Linear, exponential, logarithmic, polynomial, rational, and trigonometric functions, conic sections, binomial, surds, graphs and transformations of graphs, equation- and system-solving, and other symbolic-manipulation topics.

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6
votes
5answers
250 views

Show that $\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
0answers
27 views

find the complex number $z^4$

Let $z = a + bi$ be the complex number with $|z| = 5$ and $b > 0$ such that the distance between $(1 + 2i)z^3$ and $z^5$ is maximized, and let $z^4 = c + di$. Find $c+d$. I got that the ...
1
vote
4answers
172 views

If real number x and y satisfy $(x+5)^2 +(y-12)^2=14^2$ then find the minimum value of $x^2 +y^2$

Problem : If real number x and y satisfy $(x+5)^2 +(y-12)^2=14^2$ then find the minimum value of $x^2 +y^2$ Please suggest how to proceed on this question... I got this problem from ...
2
votes
3answers
31 views

Proving $|x+y|=|x|+|y| \iff x\cdot y \geq 0$

Prove: $|x+y|=|x|+|y| \iff x\cdot y \geq 0$. $|x+y|=|x|+|y| \iff x+y=x+y$ and $-(x+y)=-x-y \iff \{x,y\}\geq 0$ and $\{x,y\}\leq 0 \iff x\cdot y\geq 0$ in both cases.
1
vote
2answers
27 views

Remainder Dividing Repunits

If n= 11111....1 ( 1 repeated 123 times.) Then find the remainder when n is divided by 271 ? I know I can write this in the form of a sum of a gp but it doesn't help to find the remainder... Any ...
0
votes
3answers
50 views

Find all solutions to the equation. $7 \sin^2x - 14 \sin x + 2 = -5$

I got this question wrong on a test and I want to see what I did wrong so I don't get this type of question wrong again.
2
votes
9answers
193 views

How is $x \leq x^2$ false?

There's an equation that says $$x \leq x^2$$ and $x \in \mathbb R$. What I can solve and clearly see is that this equation would be true for any value of '$x$' but then how come my maths teacher ...
-4
votes
2answers
29 views

Basic root numbers question [on hold]

Hello I want to simplify this expression $1\over\sqrt{(2-\sqrt{5})^2}$ Thank you
1
vote
4answers
129 views

A basic root numbers question

If $\sqrt{x^2+5} - \sqrt{x^2-3} = 2$, then what is $\sqrt{x^2+5} + \sqrt{x^2-3}$?
20
votes
12answers
3k views

What do sine, tan, cos actually mean?

I know that $\sin\theta=\frac{y}{r}$ and $\cos\theta=\frac{x}{r}$. My question is: is $\sin$ a function of $\theta$, as in $\sin (\theta$)? If yes, why is there no $\theta$ on the right hand side of ...
0
votes
0answers
27 views

Determining polynomial values

I need to determine the values of $A$ and $B$ in this polynomial equation. $R(x)=x^4+Ax^3+Bx^2+10x-1$ when divided by $x+1$ the remainder is $-15$.
2
votes
6answers
141 views

Prove the inequality $|xy|\leq\frac{1}{2}(x^2+y^2)$

How can I prove the inequality $|xy|\leq\frac{1}{2}(x^2+y^2)$ I have tried substitute $x,y$ for numbers, which turns out right, but I don't know how to reason here. Thanks in advance!
0
votes
3answers
27 views

Find the number of seven digit whole numbers in which only 2 and 3 are present as digits if no two 2's are consecutive in any number?

Find the number of seven digit whole numbers in which only $2$ and $3$ are present as digits if no two $2$'s are consecutive in any number? My Approach: We can make numbers and see like: ...
0
votes
1answer
15 views

Quadratic equation roots values was positive but shown as negative

Hi, This screen capture was taken from KhanAcademy. I am an adult learner trying to revisit Algebra I/II concepts. In the video, p was calculated as 1/4 or 4. But, why was is factorized as ...
0
votes
1answer
27 views

If $500! = 2^m\cdot$N, where N is an odd positive integer, then find $m$

Problem : If $500! = 2^m\cdot$N, where N is an odd positive integer, then find $m$ My approach : Shall we need to expand $500!$ and then find prime factors and see what is the power of 2 in that ...
16
votes
6answers
510 views

$\log_9 71$ or $\log_8 61$

I am trying to know which one is bigger :$$\log_9 71$$ or $$\log_8 61$$ how can i know without using a calculator ?
4
votes
1answer
151 views

Richardson's theorem for constants

It's known that there is no algorithm for deciding for any elementary function is it identically zero or not (http://en.wikipedia.org/wiki/Richardson%27s_theorem ). But if I consider only constants - ...
2
votes
3answers
73 views

In what conditions a quadratic function has an integer value of $f(x)$ where $x$ is also an integer?

