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.

learn more… | top users | synonyms

16
votes
3answers
389 views

A sum of fractional parts.

I am looking to evaluate the sum $$\sum_{1\leq k\leq mn}\left\{ \frac{k}{m}\right\} \left\{ \frac{k}{n}\right\} .$$ Using matlab, and experimenting around, it seems to be $\frac{(m-1)(n-1)}{4}$ when ...
9
votes
1answer
116 views

How could I improve this approximation?

In a computer application, I need to solve trillions of times an equation which can be reduced to $$f(x)=\sin(x)-a x=0$$ Newton methods (quadratic and higher orders) are used for the solution. ...
5
votes
2answers
161 views

Math Olympiads: GCD of terms in a sequence equals GCD of their indices.

The sequence $a_1 ,a_2 ,a_3 ,...$ of positive integers satisfies $\text{gcd}(a_i ,a_j ) = \text{gcd} (i, j)$ for $i \neq j$. Prove that $a_i = i$ for all $i$. Source: Russian Mathematical Olympiad, ...
4
votes
2answers
357 views

How to solve an overdetermined system of point mappings via rotation and translation

I have a set of points in one coordinate system $P_1, \ldots, P_n$ and their corresponding points in another coordinate system $Q_1, \ldots , Q_n$. All points are in $\mathbb{R}^3$. I'm looking for a ...
3
votes
2answers
217 views

Find the number of digits of $2013^{2013}$?

Is is possible to find the number of digits of $2013^{2013}$ without a calculator?
3
votes
3answers
404 views

Fractions in Ancient Egypt

In ancient Egypt, fractions were written as sums of fractions with numerator 1. For instance,$ \frac{3}{5}=\frac{1}{2}+\frac{1}{10}$. Consider the following algorithm for writing a fraction ...
3
votes
1answer
84 views

Imperfect digit-to-digit invariants in Base $10$

$3435 = 3^3 + 4^4 + 3^3 + 5^5$ is an example of a perfect digit-to-digit invariant. Fact: The number of PDDIs is finite for any given base; in particular, for base $10$. Question: Working over base ...
3
votes
1answer
115 views

$0 = \left(\sqrt{p^2+m^2}-\sqrt{k^2+p^2+2\cdot k\cdot p\cos(\theta)}\right)^2 -k^2-m^2$ solving for $k$

This question is related to $\delta(f(k))$ concerning the Dirac-delta. OK I know this might seem trivial but the result is very very important to me so I want to check with you if my logic seems ...
3
votes
1answer
1k views

Putnam 2012 B3 - Tournament combinatorics

A round-robin tournament among $2n$ teams lasted for $2n-1$ days, as follows. On each day, every team played one game against another team, with one team winning and one team losing in each of the $n$ ...
3
votes
2answers
371 views

$W$ white balls, $B$ black balls, adding $K$ of the resultant color each iteration

The problem is stated as follows. We have a box with $W$ white balls and $B$ black ones. Repeat $N$ times: each iteration a ball is taken out (uniformly), and put back along with $K$ (constant) more ...
2
votes
2answers
707 views

calculate the limit of this sequence $\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1..}}}}$ [duplicate]

Possible Duplicate: $\sqrt{c+\sqrt{c+\sqrt{c+\cdots}}}$, or the limit of the sequence $x_{n+1} = \sqrt{c+x_n}$ i am trying to calculate the limit of ...
2
votes
1answer
261 views

Expected value of randomly distributed projection

Problem: Find the mathematical expectation of the area of the projection of a cube with edge of length 1 onto a plane with an isotropically distributed random direction of projection. Source: ...
1
vote
2answers
94 views

Formula for solving for Cx and Cy…

I'm trying to create a formula to find the third point in a triangle based on two known points and three known sides. Known Sides: $AB, BC, AC$ Known Points: $A(x, y), B(x, y)$ Unknown Points: ...
1
vote
1answer
371 views

Multiplication Table with a frame and picture of equal sum

Is there an $n \times n$ multiplication table such that if you form a border of width $k$ ("the frame") and sum its elements, the total will equal the sum of the remaining elements ("the picture")? ...
1
vote
4answers
177 views

Looking for a simple problem for math demonstration

I'm holding a 3-5 minute speech next week on mathematical problem solving, and how it makes me happy, to 15-20 non-mathematicians. As a part of it, I had thought about demonstrating two problems, but ...
0
votes
1answer
57 views

Greatest common divisor of $3$ numbers

Let $a,b, c$ belong to $\mathbb Z$ such that $(a,b,c) \neq (0,0,0)$. Define the [highest common factor] greatest common divisor ${\rm gcd}(a, b, c)$ to be the largest positive integer that divides $a, ...
0
votes
4answers
202 views

How many positive and even factors does $2013!$ have?

