bc - ad)(ac - bd)(ab-cd)$$ lies strictly between $96100$ and $98000$. Determine the value ...
2
votes
1answer
67 views

Putnam inspired problem

The following is a beautiful problem from Putnam 2003 minimize $|\sin x + \cos x + \tan x + \csc x + \sec x + \cot x|$ I was thinking about a small variation of the above problem minimize $|\sin ...
1
vote
2answers
28 views

Connecting square vertexes with minimal road

I have four cities in $A=(0,0),B=(1,0),C=(1,1),D=(0,1)$. I am asked to build the shortest motorway to connect the cities. How can I do that? I was thinking that first I need some compactness argument ...
4
votes
1answer
179 views

Inequality in triangle involving side lenghs, medians and area

A, B and C are the vertices of a triangle. Denote $m_a$, $m_b$ and $m_c$ the medians from A, B and C. Prove the inequality: $$\sum_{cyc}{a^2bcm_a}\geq\sum_{cyc}{cS(a^2+b^2)}$$where a, b and c are the ...
1
vote
1answer
57 views

Prove the derivative

Let $f(x) = (x^2-1)^{\frac{1}{2}}, x>1$. How do I prove that the $n$th derivative of $f(x) > 0$ for odd $n$, and the $n$th derivative of $f(x) < 0$ for even $n$?
5
votes
3answers
112 views

An inequality for sides of a triangle

Let $ a, b, c $ be sides of a triangle and $ ab+bc+ca=1 $. Show $$(a+1)(b+1)(c+1)<4 $$ I tried Ravi substitution and got a close bound, but don't know how to make it all the way to $4 $. I am ...
7
votes
7answers
269 views

$211!$ or $106^{211}$:Which is greater?

A BdMO question: Let $a=211!$ and $b=106^{211}$. Show which is greater with proper logic. By matching term by term,it is pretty easy to note that $106!<106^{106}$ $106^{105}<107\cdot ...
1
vote
2answers
55 views

Given $x^2 + y^2 + z^2 = 3$ prove that $x/\sqrt{x^2+y+z} + y/\sqrt{y^2+x+z} + z/\sqrt{z^2+x+z} \le \sqrt3$

Given $x^2 + y^2 + z^2 = 3$ Then prove that $${x\over\sqrt{x^2+y+z}} + {y\over\sqrt{y^2+x+z}} + {z\over\sqrt{z^2+x+y}} \le \sqrt 3$$ I tried using Cauchy-Schwartz inequality but the inequality is ...
8
votes
4answers
108 views

If $a,b,c,d>0$ and $a+b+c+d=4$, then $\frac{1}{11+a^2}+\frac{1}{11+b^2}+\frac{1}{11+c^2}+\frac{1}{11+d^2} \leq \frac {1}{3}$

Prove if $a,b,c,d>0$ and $a+b+c+d=4$, then $$\dfrac{1}{11+a^2}+\dfrac{1}{11+b^2}+\dfrac{1}{11+c^2}+\dfrac{1}{11+d^2} \leq \dfrac {1}{3}$$ This was an Inequality Olympiad Problem1. I proved by ...
-1
votes
1answer
99 views

$a^3+b^3+c^3 + 21abc \geq 3$ for $(a+b)(a+c)(b + c) = 1$ and $a,b,c>0$

$a, b, c \gt 0$ and $(a+b)(a+c)(b + c) = 1$ Prove that $a^3+b^3+c^3 + 21abc \geq 3$ In this problem I spotted one trick $(a+b)(a+c)(b + c) = 1 \Leftrightarrow \\(a \sqrt{b+c})^2+(b\sqrt{a+c})^2+(c ...
1
vote
0answers
39 views

Generalized inequality with parameters $\alpha, \beta$

Let $d$ be a positive integer, and let $\alpha, \beta$ be positive real numbers such that $\alpha+\beta=1$. Consider the inequality in $k$ variables $x_1, x_2, …, x_k$, $$ \alpha \cdot \sum_{i=1}^k ...
2
votes
2answers
74 views

How to prove the inequality: $\frac{(1+x)^2}{2x^2+(1-x)^2}+\frac{(1+y)^2}{2y^2+(1-y)^2}+\frac{(1+z)^2}{2z^2+(1-z)^2}\leq 8$

Prove that: $$\frac{(1+x)^2}{2x^2+(1-x)^2}+\frac{(1+y)^2}{2y^2+(1-y)^2}+\frac{(1+z)^2}{2z^2+(1-z)^2}\leq 8$$ subject to the constraints: $$x,y,z >0$$ and $$x+y+z=1.$$
2
votes
2answers
75 views

