Use this tag when you want to determine the thinking that is needed to solve a certain type of problem, as opposed to looking for a specific answer to a question.

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2answers
171 views

The number of numbers whose digits are different and add up to 36

All the digits of a number are different, the first digit is not zero, and the sum of the digits is 36. There are N × 7! such numbers. What is the value of N? How should I approach this problem? I ...
0
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3answers
44 views

What can you say about a number with remainder 1 and 2 when divided by 3 and 4 respectively?

I was trying to solve a problem which states: How many two-digit numbers have remainder 1 when divided by 3 and remainder 2 when divided by 4? and solved it by writing down individual numbers... ...
1
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3answers
92 views

Solve the system of equations $\begin{cases} xy-2y-3 &=\sqrt{y-x-1}+\sqrt{y-3x+5} \\ (1-y)\sqrt{2x-y}+2(x-1) &=(2x-y-1)\sqrt{y}. \end{cases}$

Solve the following system of equations ($x,y \in \Bbb R$): $$\begin{cases} xy-2y-3 &=\sqrt{y-x-1}+\sqrt{y-3x+5} \\ (1-y)\sqrt{2x-y}+2(x-1) &=(2x-y-1)\sqrt{y}. \end{cases}$$ I think this ...
3
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5answers
105 views

Solve the equation $x(\log \log k - \log x) = \log k$

I want to solve this equation by expressing $x$ in function of $k$. Is it possible? Thanks.
3
votes
4answers
341 views

If $x\cos(\theta)-\sin(\theta)=1$ then what is the value of $x^2+(1+x^2)\sin(\theta)=1$

The question given is, If $x\cos(\theta)-\sin(\theta)=1$ then find the value of $x^2+(1+x^2)\sin(\theta)$. There are four options given $1$, $-1$, $0$ and $2$. I tried using $\sin^2+\cos^2=1$. I ...
4
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5answers
2k views

If $x^3+y^3=72$ and $xy=8$ then find the value of $x-y$.

I recently came across a question, If $x^3+y^3=72$ and $xy=8$ then find the value of $(x-y)$. By trial and error I found that $x=4$ and $y=2$ satisfies both the conditions. But in general how ...
5
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0answers
120 views

Approximating $\pi$ by an expression of the form $\sqrt{\sqrt{ \cdots \sqrt{ n!! \cdots !}}}$

Here is a problem that appeared as a prize challenge in a periodical for science students, back when I was a student: Find an approximation of $\pi$ formed of the numbers $0$ through $9$, each used ...
1
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0answers
83 views

Minimization of a multivariate quadratic equation

I am interested in the minimum of a general multivariate quadratic equation for non-negative real numbers: $$ \begin{aligned} & \underset{x_i}{\text{minimize}} & & \sum^{n}_{i=1}\sum^{n}_{...
1
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1answer
71 views

How to apply Euler's Formula in topology to this problem?

Prove that it is impossible to make a football out of exactly 9 squares and $m$ octagons, where $m \ge 4$. (In this context, a “football” is a convex polyhedron in which at least 3 edges meet at ...
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0answers
21 views

Any way to factor, collect variable from this equation?

For a sum of quadratic solutions, is there any possible way to factor out the variable $P$ from the following real function? $QT$ is also a variable, and If it matters, $P > 0$ and all indexed $...
1
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2answers
180 views

How to shade one square to make the figure symmetrical about exactly one axis? [closed]

I find it hard to understand the answer, which says How to understand the answer given? Why does it say 'only line of symmetry possible is the diagonal through $1$ and $5$? What if I shade $2$ and $3$...
0
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1answer
68 views

how many hexagons can be found in the graph?

At first sight I thought that this question requires considering different cases, but I find it difficult to convince myself as to how to start this question. Could someone please give me some hints?
1
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1answer
38 views

Solving natural log equations

The equation $\ln(|y+1|)= x-2$ where you solve for $y$, I am just unsure of how the absolute value plays into this. I am assuming that I would convert to exponential form to get $|y+1|=e^{x-2}$ and ...
5
votes
3answers
495 views

Problem Solving Positive Integers

This is a very interesting word problem that I came across in an old textbook of mine. So I know the maximum value of the HCF has to be a factor of $540$ and mayhaps the Euclidean Algorithm, but other ...
1
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2answers
33 views

A question on inequality and differentiation of logarithms

Show by differentiating that $\ln x$ is a concave function of $x$. Deduce that if $p,q,x,y$ are positive real numbers with ${1\over p}+{1\over q}=1$, then $$xy \lt {x^p\over p}+{y^q\over q}$$ I can ...
12
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2answers
493 views

Show divisibility by 7

I was stuck at this question: Suppose $a^2+b^2=c^2$ for $a,b,c \in \mathbb Z$, and neither $a$ nor $b$ is a multiple of 7. Show that $a^2-b^2$ is a multiple of 7 I tried to write $b^2$ as $c^2-a^...
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1answer
53 views

How to show the average x-coordinates of four collinear points on the curve is a constant?