How many positive and even factors does $2013!$ have? So I know that $2013 = 2\times1006 + 1$ So, does that mean $2013!$ has $1006$ even factors?
0
votes
1answer
200 views

Is it possible to derive the CDF of $Z$?

Assume that $X_i$, $Y_k$, $i=0,\ldots,N$, $k=1,\ldots,K$ are non-negative independent non-identically distributed random variables. Let us define the random variable $Z$ as \begin{align} ...
4
votes
2answers
121 views

Egg drop problem

Suppose that you have an $N$-story building and plenty of eggs. An egg breaks if it is dropped from floor $T$ or higher and does not break otherwise. Your goal is to devise a strategy to determine ...
4
votes
4answers
2k views

Using + - * / operators and 4 4 4 4 digits find all formulas that would resolve to 1 2 3 4 5 6 7 8 9 10

I had a conversation with a colleague of mine and he brought up an interesting problem. Using the + - * / operators and four 4 4 4 4 digits, create an algorithm that will output all the formulas that ...
4
votes
2answers
334 views

prove the divergence of cauchy product of convergent series $a_{n}:=b_{n}:=\dfrac{(-1)^n}{\sqrt{n+1}}$

i am given these series which converge. $a_{n}:=b_{n}:=\dfrac{(-1)^n}{\sqrt{n+1}}$ i solved this with quotient test and came to $-1$, which is obviously wrong. because it must be $0<\theta<1$ so ...
4
votes
4answers
272 views

Proof of $n^2 \leq 2^n$.

I am trying to prove that $n^2 \leq 2^n$ for all natural $n$ with $n \ne 3$. My steps are: induction base case: $n=0:$ $0² \leq 2⁰$ which is okay. inductive step: $n \rightarrow n+1:$ ...
3
votes
3answers
124 views

Is this question of sequence a Mathematical one, i.e. does it have objectively only one answer for each subpart.

This question is taken from 11th class Math book. Look at this question: At the very first glance one can tell that all the three sequences are G.P But! by using interpolation(as this answer ...
3
votes
2answers
157 views

Avoid more than one duplicate opponent

OK, I'm not sure if I can explain this: I have 12 players I want that each player play 3 times Each game is of 3 vs 3 players In each game each player plays with 2 different team members (no ...
3
votes
2answers
174 views

Proving there are no integer solutions for $3x^2=9+y^3$

Prove there are no $x,y\in\mathbb{Z}$ such that $3x^2=9+y^3$. Initial proof Let us assume there are $x,y\in\mathbb{Z}$ that satisfy the equation, which can be rewritten as $$3(x^2-3)=y^3.$$ So, ...
2
votes
1answer
215 views

equations solved with Newton's method by finding the zeros of functions?

I found this statement in one paper I read recently: This problem can be solved by finding the zero of functions: ...
2
votes
2answers
74 views

Accumulation points of $\{ \sqrt{n} - \sqrt{m}: m,n \in \mathbb{N} \}$

This is my first post on MSE, so, pardon me if I'm not used to the site's rule yet. I'm trying to prepare myself for competitions in the future and I'm trying to improve my problem solving skills. ...
2
votes
3answers
170 views

Where is the lost dollar?

Somebody explained me this problem, but I am not sure to understand what is wrong. ...
2
votes
2answers
88 views

solve for m by rewriting the equation (transposition)

In the following equation how would I rewrite the equation to solve for $m$? $$z=\frac{-4m-8+\sqrt{(4m+8)^2+4(4(mx+y-4m-4))}}{8}$$ when $x=66$ and $y=22$ and $z=10$
2
votes
1answer
194 views

Searching for the value of $p_5$

Reference post: click here Given, \begin{eqnarray} &&\Delta p_5-p_5+3S^2p_5 +\frac{SZ}{576\sqrt{\lambda}}(3Z-5S^3) \left(\frac{15g_5}{\lambda^2}+1\right)^2\nonumber\\ ...
2
votes
5answers
687 views

Different ordered triples $(a,b,c)$ of non-negative integers

How many different ordered triples $(a,b,c)$ of non-negative integers are there such that $a+b+c=50$? I tried to list the possibilities but the list is way too long, I know how to find the ordered ...
2
votes
3answers
159 views

Solve equations using the $\max$ function

How do you solve equations that involve the $\max$ function? For example: $$\max(8-x, 0) + \max(272-x, 0) + \max(-100-x, 0) = 180$$ In this case, I can work out in my head that $x = 92.$ But what is ...
1
vote
0answers
43 views

bounding the sum of squares of lengths of a quadrilateral inscribed in a unit square

Consider this nice little problem: if $ABCD$ is a quadrilateral inscribed in a unit square, then $$2\leq AB^2+BC^2+CD^2+DA^2\leq4$$ (Evidently this is problem 1 on paper 1 of the 1989 Irish ...
1
vote
0answers
33 views