The floor of a product of fractions

Evaluate: $ \displaystyle \Bigg \lfloor \prod_{n=0}^{248} \frac{33+8n}{29+8n} \Bigg \rfloor= \Bigg \lfloor \frac{33}{29} \times \frac{41}{37} \times \frac{49}{45} \times\ ...\ \times ...
4
votes
2answers
61 views

Minimum value of: $x^7(yz-1)+y^7(zx-1)+z^7(xy-1)$

$x$, $y$ and $z$ are positive reals such that $x+y+z=xyz$. Find the minimum value of: $$x^7(yz-1)+y^7(zx-1)+z^7(xy-1)$$ I put it in the form $x^6y +x^6z+y^6x+y^6z+z^6x +z^6y$. I tried AM-GM but ...
0
votes
1answer
127 views

Let $f(x)=\exp(-a|x|)$ and $a>0$. Show that there exists $C$ and $\alpha$ such that $|f(x)-f(y)|\le\frac{C|x-y|}{1+x^2}$ for $|x-y|\le\alpha$.

Let $f(x)=\exp(-a|x|)$ and $a>0$. Show that there exists $C$ and $\alpha$ such that $$|f(x)-f(y)|\le\frac{C|x-y|}{1+x^2}$$ for $|x-y|\le\alpha$. From the mean value theorem, given any $x,y$ with ...
0
votes
4answers
70 views

How to prove this ineqality

prove that $1 \leq \frac{1}{1001} + \frac{1}{1002} + ......+\frac{1}{3001} \leq \frac{4}{3} $ it seems from some Olympiad. i tried using sum of series etc. but could not get it.
2
votes
0answers
44 views

Prove $\sup_{0\le x\le 1}|f(x)|\le\int_0^1(|f(t)|+|f'(t)|)dt$

Let $f\in C^1([0,1])$. Prove the following: $$\sup_{0\le x\le 1}|f(x)|\le\int_0^1(|f(t)|+|f'(t)|)dt$$ and $$|f(1/2)|\le\int_0^1(|f(t)|+\frac12|f'(t)|)dt$$ Note that ...
1
vote
0answers
63 views

Prove that $0\le\frac1{b-a}\int_a^b|f(x)|dx-\left|\frac1{b-a}\int_a^bf(x)dx\right|\le\frac{b-a}3\sup_{a\le x\le b}|f'(x)|$

Let $f'$ be integrable. Prove that $$0\le\frac1{b-a}\int_a^b|f(x)|dx-\left|\frac1{b-a}\int_a^bf(x)dx\right|\le\frac{b-a}3\sup_{a\le x\le b}|f'(x)|$$ Source: ...
7
votes
1answer
141 views

Bound on $|f(x)|^2 + |f'(x)|^2$

Let $f\in C^2(\mathbb{R})$ be a twice differentiable function satisfying $$|f(x)|^2\le a$$ and $$|f'(x)|^2 + |f''(x)|^2\le b$$ for all real $x$, where $a$ and $b$ are positive constants. Prove that ...
2
votes
3answers
88 views

For any real numbers $a,b,c$ show that $\displaystyle \min\{(a-b)^2,(b-c)^2,(c-a)^2\} \leq \frac{a^2+b^2+c^2}{2}$

For any real numbers $a,b,c$ show that: $$ \min\{(a-b)^2,(b-c)^2,(c-a)^2\} \leq \frac{a^2+b^2+c^2}{2}$$ OK. So, here is my attempt to solve the problem: We can assume, Without Loss Of Generality, ...
6
votes
5answers
151 views

Show that $\displaystyle \frac{xy}{z} + \frac{xz}{y} + \frac{yz}{x} \geq x+y+z $ by considering homogeneity

Well, I'm preparing for an undergrad competition that is held in April and because of that I've been trying to solve the inequalities I find on the internet. I found this problem: $$\displaystyle ...
1
vote
1answer
21 views

Interesting continued fraction problem $|r_i-u_0/u_1|\le\frac1{k_ik_{i+1}}$

Let $u_0/u_1$ be a rational number in lowest terms, and write $u_0/u_1=\langle a_0, a_1,...,a_n\rangle$ in standard continued fraction notation. Show that if $0\le i<n$, then ...
1
vote
1answer
96 views

How find this maximum of $P_{1}+P_{n}$

Question $n$ students attend a test of $m$ problems where $m, n \ge 2$. The scoring rule for each problem is: If $x$ students answer a problem incorrectly, then a correct answer worth $x$ points ...
1
vote
1answer
63 views

Prove $1 + \sum_{i=0}^n(\frac1{x_i}\prod_{j\neq i}(1+\frac1{x_j-x_i}))=\prod_{i=0}^n(1+\frac1{x_i})$

Prove the identity $$1 + \sum_{i=0}^n \left(\frac1{x_i}\prod_{j\neq i} \left(1+\frac1{x_j-x_i} \right) \right)=\prod_{i=0}^n \left(1+\frac1{x_i} \right)$$ and hence deduce the inequality in Problem ...
8
votes
1answer
150 views

