262 votes

A goat tied to a corner of a rectangle

Parts of three different circles.
Seyed's user avatar
  • 8,878
140 votes

Is being good at mathematic contests necessary to pursue a career in mathematics or physics?

Mathematics takes place at different time-scales. If you can solve a problem in $5$ minutes that others need an hour to solve, you can probably get a good job. If you can solve a problem in a month ...
Robert Israel's user avatar
113 votes

7 fishermen caught exactly 100 fish and no two had caught the same number of fish. Then there are three who have together captured at least 50 fish.

Let's work with the lowest four numbers instead of the other suggestions. Supposing there is a counterexample, then the lowest four must add to at least $51$ (else the highest three add to $50$ or ...
Mark Bennet's user avatar
  • 99.4k
109 votes

Prove that at a party of $25$ people there is one person knows at least twelve people.

If everyone knows everyone, then you are done. Otherwise choose two people, A and B say, who don't know each other. These two people are part of $23$ triples. In each of these triples, either A ...
paw88789's user avatar
  • 39.9k
81 votes

Witty functional equation

We have $\displaystyle f(1-x)=\frac{2}{4^{1-x}+2}=\frac{2\cdot 4^x}{4+2\cdot 4^x}=\frac{4^x}{2+ 4^x}$ $\displaystyle f(x)+f(1-x)=\frac{2}{4^x+2}+\frac{4^x}{2+ 4^x}=1$ The required sum is $$\frac{...
CY Aries's user avatar
  • 23.3k
74 votes

Is being good at mathematic contests necessary to pursue a career in mathematics or physics?

It's nice to have awards from contests on your grad school, etc. applications (and it's something to be legitimately proud of), but they have little resemblance to what mathematicians or physicists do ...
anomaly's user avatar
  • 24.9k
74 votes

Find $xy+yz+zx$ given systems of three homogenous quadratic equations for $x, y, z$

We can obtain $yz+zx+xy=2$ simply by finding the values of $x$, $y$, and $z$. We are given $$y^2+yz+z^2=1,\qquad(1)$$$$z^2+zx+x^2=4,\qquad(2)$$$$x^2+xy+y^2=3,\,\qquad(3)$$with $x,y,z>0$. ...
John Bentin's user avatar
  • 17.9k
72 votes

How can I answer this Putnam question more rigorously?

Why not write it the other way round? The polynomial function $$F(x)=\sum_{k=0}^n\frac{a_k}{k+1}x^{k+1} $$ is a differentiable function $\Bbb R\to\Bbb R$ with derivative $$F'(x)=\sum_{k=0}^na_kx^k.$$...
Hagen von Eitzen's user avatar
67 votes

Show that there are infinitely many powers of two starting with the digit 7

I was intrigued. Not being smart enough to get a theoretic proof I used brute force. Definitely ugly. Only redeeming aspect is it is constructive. Here is how I proceeded. Remark: If two numbers are ...
P Vanchinathan's user avatar
65 votes

Drunk man with a set of keys.

The key thing here is this: let $T$ be the number of tries it takes him to open the door. Let $D$ be the event that the man is drunk. Then $$ P(D\mid T=3)=\frac{P(T=3, D)}{P(T=3)}. $$ Now, the event ...
Nick Peterson's user avatar
57 votes

How to prove that a very large number is not prime

We have the number $10^{20}+1$. Whenever we have something in this kind of form, we need to find an odd factor of the exponent. In this case $5 \mid 20$, so we can use $5$ as the factor. Now, we can ...
Noble Mushtak's user avatar
57 votes

What is the algebraic intuition behind Vieta jumping in IMO1988 Problem 6?

At the heart of these so-called "Vieta-jumping" techniques are certain symmetries (reflections) on conics. These symmetries govern descent in the group of integer points of the conic. If you wish to ...
Bill Dubuque's user avatar
56 votes

How many ways can I go from 1 to 10 in the following diagram?

You can think the problem as start with the $10$ and follow the numbers in order until you reach $1$. Then there is two ways for each step: up or left. Then there are $2^{10-1}=2^9=512$ ways. Done!
Culver Kwan's user avatar
  • 2,775
55 votes

$\int_{0}^{\frac{\pi}{4}}\frac{\tan^2 x}{1+x^2}\text{d}x$ on 2015 MIT Integration Bee

There must be some problem here. Note that on $[0, \pi/4]$, $$\frac{\tan^2 x}{1+x^2} \le \frac{\tan^2 x}{1 + 0^2} = \tan^2 x,$$ therefore the definite integral is bounded above as follows: $$0 \le \...
heropup's user avatar
  • 129k
55 votes

Fewest steps to reach $200$ from $1$ using only $+1$ and $×2$

Look at what the operations $[+1]$ and $[\times 2]$ do to the binary expansion of a number: $[\times 2]$ appends a $0$, and increases the length by one, leaving the total number of $1$'s unchanged; ...
TonyK's user avatar
  • 63.3k
53 votes

Which is larger, $\sqrt[2015]{2015!}$ or $\sqrt[2016]{2016!}$?

