Problems from or inspired by mathematics competitions. Questions regarding mathematics competitions.

learn more… | top users | synonyms (2)

10
votes
4answers
3k views

Undergraduate/High-School-Olympiad Level Introductory Number Theory Books For Self-Learning

I don't know whether the books metioned in Best ever book on Number Theory are beyond undergraduate/high-school-olympiad level. Please recommend your favourite.
15
votes
6answers
14k views

Expected Number of Coin Tosses to Get Five Consecutive Heads

A fair coin is tossed repeatedly until 5 consecutive heads occurs. What is the expected number of coin tosses?
12
votes
4answers
625 views

“If $1/a + 1/b = 1 /c$ where $a, b, c$ are positive integers with no common factor, $(a + b)$ is the square of an integer”

If $1/a + 1/b = 1 /c$ where $a, b, c$ are positive integers with no common factor, $(a + b)$ is the square of an integer. I found this question in RMO 1992 paper ! Can anyone help me to prove ...
8
votes
5answers
821 views

How many rationals of the form $\large \frac{2^n+1}{n^2}$ are integers?

This was Problem 3 (first day) of the 1990 IMO. A full solution can be found here. How many rationals of the form $\large \frac{2^n+1}{n^2},$ $(n \in \mathbb{N} )$ are integers? The possible ...
29
votes
4answers
2k views

Integral Contest

Before you answer this OP, please read all the terms and conditions below. Thank you... Today I hold an unofficial little contest on brilliant.org. Now, I will hold it here on Math S.E. It's just for ...
61
votes
6answers
8k views

Studying for the Putnam Exam

This is a question about studying for the Putnam examination (and, secondarily, other high-difficulty proof-based math competitions like the IMO). It is not about the history of the competition, the ...
17
votes
5answers
2k views

Probability of random integer's digits summing to 12

What is the probability that a random integer between 1 and 9999 will have digits that sum to 12? As a user suggested, I could make a spreadsheet and count them, but is there a quicker way to do ...
16
votes
4answers
2k views

The number $2^{29}$ has exactly $9$ distinct digits. Which digit is missing?

The number $2^{29}$ has exactly $9$ distinct digits. Which digit is missing? I came across this question in a math competition and I am looking for how to solve this question without working it ...
22
votes
2answers
2k views

How do people come up with difficult math Olympiad questions?

The problems that appear in difficult math competitions such as the IMO or the Putnam exam are usually very difficult and require some ingenuity to solve. They also usually don't look like they can be ...
4
votes
3answers
400 views

Prove $a^ab^bc^c\ge (abc)^{\frac{a+b+c}3}$ for positive numbers.

Prove that the following inequality holds $$a^a b^b c^c\ge (abc)^{\frac{a+b+c}{3}}$$ if $a,b,c$ are positive. I'm not sure how to handle these kinds of powers. Are there any "famous" but not ...
7
votes
2answers
220 views

Compositeness of $n^4+4^n$ [duplicate]

My coach said that for all positive integers $n$, $n^4+4^n$ is never a prime number. So we memorized this for future use in math competition. But I don't understand why is it?
2
votes
2answers
665 views

Euclidean Geometry Intersection of Circles

Two circles intersect in the Cartesian Coordinate system at points $A$ and $B$. Point $A$ lies on the line $y=3$. Point $B$ lies on the line $y=12$. These two circles are also tangent to the x-axis at ...
0
votes
3answers
498 views

Prove that $a^3+b^3+c^3 \geq a^2b+b^2c+c^2a$

Let $a,b,c$ be positive real numbers. Prove that $a^3+b^3+c^3\geq a^2b+b^2c+c^2a$. My (strange) proof: $$ \begin{align*} a^3+b^3+c^3 &\geq a^2b+b^2c+c^2a\\ \sum\limits_{a,b,c} a^3 &\geq ...
71
votes
6answers
2k views

Contest problem about convergent series

The following is probably a math contest problem. I have been unable to locate the original source. Suppose that $\{a_i\}$ is a set of positive real numbers and the series $$\sum_{n = 1}^\infty ...
24
votes
4answers
1k views

Prove that $2^{2^{\sqrt3}}>10$

With a computer or calculator, it is easy to show that $$ 2^{2^\sqrt{3}} = 10.000478 \ldots > 10. $$ How can we prove that $2^{2^{\sqrt3}}>10$ without a calculator?
15
votes
2answers
513 views

Integral $\int_0^\pi \theta^2 \ln^2\big(2\cos\frac{\theta}{2}\big)d \theta$.

I am trying to calculate $$ I=\frac{1}{\pi}\int_0^\pi \theta^2 \ln^2\big(2\cos\frac{\theta}{2}\big)d \theta=\frac{11\pi^4}{180}=\frac{11\zeta(4)}{2}. $$ Note, we can expand the log in the integral to ...
15
votes
2answers
691 views

Problem 6 - IMO 1985

For every real number $x_1$ construct the sequence $x_1,x_2,x_3,\ldots$ by setting $x_{n+1}=x_n(x_n+\frac{1}{n})$ for each $n \ge 1$. Prove that there exists exactly one value of $x_1$ for which $0 ...
10
votes
2answers
214 views

No cont function $f\colon\mathbb{R}\to\mathbb{R}$ with $f(x)$ rational $\iff f(x+1)$ irrational.