An exponential “rearrangement” inequality: $x^x+y^y>x^y+y^x$

Let $x,y$ be distinct real numbers greater than $0$. Prove $$x^x+y^y>x^y+y^x .$$ Source: I think it comes from a Russian test given in 1991, but I haven't been able to verify this.
12
votes
1answer
309 views

Prove $\left|\sum_{k=2001}^{m}a_{k}\sin{(kx)}\right|\le 1+\pi $ ,$m\ge 2001,x\in R$

let $\{a_{n}\}$ is non-increasing postive sequence;show that if for $n\ge 2001,na_{n}\le 1$, then for any positive integer numbers $m\ge 2001,x\in R$, we have ...
4
votes
1answer
136 views

$a,b,c>0,a+b+c=21$ prove that $a+\sqrt{ab} +\sqrt[3]{abc} \leq 28$

$a,b,c>0,a+b+c=21$ prove that $a+\sqrt{ab} +\sqrt[3]{abc} \leq 28$ I have tried to use AM-GM inequality, but get no result as follows: $$a+\sqrt{ab}+\sqrt[3]{abc}\leq ...
7
votes
2answers
186 views

How prove this inequality generalized from 1969 IMO problem 6

Let $x_{1},x_{2},y_{1},y_{2},z_{1},z_{2},w_{1},w_{2} $ are all positive numbers, and such $$x_{1}y_{1}z_{1}-w^3_{1}>0,\; \text{ and }\;x_{2}y_{2}z_{2}-w^3_{2}>0.$$ show that ...
2
votes
1answer
42 views

A $4$ variable inequality

If $a,b,c,d$ are positive numbers such that $c^2+d^2=(a^2+b^2)^3$, prove that $$\frac{a^3}{c} + \frac{b^3}{d} \ge 1,$$ with equality if and only if $ad=bc$. Source: Don Sokolowsky, Crux ...
4
votes
1answer
94 views

An $n$th root inequality: $\sqrt[n]{n} < 1 + \sqrt{2/n}$

Prove that for any positive integer $n$, $$n^{1/n} < 1 + \sqrt{\frac{2}{n}}.$$ This due to Victor Linis, Eureka, Vol. 2, No. 2, February 1976, p. 29. Hint:
31
votes
2answers
1k views

A generalization of IMO 1983 problem 6

Note: This question has a bounty that will expire in just a few days. Let $a,b,c$ and $d$ be the lengths of the sides of a quadrilateral. Show that $$ab^2(b-c)+bc^2(c-d)+cd^2(d-a)+da^2(a-b)\ge 0$$ ...
6
votes
1answer
266 views

Prove that: $\dfrac{1}{a+3}+\dfrac{1}{b+3}+\dfrac{1}{c+3}+\dfrac{1}{d+3}\leq1$

Let $a$, $b$, $c$ and $d$ are non-negative numbers such that $abc+abd+acd+bcd=4.$ Prove that: $\dfrac{1}{a+3}+\dfrac{1}{b+3}+\dfrac{1}{c+3}+\dfrac{1}{d+3}\leq1$ I simplified it and it turns out that ...
13
votes
1answer
181 views

integral inequality $\int_0^a \left(\frac{f(x)}{2x}\right)^2 dx \le \int_0^a (f'(x))^2 dx$

Let $f:[0,a]\rightarrow\mathbb{R}$ be continuous differentiable function satisfying $f(0)=0$. Prove the following inequality $$\int_0^a \left(\frac{f(x)}{2x}\right)^2 dx \le \int_0^a (f'(x))^2 dx$$ ...
4
votes
1answer
118 views

Probability that the first digit of $2^{n}$ is 1

Let $a_{n}$ be the number of terms in the sequence $2^{1},2^{2},\cdots ,2^{n}$ which begins with digit 1. Prove that $$\log2 -\frac{1}{n}<\frac{a_{n}}{n}<\log2\text{ (log base is 10)}$$ ...
5
votes
1answer
111 views

Maximizing an unusual function (Putnam 1996)

“Fish," he said, "I love you and respect you very much. But I will kill you dead before this day ends.” -- Ernest Hemingway, The Old Man and the Sea I have, with varying degrees of concentration, ...
0
votes
2answers
42 views

Prove that nearly all positive integers are equal to $a + b + c$ where $a | b$ and $b | c$, $a \lt b \lt c$

If a positive integer $n$ is equal to $a + b + c$ where $a | b$, $b | c$ and $a \lt b \lt c$, let it be called "faithful". Prove that nearly all numbers are faithful and list the non-faithful ...
8
votes
1answer
137 views

How prove this $ab+bc+cd\le\dfrac{5}{4}$

let $a,b,c,d\in \Bbb R$ and $a,b,c,d>-1,a+b+c+d=0$ prove that $$ab+bc+cd\le\dfrac{5}{4}$$ I have this solution if $b\le c$, then $$ab+bc+cd=a(b-c)-c^2\le ...
1
vote
2answers
65 views

Is $u_n\le(1-a)^n\forall n\in\mathbb{N}$?