Show that if four distinct points of the curve $y=2x^4+7x^3+3x-5$ are collinear then their average x-coordinate is some constant k. Find k. Shall I use vector to calculate their x-coordinate, or ...
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3answers
93 views

Algebra 2 Help??!… [closed]

use the discriminant to find the number and kind of solutions for the following equation. $$ 9x^2+12x=-4 $$
2
votes
1answer
110 views

Quartic equation or Sextic equation? And how to solve it?

In this arxiv paper (p. 11, eq. (3.2)) the authors claim that equation (3.2) is ... a quartic equation [...] which can be solved explicitly. The equation in question is \begin{equation} %\tag{...
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1answer
89 views

Showing that an oscillator has its amplitude reduced after completing half-cycle

Consider a mass $m$ at position $x(t)$ on a rough horizontal table attached to the origin by a spring with constant $k$ (restoring force $-kx$) and with a dry friction force $f$ $$\begin{cases} f=F,...
2
votes
4answers
127 views

Lawn mowing problem solving

Kate can mow the lawn in 45 minutes. Kate's sister takes twice as long to mow the same lawn. If they both have a mower and mow the lawn together, how many minutes will it take them? I know the answer ...
3
votes
3answers
147 views

Which is greater as $n$ gets larger, $f(n)=2^{2^{2^n}}$ or $g(n)=100^{100^n}$?

It is the first time I met such a question: Which is greater as $n$ gets larger, $f(n)=2^{2^{2^n}}$ or $g(n)=100^{100^n}$? Intuitively I think $f(n)$ would gradually become larger as $n$ gets ...
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2answers
68 views

A tricky diophantine equation with factorials

I am being unable to solve this diophantine equation. Does anyone have any suggestions. Let $n$ and $m$ both be non-negative integers. Find all solutions to $$n(nm - 2)! = (n!)^m$$ How would one ...
2
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2answers
74 views

Show that $1+(x_1x_2…x_n)^{\frac{1}{n}} \leq [(1+x_1)(1+x_2)…(1+x_n)]^{\frac{1}{n}}$

Show that $1+(x_1x_2...x_n)^{\frac{1}{n}} \leq [(1+x_1)(1+x_2)...(1+x_n)]^{\frac{1}{n}}, \forall x_i \geq 0, i = 1,2,3...,n$ So, I have to make this function something like this: $f(t_1x_1+t_2x_2+....
10
votes
4answers
223 views

A circle with $500$ points in its interior

Given any $1000$ points in the plane, show that there is a circle which contains exactly $500$ of the points in its interior, and none on its circumference. How do I approach this problem? I feel it ...
12
votes
1answer
106 views

Does there exist a polynomial $f(x)$ with real coefficients such that $f(x)^2$ has fewer nonzero coefficients than $f(x)$?

I saw this problem on a problem set and I have absolutely no idea how to proceed in a feasible way. Does there exist a polynomial $f(x)$ with real coefficients such that $f(x)^2$ has fewer nonzero ...
3
votes
2answers
54 views

How To Tackle Trigonometric Proofs involving $4$th and $6$th powers?

How do I prove that $\cos^4A - \sin^4A+1=2\cos^2A$ $\cos^6A + \sin^6A =1-3\sin^2A\cdot\cos^2A$ I was going through a very old and very rich book of Plane Trigonometry to build a nice foundation for ...
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0answers
58 views

A harder long division puzzle than the first; what should “Algebra I” solution look like?

Here's another problem, significantly harder than the first, but still accessible to target audience. The statement of the problem (i.e., northwest corner only) comes from a PennyDell puzzle magazine: ...
6
votes
3answers
255 views

“Long-division puzzles” can help middle-grade-level students become actual problem solvers, but what should solution look like?

This is my first post. I hope it's acceptable. EDIT Since there are people to whom such notation is foreign, I will point out that the problem represents KRRAEE / KMS, where PEI is the quotient and ...
2
votes
3answers
106 views

Application of Euler's theorem apart from finding last digits of huge numbers

I am looking for clever applications of Euler's Theorem. On browsing the internet, I see that nearly all the applications of the theorem asks for finding last few digits of a huge number. The only ...
10
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2answers
311 views

Real Induction Over Multiple Variables?

I've seen in several different places* that one can use normal mathematical induction to prove the truth of a statement that relies not on just one variable (say, $x$,) but multiple variables (for ...
4
votes
2answers
187 views

Is the following PDE boundary value problem well-posed?

My Question Is the following Poisson boundary value problem well-posed, as stated? If so, how could I go about solving it? If not, what would it need to be well-posed? Does it satisfy the "...
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0answers
35 views

Looking for problems which can be solved by the similar technique

While browsing on internet for different proofs of Fermat's theorem on sums of two squares, I came across Zagier's "one-sentence proof" which seems to be the most elegant and short proof. It invokes a ...
3
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1answer
175 views

Diophantus' Lifespan

Today I saw Diophantus' Epitaph. For those of you who don't know it and don't feel like googling: 'Here lies Diophantus,' the wonder behold. Through art algebraic, the stone tells how old: 'God ...
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1answer
43 views

For $n \geq 2$, find $\theta_n, \theta_n > 1$ s.t. $-\log(1-\frac{1}{n}) = \frac{1}{n} + \frac{\theta_n}{2n^2}$

For $n \geq 2$, show that $\exists$ a number $\theta_n, \theta_n > 1$ such that $-\log(1-\frac{1}{n}) = \frac{1}{n} + \frac{\theta_n}{2n^2}$ $\lim_{n\to \infty} \theta_n$ My attempt: I am not ...
0
votes
1answer
82 views

How can I solve this recurrence problem?