EDITED Sorry, the question was wrong. Please forgive me for this. Suppose a quadratic function $f(x) = ax^2+bx+c$, what I want to know is if in an integer $x$, say $x=1, x=2, x=3, ...$, the function ...
5
votes
5answers
135 views

$f(x) =ax^6 +bx^5+cx^4+dx^3+ex^2+gx+h $ find f(7)

Problem : $f(x) =ax^6 +bx^5+cx^4+dx^3+ex^2+gx+h$ Given that : $f(1)= 1, f(2) =2 , f(3) = 3, f(4) =4, f(5)=5, f(6) =6$ find $f(7) =?$ My approach: We can put the values of $f(1) = 1$ in the ...
0
votes
1answer
45 views

Vibrating water container problem

I am struggling with this seemingly difficult question: "A water-filled container is sitting still on a platform as shown. Suddenly, the platform starts shaking vertically due to the action of a ...
3
votes
2answers
57 views

Finding the equation of a circle.

A circle of radius $2$ lies in the first quadrant touching both axis. Find the equation of the circle centered at $(6,5)$ and touching the above circle externally. Let me share how I answered this ...
2
votes
2answers
88 views

Understanding Bernoulli's Inequality

Here is the proof, I understand it for the most part, but have some specific questions to help me understand better. In the induction step: Line 2, we multiply $(1+x)$ to balance the comparison so ...
1
vote
0answers
26 views

Proving an inequality involving discrete variables

I'm trying to show that the following inequality holds $$ \frac{1-x^{n}}{1-x^{n+1}}\geq\frac{\sum_{i=0}^{n-2}x^{i}(1-x_{1}^{n-(i+1)})}{\sum_{i=0}^{n-1}x^{i}(1-x_{1}^{n-i})}, $$ where $n$ is a ...
2
votes
1answer
46 views

Find a constant $C$ such that $ \Bigg| \frac{\prod_{i=0}^{k-1} (n-i) }{n^k} - 1 \Bigg| \leq \frac{C}{n}, \forall k \leq n$

Consider the following: $$ \Bigg| \dfrac{\prod_{i=0}^{k-1} (n-i) }{n^k} - 1 \Bigg| \leq \frac{C}{n}, \forall k \leq n $$ How to find an expression for $C$ independent of $k$ and thus $n$? It arises ...
2
votes
5answers
72 views

Find the equation of the circle.

Find the equation of the circle whose radius is $5$ which touches the circle $x^2 + y^2 - 2x -4y - 20 = 0$ externally at the point $(5,5)$
0
votes
1answer
24 views

3 Points in 3D Space to Develop an Arc or Circle

Background: I'm a Robotics Engineer and I am trying to develop a more flexible, modular, and robust program for our welding robots, which will minimize teaching time for new robots and also minimize ...
1
vote
2answers
41 views

Finding the intersection of a line and standard wave function

Let's say I have two functions: $f(x)=5cos(x)$ $f(x)=4$ The line lies in between the range of the wave function so there will be two intersections for each period of the wave. I want to find ...
1
vote
0answers
57 views

Find the number of “p-safe numbers”

For a positive integer $p$, define the positive integer $n$ to be $p$-safe if $n$ differs in absolute value by more than $2$ from all multiples of $p$. For example, the set of $10$-safe numbers is ...
53
votes
17answers
22k views

What is the most elegant proof of the Pythagorean theorem?

The Pythagorean Theorem is one of the most popular to prove by mathematicians, and there are many proofs available (including one from James Garfield). What's the most elegant proof? My favorite ...
0
votes
2answers
63 views

Find all the possible real values for $a,b,c,d$.

Let pairs $(a,c)$ and $(b,d)$ be roots of the equations $x^2 + ax - b = 0$ and $x^2 + cx + d = 0$ respectively. Find all possible real values for $a,b,c,d$.
1
vote
1answer
410 views

Algorithm to find the maximum/minimum of a polynomial without graphing.

For a quadratic equation of the form $y=ax^2+bx+c$ the max/min occurs at $x=-\frac{b}{2a}$. Is there any hard and fast equation like this for polynomials of degree $\geq 4$?. For such polynomials the ...
0
votes
1answer
18 views

equation solving approach for the given equation

$y^4-10y^3+5y^2+100y+100=0$ how to solve this equation ? what will be the approach to solve this question. Breaking $100$ into $84$ and $16$ and then solving could be one .
1
vote
2answers
57 views

This expression is always a perfect square [on hold]

How to show that for $x,y\in \Bbb R$, the expression $xy+\left(\frac{x-y}{2} \right)^2$ is always a perfect square? For example $x=7, y=3$, $7\times 3+\left(\frac{7-3}{2} \right)^2=25=5^2$
2
votes
2answers
29 views