Consider the sequence $\{u_n\}$ where $u_0=1,u_1=1-a$ for some $0< a < 1/4$, and $u_{n+2} = u_{n+1}-au_n$. Is $u_n\le(1-a)^n\forall n\in\mathbb{N}$?
4
votes
3answers
224 views

Verifying a proof that if $x,y,z \geq 0$ and $x+y+z = 1$, then $0 \le xy + yz + zx - 2xyz \le \frac{7}{27}$

I was working some recreational problems from a book (The Art and Craft Of Problem Solving, Zeitz) and came across one from the '84 IMO: Suppose that $x, y, z$ are non-negative reals, with $x + y ...
4
votes
0answers
109 views

Inequality problem with factorials

I am not sure if this kind of "question" is welcome on MSE. Here is an olympiad-like problem that I would like to share with you: Let $a,b,c$ be nonnegative integers. Prove that $$ ...
2
votes
2answers
111 views

Minimum Value of expression

Given that $x$, $y$ and $z$ are positive real numbers satisfying $xyz=32$, find the minimum value of: $$x^2+4xy+4y^2+2z^2$$ Perhaps AM-GM and manipulation but I'm not quite sure how? Source BMO.
2
votes
1answer
60 views

If $H$ has an $a\times b$ submatrix of all $1$s, please prove that $ab\le n$.

Let $H$ be an $n\times n$ matrix with entries $\pm1$. Its rows are mutually orthogonal. If $H$ has an $a\times b$ submatrix of all $1$s, please prove that $ab\le n$.
2
votes
3answers
77 views

Help with inequality please

Once again I have come across an olympiad-type problem which probably requires some sort of insight even though it looks simple. The question is as follows: Let $a$, $b$ and $c$ be positive real ...
10
votes
4answers
792 views

How to prove this inequality? $ab+ac+ad+bc+bd+cd\le a+b+c+d+2abcd$

let $a,b,c,d\ge 0$,and $a^2+b^2+c^2+d^2=3$,prove that $ab+ac+ad+bc+bd+cd\le a+b+c+d+2abcd$ I find this inequality are same as Crux 3059 Problem.
11
votes
4answers
373 views

Proving the inequality $\arctan\frac{\pi}{2}\ge1$

Do you see any nice way to prove that $$\arctan\frac{\pi}{2}\ge1 ?$$ Thanks! Sis.
7
votes
1answer
98 views

Geometric inequality with a triangle

The positive real numbers $x,y,z$ are the side lengths of a triangle iff $$x^2 + y^2 + z^2 < 2\sqrt{x^2y^2 + y^2z^2 + z^2x^2}$$
0
votes
1answer
54 views

Geometric inequality regarding a tetrahedron

The circumradius of a tetrahedron $ ABCD$ is $ R$, and the lenghts of the segments connecting the vertices $ A,B,C,D$ with the centroids of the opposite faces are equal to $ m_a,m_b,m_c$ and $ m_d$, ...
13
votes
4answers
590 views

Showing that $ |\cos x|+|\cos 2x|+\cdots+|\cos 2^nx|\geq \dfrac{n}{2\sqrt{2}}$

For every nonnegative integer $n$ and every real number $ x$ prove the inequality: $$\sum_{k=0}^n|\cos(2^kx)|= |\cos x|+|\cos 2x|+\cdots+|\cos 2^nx|\geq \dfrac{n}{2\sqrt{2}}$$
11
votes
1answer
258 views

Prove that $\frac{\pi}{4}\le\sum_{n=1}^{\infty} \arcsin\left(\frac{\sqrt{n+1}-\sqrt{n}}{n+1}\right)$

Prove that $$\frac{\pi}{4}\le\sum_{n=1}^{\infty} \arcsin\left(\frac{\sqrt{n+1}-\sqrt{n}}{n+1}\right)$$ EDIT: inspired by Michael Hardy's suggestion I got that $$\arcsin ...
4
votes
1answer
201 views

$x^2+y^2=z^2(1+xy)$ prove $z=\min \{x;y;z\}$ (with $x,y,z \in \mathbb{Z^+}$)

$x,y,z \in \mathbb{Z^+}$ such that $x^2+y^2=z^2(1+xy)$. Prove $z=\min \{x;y;z\}$ $$x^2+y^2=z^2(1+xy) \iff xy = \frac{x^2+y^2} {z^2} - 1$$. Assum $z>y \implies xy < x^2/z^2$, we have $xy \in Z ...