2
votes
1answer
43 views

Sum involving integer part and cosine function

How to find the close form of sum and eliminate $k$? $$ \sum_{k=1}^{n} \frac{n \left[ \cos \left( \frac{n}{k}- \left[\frac{n}{k} \right]\right) \right]}{k} $$
3
votes
2answers
65 views

Elementary algebra problem

Consider the following problem (drawn from Stanford Math Competition 2014): "Find the minimum value of $\frac{1}{x-y}+\frac{1}{y-z}+ \frac{1}{x-z}$ for for reals $x > y > z$ given $(x − y)(y − ...
3
votes
1answer
78 views

Solve in $\mathbb{R}$: $(x^2-3x-2)^2-3(x^2-3x-2)-2-x=0$

Solve in $\mathbb{R}$: $(x^2-3x-2)^2-3(x^2-3x-2)-2-x=0$ I'm supposed to solve this equation. It's from a math contest so solving it by hand would be preferable (no quartic formulas). I thought ...
1
vote
4answers
81 views

Solve the following equation: $\frac{1}{x^2}+\frac{1}{(4-\sqrt{3}x)^2}=1$

Solve the following equation: $$\frac{1}{x^2}+\frac{1}{(4-\sqrt{3}x)^2}=1$$ I know it's from a Math Olympiad but I don't know which and I couldn't find it on the internet. Expanding everything ...
0
votes
2answers
44 views

Roots Of Monic Cubic

I'm currently preparing for the USA Mathematical Talent Search competition. I've been brushing up my proof-writing skills for several weeks now, but one area that I have not been formally taught about ...
5
votes
2answers
171 views

Help with complicated functional equation

Problem: Let $T=\{(p,q,r)\mid p,q,r \in \mathbb{Z}_{\geq0}\}$. Find all functions $f:T\to \mathbb{R}$ such that: $$f(p,q,r)=\\ =\begin{cases} 0, & \text{ if } pqr = 0 \\ 1 + ...
11
votes
1answer
147 views

Cyclic system of equations

Consider the system of equations $$ \begin{align*} x^2+(1-y)^2&=a\\ y^2+(1-z)^2&=b\\ z^2+(1-x)^2&=c\\ \end{align*} $$ Compute $x(1-x)$ in terms of $a,b,c$. Edit: The question should say ...
9
votes
2answers
248 views

If $f$ is a strictly increasing function with $f(f(x))=x^2+2$, then $f(3)=?$

Bdmo 2014 regionals(a tweaked version of question): If $f$ is a strictly increasing function over the reals with $f(f(x))=x^2+2$, then $f(3)=?$ Obviously,$f(3)=f(1)^2+2$ but I can't see where we ...
4
votes
2answers
103 views

Evaluating $\sum_{n=1}^{99}\sin(n)$ [duplicate]

I'm looking for a trick, or a quick way to evaluate the sum $\displaystyle{\sum_{n=1}^{99}\sin(n)}$. I was thinking of applying a sum to product formula, but that doesn't seem to help the situation. ...
0
votes
1answer
57 views

Understanding 2012 AMC 12B #23

Monic quadratic polynomial $P(x)$ and $Q(x)$ have the property that $P(Q(x))$ has zeros at $x=-23$, $-21$, $-17$, and $-15$, and $Q(P(x))$ has zeros at $x=-59$,$-57$,$-51$ and $-49$. What is ...
7
votes
3answers
366 views

Math Algebra Question with Square Roots

For $a\ge \frac{1}{8}$, we define, $$g(a)=\sqrt[\Large3]{a+\frac{a+1}{3}\sqrt{\frac{8a-1}{3}}}+\sqrt[\Large3]{a-\frac{a+1}{3}\sqrt{\frac{8a-1}{3}}}$$ Find the maximum value of $g(a)$. I ...
2
votes
2answers
76 views

Algebra Manipulation Contest Math Problem

The question was as follows: The equations $x^3+Ax+10=0$ and $x^3+Bx^2+50=0$ have two roots in common. Compute the product of these common roots. Because $x^3+Ax+10=0$ and $x^3+Bx^2+50=0$ it means ...
6
votes
4answers
156 views

Prove that there is an integer $n$ such that $n^{1992}$ starts with $1992$ one's.

