Questions tagged [problem-solving]

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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-3
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0answers
18 views

Determine the nature of the given sequence. [closed]

Which of the above options is correct?I have tried to solve it but cannot think of a proper way.This is a problem from IIT-JAM.
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1answer
100 views

Is it true that $\gcd(p+1,pk)=1$. [closed]

Let $p$ be a prime. Consider the statement $\gcd(p+1,pk)=1$. The statement does not seem to be true for all $k$. In particular, take $k=3,p=2,\gcd(2+1,2(3))=3>1$. So is this statement true if $\gcd\...
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44 views

prove that $\underbrace{\sqrt{2+\sqrt{2+\sqrt{2+\dots+\sqrt{2}}}}}_{\text{n radicals}}=2\cos{\frac{\pi}{2^{n+1}}}$ [duplicate]

Provide a complete proof for the statement: prove that $\underbrace{\sqrt{2+\sqrt{2+\sqrt{2+\dots+\sqrt{2}}}}}_{\text{n radicals}}=2\cos{\frac{\pi}{2^{n+1}}}$
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10 views

Analysis of a particular instance of a non-linearsystem of equations

Consider a vector space $V \in \mathbb{R}^n$. Given a set of $\frac{1}{2}n(n-1)$ square matrices $\{\boldsymbol{A}^{(ij)}\}_{ij}$ (where $(\boldsymbol{A}^{(ij)})^T = \boldsymbol{A}^{(ji)}$), how do we ...
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1answer
66 views

Pr. 5 of chapter 1 of Walter Rudin.

Pr 5. Let $A$ be a nonempty set of real no. Which is bdd below. Let $-A$ be set of all no. $-x; x \in A$. Prove $\inf(A)=-\sup(-A)$. My attempt: If $x>0, \inf(A) \ge 0$ & by def $\inf(A) \le x$...
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2answers
133 views

Probability that the sum of two integers in $\{1,\dots,n\}$ equals a perfect square

I found the following question in MIT OCW's Mathematical Problem Solving and I'd like to know if my solution is ok: "Let $p_n$ be the probability that $c+d$ is a perfect square when the integers $...
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2answers
89 views

What does “parity pattern” mean?

Five lattice points are chosen in the plane lattice. Prove that you can always choose two of these points such that the segment joining these points pass through another lattice point. (The lattice ...
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0answers
117 views

Proof of $ASA$ criterium

I am studying affine geometry and I found this problem in which I am having some trouble solving it. Let $\mathcal{A}$ be an affine space over an Euclidian space $E$ of dimension $2$ and let $A,A’,B,B’...
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1answer
37 views

Wave equation, finding a constant that solves the equation?

Reading an elementary book, I came across this problem: Let $$f_{xx}+f_{yy}=f_{tt}$$ We have the solution: $$f(x,y,t)=\sin(nx)\cos(nt)+\sin(my)\cos(mt)+\sin(nx+my)\cos(kt)$$ Where $m$ and $n$ are ...
3
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1answer
53 views

Let $R$ be the set $\mathbb{R}$ of all real numbers with the co-finite topology. How can one show that $R$ is contractible?

I have recently started learning Algebraic Topology, and more specifically, homotopy theory. That is where I encountered this question: Let $R$ be the set $\mathbb{R}$ of real numbers, with the co-...
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2answers
36 views

Car A is 50km east of car B and begins moving west at 50km/h. Simultaneously, car B begins moving north at 90km/h. Find:

a) The closest distance the cars will be from each other in kilometers b) What time "t" in hours will this distance occur So this is a practice question from my grade 12 calculus textbook. ...
3
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1answer
67 views

Let $a = 2019^{1009} , b = 2019!$ and $c = 1010^{2019}$

Let $a = 2019^{1009} , b = 2019!$ and $c = 1010^{2019}$. Arrange $a,b,c$ in ascending order. What I Tried:- I have solved one of the parts. We have $2019 < 1010^2$. $\implies 2019^{1009} < ({...
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2answers
52 views

Two Squares $S_1$ and $S_2$, are inscribed in the triangle $ABC$ as $S_1$ and $ABC$ share a common vertex $C$ and $S_2$ has one of its sides on $AB$.