Parallelogram ABCD

There's a parallelogram $ABCD$. I'm given point $A(3,12)$ and point $B(-1,5)$. Given the equations of the lines $BC$ and $AC$ are $y=8x+13$ and $y=3x+3$ respectively. How to find the coordinates of ...
1
vote
1answer
61 views

How to find x,y,z such that $\frac{1800}{x}=a$, $\frac{1800}{y}=b$, $\frac{1800}{z}=c$, $\frac{1800}{a+b+c}=d$

I'm really fascinated by how questions and problems are designed in mathematics. So, I was designing a simple word problem, and in the course I fell into this situation: a,b,c,d are natural numbers. ...
-2
votes
3answers
117 views

Harder-Than-Seems Inverse of $f(x)=x^3-x-12$?

This may seem simple but I have had long days of frustration with finding the inverse of this: $$f(x)=x^3-x-12.$$ I got this on some homework and it did not ask for the inverse. However I wanted to ...
2
votes
0answers
47 views

Is there a closed-form solution (even approximated) to this inequality?

I have the following function: $f(x, \theta) = (1-\theta)(x+1)^{-\theta}\left[ \frac{2-2\theta}{1- 2\theta} (N^{1-2\theta} - (x+1)^{1-2\theta}) - (x+1)^{-\theta}(N^{1-\theta} - (x+1)^{1-\theta}) ...
-2
votes
1answer
51 views

Algebraic Problem regarding Cubes. [on hold]

If $a^3 + b^3 + c^3 = 3abc$ where $a \ne b \ne c$, find the value of: $(a + b + c)$
1
vote
2answers
47 views

Intuition: inverse function

Disclaimer: I'm a beginner with inverse functions. Can anyone explain what I'm doing wrong here? I'd like to avoid using "y" -- that is, I want to show everything in terms of x and f(x). Thanks!
14
votes
7answers
1k views

Why do remainders show cyclic pattern?

Let us find the remainders of $\dfrac{6^n}{7}$, Remainder of $6^0/7 = 1$ Remainder of $6/7 = 6$ Remainder of $36/7 = 1$ Remainder of $216/7 = 6$ Remainder of $1296/7 = 1$ This pattern of ...
0
votes
3answers
47 views

If a given # is $70$% of $X$. How do you determine what $X$ is? [duplicate]

Given I have the number $50,000$ which is $70$% of $X$. How do I calculate what $X$ is without guessing. Thanks
-1
votes
1answer
12 views

How to find the bearing and velocity of a boat on a flowing river

Point A is on the west bank of a river and point C is directly across from it on the east bank. The river is 648.6 meters wide and flows south at 2.45 km/hr. A boat wants to cross the river from point ...
1
vote
3answers
64 views

how can i prove this trigonometry equation

I need help on proving the following: $$\frac{\cos {7x} - \cos {x} + \sin {3x}}{ \sin {7x} + \sin {x} - \cos {3x} }= -\tan {3x}$$ So far I've only gotten to this step: $$\frac{-2 \sin {4x} \sin {3x} ...
1
vote
1answer
39 views

How to solve a equation task

I have found a equation task that is: y=0,5x+2,5 y=3x-1 The answer and its process is: ...
2
votes
3answers
192 views

Unclear step in the proof of half-angle formula for tangent

I wonder how could $$2\cos^2\left(\frac a2\right)$$ be transformed into $$1+\cos(a)$$ This is from a step in my textbook's proof of the tangent half-angle formula: $$tan\left(\frac a2\right) = .. ...
-9
votes
1answer
42 views

Algebra 2 help! [on hold]

Write the equation of a circle with the given center and radius. Center (-2,3); Radius 8
1
vote
2answers
33 views

How to show that this interesting difference of products is $O \left( \frac{1}{n^2} \right) $

Let $k \leq n$. Consider the following difference of products: $$ \prod_{i=1}^{k-1} \left( 1 - \frac{i}{n+1} \right) - \prod_{i=1}^{k-1} \left( 1 - \frac{i}{n} \right)$$ For $n=1,2,3$, this is ...
-3
votes
1answer
56 views

Help me understand probability..I don't get it!! Exam tomorrow

Postcodes can be made from 4 digits --> 1234 How many different postcodes beginning with 2 are possible? How do I truly understand probability questions? Nothing beyond the level of ordered / ...
3
votes
3answers
100 views

High computation in probability

Six men and some number of women stand in a line in random order. Let $p$ be the probability that a group of at least four men stand together in the line, given that every man stands next to at ...
0
votes
1answer
24 views

How to prove that $(a-b) \mod N = a \mod N + ((-b) \mod N)$?

I've gone through the similar post Modulo of a negative number . But that post is not about proof and I'm asking for the proof in general. This question is another follow up question of my previous ...