This was taken from an old Brazilian Mathematical Olympiad (1992). As the title says, we're supposed to prove that there is an integer $n$ such that $n^{1992}$ starts with $1992$ one's (in the ...
2
votes
4answers
89 views

Algebraic Solving Contest Problem

The problem is as follows If $x^2+x-1=0$, compute all possible values of $\frac{x^2}{x^4-1}$ This was a no-calculator 10 min for 2 problem format contest. I started by using quadratic formula, but ...
2
votes
1answer
50 views

Algebraic maximum and minimum based on a constraint

Suppose $a,b,c$ are real numbers such that $a^2b^2 + b^2c^2 + c^2a^2 = k$, where $k$ is a constant. Then the set of all possible values of $abc(a+b+c)$ is? I attempted writing the constraint in the ...
0
votes
1answer
28 views

Algebra Value based on condition provided

Let $a, b, c$ be distinct real numbers such that $a^2 - b = b^2 - c = c^2 - a$ Then $(a+b)(b+c)(c+a)$ equals? I attempted manipulations with that condition provided, but then I'm unable to go ...
5
votes
2answers
107 views

A different type binomial expansion problem

Suppose we have $$(1+x+x^2)^n = a_0 + a_1 x + a_2 x^2 + \cdots + a_{2n} x^{2n}.$$ What will be the value of $a_0^2 - a_1^2 + a_2^2 - \cdots + a_{2n}^2$? The answer is $a_n$, but I can't solve it. ...
0
votes
1answer
91 views

If $x + \frac{1}{x} = k$, what's the value of this sum?

Friends, if $x + \frac{1}{x} = k$, with $k$ positive real number, what's the value of $$1+ x^6 + x^{12} + x^{18} + x^{24}+x^{30}$$ I tried with the substitution $u= x^6$: $$a= 1+ u + u^2 + u^3 + u^4 ...
1
vote
3answers
67 views

Ratio Math Problem

This is from a competition math problem I had recently that I just couldn't figure out. If $ (x+y):(y+z):(x+z) = 1:2:4$ and $x+y+z=35$ compute the value of x. I can tell that $7*(x+y)=2x+2y+2z$ ...
0
votes
3answers
83 views

$1-x+x^2-x^3+. . .-x^{17}=a_0+a_1y+a_2y^2+. . .+a_{17}y^{17},y=x+1$

This is a previous AIME question. $1-x+x^2-x^3+. . .-x^{17}=a_0+a_1y+a_2y^2+. . .+a_{17}y^{17},y=x+1$. Then what is $a_{17}$? Is anything wrong with the following method? $1-x+x^2-x^3+. . ...
0
votes
1answer
60 views

Trigonometric eq.

The equation $3\sin(x)+4\cos(x)=5$ is well-known. The equation $3\sin^m(x)+4\cos^n(x)=5$ where $m$ and $n$ are non-negative integers is much more interesting.. I would like to see a nice, elementary ...
2
votes
1answer
127 views

IMO problem 4, $1998$

Determine all pairs $(a, b)$ of positive integers such that $ab^{2} + b + 7$ divides $a^{2}b + a + b$. I really have no idea where to start with this. This is the first IMO problem that I attempted, ...
-2
votes
2answers
128 views

algebra , JEE-IIT entrance test sample questions [closed]

$x$ is a real number. If $x^3+1/x^3=52$, find the value of $x^5+1/x^5$.
7
votes
7answers
288 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 ...
2
votes
5answers
121 views

Prove that there exist infinitely many pythagorean integers $a²+b²=c²$

Prove that there exist infinitely many Pythagorean integers $a²+b²=c²$ My key idea is to show that there exists infinitely many integers that can be the length of the sides of a right triangle, but ...
2
votes
1answer
76 views

How to prove $\binom{2n}{n}\frac{1}{n+1} = \prod \limits_{i = 2}^n \frac{2i-1}{i+1} $?