Let $ABC$ be a right triangle with $\angle C = 90^\circ$. Two Squares $S_1$ and $S_2$, are inscribed in the triangle $ABC$ such that $S_1$ and $ABC$ share a common vertex $C$ and $S_2$ has one of its ...
2
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3answers
106 views

Let $(m,n)$ be the pair of integers satisfying $2(8n^3 + m^3) + 6(m^2 - 6n^2) + 3(2m + 9n) = 437.$ Find the sum of all possible values of $mn$.

Let $(m,n)$ be the pair of integers satisfying $2(8n^3 + m^3) + 6(m^2 - 6n^2) + 3(2m + 9n) = 437.$ Find the sum of all possible values of $mn$. What I Tried:- Given this expression I had no idea to ...
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2answers
98 views

In a $\triangle ABC$, $\angle A = 30^\circ, BC = 13.$

In a $\triangle ABC$, $\angle A = 30^\circ, BC = 13.$ Given two circles $\gamma_1,\gamma_2$ with radius $r_1,r_2$ respectively, contain $A$ and touch the side $BC$ at $B$ and $C$ respectively. Find $...
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1answer
54 views

Number of ways to arrange people in a row with two restrictions (permutations).

The question is as follows: In a group of 9 people, there are 4 Gujaratis and 5 Marwadis. Of the Gujaratis, 2 are vegetarian and the rest are non-vegetarians. Of the Marwadis, 2 are vegetarian and ...
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0answers
38 views

How Does One Come Up With This Proof of The Triangle Inequality of Complex Integrals?

I saw this proof in the textbook and wonder what might have motivated it? Set $\theta = \arg \int^b_a f(t) \,dt$ then $$\left|\int^b_a f(t) \,dt\right|=e^{-i\theta}\int^b_a f(t) \,dt = \Re e^{-i\theta}...
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2answers
50 views

If $z$ is a complex number and $|z| = 1$ and $z^2 \neq 1$. Then $\frac{z}{1-z^2}$ lies on :- [closed]

If $z$ is a complex number and $|z| = 1$ and $z^2 \neq 1$. Then $\frac{z}{1-z^2}$ lies on :- $(a)$ A line not through origin. $(b)$ $|z| = 2$. $(c)$ $x$-axis. $(d)$ $y$-axis. What I Tried:- From ...
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0answers
22 views

Working out probabilities of a singular instance and extrapolating to larger numbers

I'm trying to work out out that if, for example, you were doing a test, and the chances of the test returning a false positive or false negative were worked out to be a specific percentage, say 20%, ...
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1answer
86 views

Determine all solutions of the equation in $\mathbb{R}, (x^2 + 3x – 4)^3 + (2x^2– 5x + 3)^3 = (3x^2 – 2x – 1)^3$

What are the steps you follow while solving for variables in an equation? From easy to difficult or tricky or exhausting, how do you solve for variables in an equation. Is there any algorithm or ...
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0answers
58 views

Midpoints of Three Straight Lines are Collinear. [duplicate]

In quadrilateral $ABCD$, let $AB$ and $CD$ meet at $E$ and $AD$ and $BC$ meet at $F$. Then prove that the midpoints of $AC,BD$ and $EF$ are collinear. What I Tried:- Here is a Picture :- First ...
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2answers
56 views

Solve for real numbers $x$ and $y$, simultaneously the equations given by $xy^2 = 15x^2 + 17xy + 15y^2$ and $x^2y = 20x^2 + 3y^2$.

Solve for real numbers $x$ and $y$, simultaneously the equations given by $$ \left\{ \begin{array}{l} xy^2 &= 15x^2 + 17xy + 15y^2 \\ x^2y &= 20x^2 + 3y^2 \end{array} \right. $$ by taking $y =...
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1answer
53 views

Prove that $X,Y,Z$ are collinear.