Solving a system of equations with fractional parts and a system with round parts

I have the following two systems of equations: $a = x_{11} - \{x_{11} + \frac{4 - \sqrt{2}}{7}b + \frac{4 - \sqrt{2}}{7}c + \frac{2\sqrt{2} - 1}{7}d\}$ $b = x_{12} - \{x_{12} + \frac{4 - ...
1
vote
2answers
68 views

Comparison between Bessel's coefficients

The spatial solution is written as $$\Phi_k(r) = r^{1-\frac{d}{2}} \left(c_1 J_{1-\frac{d}{2}}(k r) + c_2 Y_{-1+\frac{d}{2}}(kr)\right).$$ In the case $d=3$, the solutions can be written as ...
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 $\frac1a+\frac1b+\frac1c=0$?
1
vote
2answers
53 views

Problem Solving using Algebra

If Peter is $7$ years older than Sharon and John is twice as old as Peter, work out how old Peter is if the average of their ages is $19$. Thanks! :)
1
vote
4answers
273 views

solve complex equation

$x^8 = \frac{1+i}{\sqrt{3} - i} = \frac{\sqrt[8]{\frac{2}{\sqrt{2}}}(\cos \frac{\pi}{4} + i \sin{\frac{\pi}{4}})}{2 \cos \frac{\pi}{6} + i \sin \frac{3\pi}{2}}$ What's the way to solve this kind of ...
1
vote
2answers
195 views

Pack box inside “smaller” box

Now, there is a puzzle that is quite well known as far as I know, concerning the packing of rectangular boxes in 3-dimensional space. You also have a measurement of a box as the sum of the hight, ...
0
votes
1answer
31 views

Finding distance using rates of change — best approach?

The question: A man drives from state $A$ to state $B$ going $60 \frac{miles}{hour}$. Then he returns from state $B$ to state $A$, driving $45 \frac{miles}{hour}$. His total driving time is $2.5 ...
0
votes
1answer
70 views

How can I solve this problem without doing it by hand? [duplicate]

I'm dealing with the following problem in computational programming. I'm trying to find a way to build an algorithm that can quickly resolve the following problem statement without forcing me to do it ...
0
votes
0answers
59 views

Vieta jumping with non-monic polynomials

I have recently discovered Vieta jumping as a problem-solving technique. In order to teach myself about it, I have located most (all of?) the standard references, both here on MSE and "out there" (via ...
0
votes
2answers
39 views

Mod Problem solving

I can't do this last question of my homework that's due in tomorrow. Can anyone hint me on what to do? Suppose $p$ is prime and $k$ is a positive integer Show that if $p$ is odd and $x$ is an ...
0
votes
1answer
207 views

If we draw infinitely many lines on a table, can we find a triangle somewhere? [closed]

If we draw infinitely many lines on a table, can we find a triangle somewhere? We prove that there is a subgraph $C_3$ in $C_n$, which will be called a triangle. Suppose we have an infinite ...
0
votes
2answers
92 views

How to show the existence of a number with certain divisibility conditions between two multiples?

How can we show that between two even natural numbers they're exists a natural number that isn't even? How can we show that they're exists a natural number that is odd and not divisible by 3, between ...
0
votes
1answer
119 views

How to solve the problem that determines the age of Diophantus?

How to solve the problem that determines the age of Diophantus?$$$$"God gave him to be a boy for the sixth part of his life, and adding one to it twelfth part covered her cheeks fluff, He gave him the ...
0
votes
2answers
119 views

On the surface of the moon

On the surface of the moon, acceleration due to gravity is approximately 5.3 feet per second squared. Suppose a baseball is thrown upward from a height of 6 feet with an initial velocity of 15 feet ...
0
votes
1answer
42 views

Frequency determination from Dimension analysis

the time averaged total energy, $\bar E$, has the following $\varepsilon$ expansion in $D$ dimension: \begin{equation} \bar{E}=\varepsilon^{2-D}\frac{E_0}{2\lambda}+ \varepsilon^{4-D}E_1 ...
0
votes
1answer
770 views

Can this crate have even numbers in all rows and columns?

A milk crate holds 24 bottles in four rows and six columns. Can you put 18 bottles of milk in the crate so that each row and each column of the crate have an even number of bottles in it?
0
votes
5answers
113 views

What is the algorithm for solving an equation like this one?

The solutions of the equation : $\sqrt{x+2\sqrt{x-1}} + \sqrt{x-2 \sqrt{x-1}} = 2$ are: A) $x=1$; B) $x=2$; C) $x\in [1,2]$; D) $x\in \begin{bmatrix} \frac{3}{2},2 \end{bmatrix}$; E) ...