How to prove this closed form involving Catalan numbers? $$\binom{2n}{n}\frac{1}{n+1} = \prod \limits_{i = 2}^n \frac{2 \times (2i-1)}{i+1} $$ I have seen this being used here. Not sure how to derive ...
11
votes
10answers
2k views

Find five consecutive odd integers such that their sum is $55$.

So my professor asked us to do an Olympiad exercise which says that the sum of five consecutive odd integers is $55$, find those integers. But I've never seen such an exercise so it is quite new and ...
0
votes
2answers
83 views

Inequality regarding areas of triangles

BdMO Nationals 2013: There is a point O inside ∆ABC. After joining A,O; B,O and C,O extend those line and they will intersect BC, AC and AB at points D, E and F respectively. ...
5
votes
2answers
87 views

Proof that b is not divisible by 6

$$b=\left \lfloor (\sqrt[3]{28}-3)^{-n} \right \rfloor$$ The brackets mean that the number is the largest integer smaller than $(\sqrt[3]{28}-3)^{-n} $ Proof that b is never divisible by 6. I have ...
1
vote
2answers
73 views

Prove $a = b = c$, given $P_1(x) = ax^2-bx-c$ , $P_2(x) = bx^2-cx-a$, $P_3(x)=cx^2-ax-b$ and $P_1(v)=P_2(v)=P_3(v)$

Prove $a = b = c$, given $P_1(x) = ax^2-bx-c$, $P_2(x) = bx^2-cx-a$, $P_3(x)=cx^2-ax-b$ and $P_1(v)=P_2(v)=P_3(v)$ where $v$ is a real number. $a,b,c$ are non zero real numbers.
-2
votes
4answers
149 views

mental ability whiz

I got a difficult question in a national olympiad, and was not able to solve it. I can't wait for answer keys. please solve it for me! If $3a = 4b = 6c$ and $a + b + c = 27 \sqrt{29}$, then what is ...
2
votes
3answers
99 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
203 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 ...
4
votes
3answers
148 views

Find all polynomials $P(x)$ such that $2xP(x)=(x+1)P(x-1)+(x-1)P(x+1)$.

Find all polynomials $P(x)$ such that $2xP(x)=(x+1)P(x-1)+(x-1)P(x+1)$. Well, if $\deg P\le 3$ this is easy since we can deduce $P(0)=P(1)=P(-1)$ by letting $x=0,1,-1$
4
votes
2answers
229 views

Algebra question from Australia national olympiad 2013

Find all positive integers $n$ for which there are real numbers $x_1, \; x_2, \cdots,\; x_n$ satisfying $$(i) \; \; -1<x_i<1 \; for \; i=1,2, \cdots n$$ $$(ii) \; \; x_1+x_2+ \cdots +x_n=0 \; ...
3
votes
1answer
346 views

How find this value of $x,y$

let $x,y\in R$, such $$\begin{cases} \sqrt{1+(x+y)^2}=-y^6+2x^2y^3+4x^4\\ \sqrt{2x^2y^2-x^4y^4}\ge 4x^2y^3+5x^3 \end{cases}$$ find the value of $x,y$. My try: since ...
0
votes
0answers
40 views

Why is $A^{m} - 1 = (A^{m'} - 1)(A^{m'(a-1)} + A^{m'(a-2)} + … + A^{m'} + 1).$

Show that if $m$ is a multiple of $a^n$, then $(a + 1)^m -1$ is a multiple of $a^{n+1}$. Here is a solution, but I don't understand it: We use induction on $n$. For $n = 0$ we have to show that ...
7
votes
1answer
99 views