Starting with: $$\sqrt[2015]{2015!}\mid\sqrt[2016]{2016!}$$ Raise each side to the power of $2015\cdot2016$: $$2015!^{2016}\mid2016!^{2015}$$ Divide each side by $2015!^{2015}$: $$2015!^{1}\...
barak manos's user avatar
53 votes

Find $xy+yz+zx$ given systems of three homogenous quadratic equations for $x, y, z$

To my surprise, this problem can be solved using geometry. Identify the Euclidean plane with complex plane $\mathbb{C}$. and let $\omega = e^{\frac{2\pi}{3}i}$ be the cubic root of unity. Consider ...
achille hui's user avatar
51 votes

Find $S = \frac{a}{b+c}+\frac{b}{c+a} + \frac{c}{a+b}$ if values of $a+b+c$ and $\frac1{a+b}+\frac1{b+c}+\frac1{a+c}$ are given

Multiplying the given expressions together: \begin{align} \frac{47}{10} &= (a+b+c)\bigg(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\bigg) \\ \\ &= \frac{a+b+c}{a+b}+\frac{a+b+c}{b+c}+\frac{a+...
Joshua Farrell's user avatar
48 votes

Prove that there exists a triangle which can be cut into 2005 congruent triangles.

The decomposition is possible because $2005 = 5\cdot 401$ and both $5$ and $401$ are primes of the form $4k+1$. This allow $2005$ to be written as a sum of squares. $$2005 = 22^2 + 39^2 = 18^2+41^2$$ ...
achille hui's user avatar
48 votes

An interesting trigonometric integral

\begin{align*} I&= \int_0^\pi \cos^2\cos x+\sin^2\sin x \, dx\\ &= 2\int_0^{\pi/2} \cos^2\cos x+\sin^2\sin x \, dx\\ &= 2\int_0^{\pi/2} \cos^2\sin x+\sin^2\cos x \, dx \quad(x\mapsto\pi/2-...
goonfiend's user avatar
  • 2,719
46 votes

Drunk man with a set of keys.

Let's first compute the probability that he wins on the third try in each of the two cases: Sober: The key has to be one of the (ordered) five, with equal probability for each, so $p_{sober}=p_s=\...
lulu's user avatar
  • 68.3k
42 votes

Problem from the 2020 Latvian "Sophomore's Dream" competition

Let $I$ be our integral. Substitute $t=-x \Rightarrow dt = -dx$. Then: $$I=\int_{-a\pi}^{a\pi} \frac{\cos^5 t+1}{e^{-t}+1}\,dt=\int_{-a\pi}^{a\pi} \frac{e^t(\cos^5 t+1)}{e^{t}+1}\,dt=\int_{-a\pi}^{a\...
LHF's user avatar
  • 8,481
41 votes

Can a pre-calculus student prove this?

As randomgirl and Michael point out, there are counterexamples, such as $a=b=0$. However, it is true when $a \ge 1$. In principle it can be proved with just pre-calculus mathematics, but only an ...
Henry Cohn's user avatar
41 votes

What is the remainder when $6^{273} + 8^{273}$ is divided by $49$?

The binomial formula gives $$(7\pm1)^{273}={273\choose1}7^1(\pm1)^{272}+{273\choose0}7^0(\pm1)^{273}=273\cdot 7\pm1=\pm1\qquad({\rm mod}\ 49)\ ,$$ since all other terms are divisible by $7^2$. It ...
Christian Blatter's user avatar
39 votes

Knowing that for any set of real numbers $x,y,z$, such that $x+y+z = 1$ the inequality $x^2+y^2+z^2 \ge \frac{1}{3}$ holds.

Cauchy- Schwarz works: $$x^2+y^2+z^2=\frac{1}{3}(1^2+1^2+1^2)(x^2+y^2+z^2)\geq\frac{1}{3}(x+y+z)^2=\frac{1}{3}$$
Michael Rozenberg's user avatar
38 votes

Painting the plane, and finding points one unit apart

This is the subject well-known open problem called the Hadwiger-Nelson Problem. The problem asks for the exact minimum number of colors that we can color the plane with, so that no two points of ...
Caleb Stanford's user avatar
38 votes

Integer solutions to $x^3=y^3+2y+1$?

Hint: if $y>0$, then $y^3< y^3+2y+1< (y+1)^3$, so the RHS expression cannot be a perfect cube. A similar idea works if $y$ is a small enough negative number, but some negative numbers close ...
A. Pongrácz's user avatar
  • 7,368
38 votes

Putting socks and shoes on a spider

You can imagine doing this as writing a sequence, say $$3453228156467781$$ What does it mean? It means first put sock on leg $\color{red}{3}$ and on 4-th move put shoe on leg $\color{red}{3}$ ...
nonuser's user avatar
  • 89.5k
37 votes

Knowing that for any set of real numbers $x,y,z$, such that $x+y+z = 1$ the inequality $x^2+y^2+z^2 \ge \frac{1}{3}$ holds.

$x^2+y^2+z^2$ only depends on the squared distance of $(x,y,z)$ from the origin and the constraint $x+y+z=1$ tells us that $(x,y,z)$ lies in a affine plane. The problem is solved by finding the ...
Jack D'Aurizio's user avatar

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