$AD,BE$ and $CF$ are three concurrent lines meeting the sides $BC,CA,AB$ in $D,E,F$. Suppose $EF,FD$ and $DE$ meet $BC,CA,AB$ at $X,Y,Z$. Prove that $B,C$ divide $DX$ harmonically and $X,Y,Z$ are ...
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3answers
63 views

Count multiples of 7 between range using positioning

I am interested in finding the number of multiples of $7$ between $[10, 500]$ In other cases I have seen that a neat trick is to check if there is a pattern between the sequence of numbers and their ...
0
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1answer
63 views

Probability of Random Numbers in a Table Summing to $10$

Assume a table with dimensions $n$x$n$. In each of the $n^2$ spaces, a random number ($m$) such that $m\in\mathbb{N}$ and $1\leq m\le9$ will be placed. My question is two-fold: What are the odds that ...
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4answers
34 views

How was this quadratic formed?

For $[x − (a + b + c)][\frac{1}{x − a}+\frac{1}{x − b}+\frac{1}{x − c}]= 0.$ One solution is $x = a + b + c$. The notes say that the other two solutions are the roots of the quadratic equation $3x^2 − ...
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0answers
52 views

How can Abraham Wald's approach lead you to ignore crucial features of a problem?

Kindly see the red sentence below. What exactly does "that approach" mean? I don't know the term for "he peered right through to the mathematical struts and nails holding the story ...
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1answer
37 views

Suppose $M\models D(N)$. Show that $M$ has a substructure which is isomorphic to $N$

Let $N$ be an $L$-structure. The diagram of $N$, denoted $D(N)$, is the set of all quantifier-free $L_N$-sentences which are true in $N$. Suppose $M$ is a model of $D(N)$. Show that $M$ has a ...
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3answers
83 views

Randomly choosing a number and then the dealer picks $20$ from the same set

I am reading the following problem: In the game of one-spot keno, a card is purchased for $\$1$. It allows a player to choose a number from $1$ to $80$. A dealer then chooses $20$ numbers at random. ...
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4answers
105 views

Can you help me solve $2\sqrt[3]{2x+1}=x^3-1$? [closed]

I want help with this: $$2\sqrt[3]{2x+1}=x^3-1$$ I found $x = -1$ as a solution, but I don't know if it is true or not.
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1answer
23 views

FOL sentence which is true in a graph G exactly when G has no cycles of length $n $

Consider only directed graphs. Show that for each $ n \ge 1$ there is an $L_{\text{graph} } $-sentence $ \phi_n$ which is true in a graph $G $ exactly when $G $ has no cycles of length $n $. I'm not ...
2
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1answer
26 views

Let $n$ lines be drawn in the plane so that each of them intersects the others but any three lines do not coincide at any point.

a) Let $n$ lines be drawn in the plane so that each of them intersects the others but any three lines do not coincide at any point. For $n\geq 0$, let $a_n$ be the number of regions into which the ...
2
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1answer
57 views

External Angle Bisectors meeting in collinear points.

If $H$ is any point within $\Delta ABC$, prove that the external bisectors of angles AHB , BHC , CHA meet $AB,BC,CA$ respectively at three collinear points. What I Tried:- Here is a picture :- I ...
4
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5answers
179 views

Minimizing $|x-1|+|x-2|+|x-3|+…+ |x-2019|+ |x-2020|$.

The given expression $$ |x-1|+|x-2|+|x-3|+...+ |x-2019|+ |x-2020| $$ determine the greatest interval of real numbers $[a,b]$ on which the given expression has a constant value $k$. What is the value ...
3
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3answers
34 views

Determine real numbers a,b and c such that they verify a certain equation

up to this point I've determined c this way: The issue here is that I cannot figure out how to proceed the same way with the other two variables a and b. Is there something I'm missing from the start?...
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0answers
32 views

Imagine a single round round-robin chess tournament of 16 players. What is the probability that any two players meet each other within first 3 rounds?