Find the sum of the first ten terms

How do I find the sum below? $$\sum_{i=1}^{10}\frac{2i+1}{i^2(i+1)^2}$$ I think there should be a simpler way instead of just adding the ten terms up using brute force, since it's impossible to do ...
2
votes
3answers
111 views

Factor $(x+y)^7-(x^7+y^7)$

So I was doing some practice problems to prepare upcoming math contests. This is one of the problems: Factor $(x+y)^7-(x^7+y^7)$ I got zero for $(x+y)^7-(x^7+y^7)$, however, the solutions ...
3
votes
1answer
155 views

Solve the simultaneous equations $x + \frac{3x-y}{x^2+y^2} = 3 $, $ y – \frac{x+3y}{x^2+y^2} = 0$

Find all solution in $\mathbb{R}$ for the following system of equations: \begin{cases} x + \frac{3x-y}{x^2+y^2} = 3 \\ y – \frac{x+3y}{x^2+y^2} = 0 \end{cases} I've tried few method, but none ...
7
votes
2answers
213 views

Compositeness of $n^4+4^n$ [duplicate]

My coach said that for all positive integers $n$, $n^4+4^n$ is never a prime number. So we memorized this for future use in math competition. But I don't understand why is it?
4
votes
1answer
157 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 ...
11
votes
1answer
305 views

Finding $x^4 + y^4 + z^4$ using geometric series

This is a problem from the 2001 Stanford Math Tournament Algebra section. $$$$Given that $$x+y+z=3$$ $$x^2 + y^2 + z^2 = 5$$$$x^3+y^3+z^3=7$$Find $x^4+y^4+z^4$. $$$$My friend claimed that he was able ...
8
votes
4answers
188 views

equilateral triangle; $3(a^4 + b^4 + c^4 + d^4) = (a^2 + b^2 + c^2 + d^2)^2.$

In equilateral triangle ABC of side length d, if P is an internal point with PA = a, PB = b, and PC = c, the following pleasingly symmetrical relationship holds: $3(a^4 + b^4 + c^4 + d^4) = (a^2 + b^2 ...
8
votes
1answer
218 views

How to prove there exists a polynomial with degree at most $100\sqrt{nk}$ satisfying this condition

Show that for arbitrary positive integers $n,k$, there exists a polynomial $p(x)$, with degree at most $100\sqrt{nk}$, such that ...
2
votes
3answers
247 views

Show that $\gcd(a + b, a^2 + b^2) = 1\mbox{ or } 2$ [duplicate]

How to show that $\gcd(a + b, a^2 + b^2) = 1\mbox{ or } 2$ for coprime $a$ and $b$? I know the fact that $\gcd(a,b)=1$ implies $\gcd(a,b^2)=1$ and $\gcd(a^2,b)=1$, but how do I apply this to that?
1
vote
1answer
67 views

If n=(sin^2(2x))/4cos^2(x))+1/(sec^2(x)) and x=2.01307, find 2013n^2013

If $n=\dfrac{sin^2(2x)}{4cos^2(x)+\dfrac{1}{sec^2(x)}}$ and $x=2.01307$, find 2013n^2013 Your edits are wrong! These are two separate fractions not together!anymore!
1
vote
1answer
122 views

find the value of 1/(2+1/(4+1/(4+1/(…))))

the question is to find the value of this ugly non-stopping fraction $$\frac{1}{2+\frac{1}{4+\frac{1}{4+\frac{1}{\ldots}}}}$$. I have totally no clue; thanks for the help! How am I suppose to solve ...
1
vote
2answers
277 views

$\frac1a+\frac1b+\frac1c=0 \implies a^2+b^2+c^2=(a+b+c)^2$? [closed]

How to prove that $a^2+b^2+c^2=(a+b+c)^2$ given that $\frac1a+\frac1b+\frac1c=0$?
6
votes
1answer
273 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 ...