Given a function $$ f(n) = f(5n/13) + f(12n/13) + n \;\;\;\;∀n \geq 0 $$ I would like to find a function $g(n)$ such that $f ∈ Ө(g(n))$.
1
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1answer
83 views

Maths challenge problem: Why is the number of teams which require 4 substitutions 32?

I came across the following problem on a UKMT senior maths challenege: A hockey team consists of 1 goalkeeper, 4 defenders, 4 midfielders and 2 forwards. There are four substitutes: 1 goalkeeper, 1 ...
2
votes
1answer
33 views

Let $n$ be a positive integer and $S$ the set of points $(x,y)$ in the plane, where $x$ and $y$ are non-negative integers such that $x + y < n$.

Let $n$ be a positive integer and $S$ the set of points $(x,y)$ in the plane, where $x$ and $y$ are non-negative integers such that $x + y < n$. The points of $S$ are colored in red and blue so ...
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0answers
18 views

A question on the representation of all integers in terms of the sum of other interger cubes [duplicate]

The question is from a book used for transition between high school mathematics and university mathematics, which states: Prove the following statement or give a counterexample $\forall n \in \...
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1answer
28 views

More detailed explanation of how $2N_{h-2}$ becomes $2^{h/2}$?

I'm trying to learn the proof of the minimum number of nodes in an AVL tree of height h and I'm stumped on how $2N_{h-2}$ becomes $2^{h/2}$. I've read this [answer](How does $2N_{h-2}$ become $2^{h/2}$...
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1answer
49 views

Solve for b and d

Solve for b and d in the following equation. A triangle with sides $(a, a, b)$ has the same area and the same perimeter as a triangle with sides $(c, c, d)$ where $a, b, c$ and $d$ are positive ...
1
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1answer
219 views

Finding the smaller number of two given the ratio between sum, difference and product

How would you find the smaller of two numbers given the ratio between their sum, difference and product? I've been struggling with this one for a while. For example: the ratio between the sum, ...
3
votes
5answers
114 views

Why count it this way?

This is a very very elementary problem solving technique I was taught some time back. I have been using it but now looking at it, I find it kinda strange why it should be this way. Typically, the ...
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votes
1answer
370 views

A question in interview for trinity college, Cambridge

Let $M$ be a large real number. Explain why there must be exactly one root $w$ of the equation $ Mx=e^x$ with $w>1$. Why is log $M$ a reasonable approximation to $w$? Write $w = \log M +y$. Can ...
4
votes
4answers
105 views

$(x+y+z)^3-(y+z-x)^3-(z+x-y)^3-(x+y-z)^3=24xyz$?

The question given is Show that $(x+y+z)^3-(y+z-x)^3-(z+x-y)^3-(x+y-z)^3=24xyz$. What I tried is suppose $a=(y+z-x),\ b=(z+x-y)$ and $c=(x+y-z)$ and then noted that $a+b+c=x+y+z$. So the ...
4
votes
3answers
932 views

Students in a class, girls sitting with boys and boys sitting with girls

This is a very interesting word problem that I came across in an old textbook of mine. So I mused over this problem for a while and tried to look at the different ways to approach it but unfortunately ...
1
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3answers
66 views

What is the number of mappings?

It is given that there are two sets of real numbers $A = \{a_1, a_2, ..., a_{100}\}$ and $B= \{b_1, b_2, ..., b_{50}\}.$ If there is a mapping $f$ from $A$ to $B$ such that every element in $B$ has an ...
1
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5answers
151 views

If $a+b+c+d=1$ then why is the maximum value of $(a+1)(b+1)(c+1)(d+1)$ is ${\left(\frac{5}{4}\right)}^4$?

What I know is that for equations of type $x+y=8$, $xy$ attains its maximum value when $x=y$ and this can be proved by either solving the quadratic equation with completing the squares or finding the ...
0
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0answers
35 views

clarity in the solution of the following problem

$$(D^2+D)y=x^2+2x+4$$ I found the solution as $$CF=C_{1}+e^{-x}C_{2}$$ and PI=$$\left(\frac{x^3}{3}\right)+4x$$ but the solution from my teacher is PI = $$\left(\frac{x^3}{3}\right)+4x+C3$$ Where ...
2
votes
0answers
31 views

Eigen function of one Stochastic Process from the eigen function of another Stochastic Process

Let us consider a centred square integrable stochastic process $\{X_t:t\in [0,2]\}$. Also let the eigen values and the eigen function of the kernel of the covariance operator of $X_t$ are $\lambda_1\...