I am thinking of doing something like this: P(Player A and Player B meet in the first round) + P(Player A and Player B don't meet in the first round)*P(Player A and Player B meet in the second round) +...
2
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2answers
86 views

Waiting times at a bank - probability

Suppose there are 3 bank tellers, all currently occupied with a customer. You are waiting in line (and there are no other customers that will come in after you). All service times are iid exponential ...
1
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1answer
60 views

Finding the Remainder of $f(x)=(x-1)^2(x+2)Q(x)+R(x)$

So I'm a high school student and I'm stuck on a question. Please help. $f(x)=(x-1)^2Q(x)+3x+1$ $f(x)=(x+2)Q(x)+4$ $f(x)=(x-1)^2(x+2)Q(x)+R(x)$ My first approach was Making $R(x)=ax^2+bx+c$, I soon ...
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1answer
40 views

Formula for the number of binary combinations

What is the formula for knowing the number of binary combinations, where n is the total number of digits and k is the maximum number of digits 1. Only those ending in 1 will be counted. Example: n = 2,...
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1answer
72 views

Square and Quarter Circle [closed]

$ABCD$ is a square of side $18$ cm. $F$ is a point inside the square, such that $BCF$ forms an equilateral triangle. $CFA$ is a quarter circle with centre $B$. $E$ is the point on $AB$ such that the ...
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0answers
23 views

Minimum value achieved by a specific integral [duplicate]

Suppose $f:[0,1]\to \mathbb R$ be integrable such that $\displaystyle\int_{0}^{1} f(x) dx = \int_0^1 xf(x) dx= 1$. Define $\displaystyle I= \int_0^1 f(x)^2 dx$ Then find the minimum achieved by $I$. ...
3
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1answer
99 views

Why is CDF the only way to randomly select from samples?

I am confused with the concept of cumulative distribution function (CDF). I see it helps in algorithms related to sampling of data. So for instance if we have a list of values, and we randomly want to ...
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0answers
18 views

Probability and stochastic

A small store installed a diesel-powered gener- ator for emergency power outage because they function independently of electric power. The past history of the town indicates that on about 5% of days ...
0
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1answer
49 views

Understanding left and right adjoint

I'm doing some self-study and found this question. Let N be the set of natural numbers with its usual notion of ≤. There's a function f:N→N with f(x)=2x. This function doesn't have an inverse. But: ...
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0answers
36 views

Number of connected squares in a grid

Recently, the following question came to my mind: Let's say we have a $n\times m$ rectangle consisting of unit squares. We call two squares connected, if they share an edge and we want to find the ...
4
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2answers
75 views

What is it to solve an equation forward?

I'm reading a book in Monetary Economics and I don't understand a step. I have this expression: $$ \dfrac{\lambda_{t}}{P_{t}} = \beta \left( \dfrac{\lambda_{t+1} + \mu_{t+1}}{P_{t+1}} \right) $$ And ...
3
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2answers
84 views

distance of a line connecting $\ln(x)$ and $e^x$

I created this weird problem in my head which might be trivial or rather complicated but I somehow can't figure out how to solve it: Imagine you have a bar/line connecting the graphs of $e^x$ and $ln(...
0
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0answers
18 views

Times that respect clock-hand symmetry [duplicate]

A friend asked me for the time. When I looked at my analog clock (which only has a minute hand and an hour hand, no second hand), I thought of telling her the time with the hour and minute hands ...
3
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2answers
39 views

Sum of Multiple short arithmetic series

I am struggling to apply the arithmetic series formula in solving this word problem below, and any help would be appreciated: The Problem: Hercy wants to save money for his first car. He puts money in ...
1
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2answers
53 views

How to solve the system of equations like this one? [closed]

I have a hidden linear function: $f(x)=a*x+b$. For example $f(x)=2*x+3$ A hidden function $f$ was executed on some hidden input $x$=(0,1,3,4) and we have access only to output $y=(f(0)=3, f(1)=5, ...)=...

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