Prove that there are no continuous functions $f\colon \mathbb{R} \to \mathbb{R}$ with the property: For any $x \in \mathbb{R}$, $f(x)$ is a rational number if and only if $f(x+1)$ is an irrational ...
9
votes
4answers
414 views

Differentiation wrt parameter $\int_0^\infty \sin^2(x)\cdot(x^2(x^2+1))^{-1}dx$

Use differentiation with respect to parameter obtaining a differential equation to solve $$ \int_0^\infty \frac{\sin^2(x)}{x^2(x^2+1)}dx $$ No complex variables, only this approach. Interesting ...
4
votes
2answers
183 views

Integral $ \int_{-\pi/2}^{\pi/2} \frac{1}{2007^x+1}\cdot \frac{\sin^{2008}x}{\sin^{2008}x+\cos^{2008}x}dx $

I am trying to solve this integral $$ \int_{-\pi/2}^{\pi/2} \frac{1}{2007^x+1}\cdot \frac{\sin^{2008}x}{\sin^{2008}x+\cos^{2008}x}dx $$ A closed form does exist despite the looks of the integrand. ...
6
votes
1answer
332 views

Prove that $\sum_{k=1}^n \frac{2k+1}{a_1+a_2+…+a_k}<4\sum_{k=1}^n\frac1{a_k}.$

Prove that for $a_k>0,k=1,2,\dots,n$, $$\sum_{k=1}^n \frac{2k+1}{a_1+a_2+\ldots+a_k}<4\sum_{k=1}^n\frac1{a_k}\;.$$
6
votes
5answers
228 views

Show that $\displaystyle \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 ...
15
votes
2answers
809 views

Proving there are an infinite number of pairs of positive integers $(m,n)$ such that $\frac{m+1}{n}+\frac{n+1}{m}$ is a positive integer

The question is: Show that there are an infinite number of pairs $(m,n): m, n \in \mathbb{Z}^{+}$, such that: $$\frac{m+1}{n}+\frac{n+1}{m} \in \mathbb{Z}^{+}$$ I started off approaching this ...
7
votes
1answer
214 views

(USAJMO)Find the integer solutions:$ab^5+3=x^3,a^5b+3=y^3$

Find the integer solutions: $$a·b^5+3=x^3,a^5·b+3=y^3$$ This is the first problem of today's USAJMO (has finished),I only find a trival result that $x\equiv y \pmod6$ and $abxy≠0 \pmod 3$. Thanks in ...
7
votes
4answers
675 views

Seemingly invalid step in the proof of $\frac{a^2+b^2}{ab+1}$ is a perfect square?

Recall the famous IMO 1988 question 6: Suppose that $\displaystyle\frac{a^2+b^2}{ab+1}=k\in\mathbb{N}$ for some $a,b\in\mathbb{N}$. Show that $k$ is a perfect square. Solutions can be found: ...
4
votes
1answer
371 views

Interesting number theory questions

How many integers less than 1000 can be expressed in the form $$\frac{(x + y + z)^2}{xyz} $$ where $x, y, z$ are positive integers?
1
vote
0answers
311 views

Finding the number of arrangement of N people of different height such that K of them are visible from front

Moderator Note: This is a current contest question on codechef.com. [Initially, I had asked this question in stackoverflow, but someone suggested to post it here, and hence this question is ...
88
votes
4answers
3k views

A math contest problem $\int_0^1\ln\left(1+\frac{\ln^2x}{4\,\pi^2}\right)\frac{\ln(1-x)}x \ \mathrm dx$

A friend of mine sent me a math contest problem that I am not able to solve (he does not know a solution either). So, I thought I might ask you for help. Prove: ...
27
votes
7answers
4k views

Find $f(x)$ such that $f(f(x)) = x^2 - 2$

Find all $f(x)$ satisfying $f(f(x)) = x^2 - 2$. Presumably $f(x)$ is supposed to be a function from $\mathbb R$ to $\mathbb R$ with no further restrictions (we don't assume continuity, etc), but ...
23
votes
6answers
2k views

Find all polynomials $P$ such that $P(x^2+1)=P(x)^2+1$

Find all polynomials $P$ such that $P(x^2+1)=P(x)^2+1$
15
votes
3answers
2k views

The easy(?) part of IMO 2011 Problem 3

Let $f : \mathbb R \to \mathbb R$ be a real-valued function defined on the set of real numbers that satisfies $$f(x + y) \leq yf(x) + f(f(x))$$ for all real numbers $x$ and $y$. How can I prove that ...
11
votes
1answer
567 views

A Putnam Integral $\int_2^4 \frac{\sqrt{\ln(9-x)}\,dx}{\sqrt{\ln(9-x)} + \sqrt{\ln(x+3)}}.$

This is a Putnam Problem that I have been trying to solve (on and off) for two years, but I have failed. I am in Calculus BC. This problem comes from the book "Calculus Eighth Edition by Larson, ...
5
votes
4answers
371 views

Solutions to $p+1=2n^2$ and $p^2+1=2m^2$ in Natural numbers.

$$p+1=2n^2$$$$p^2+1=2m^2$$ Find positive integers $m,n$ and prime $p$ satisfying the above two equations. What would people commonly do? Subtracting both the equations. You get: ...
12
votes
1answer
591 views

Contest problems with connections to deeper mathematics

We all know that problems from for example the IMO and the Putnam competition can sometimes have lovely connections to "deeper parts of mathematics". I would want to see such problems here which you ...
11
votes
3answers
536 views

Reference for combinatorial game theory.

What is a good reference material for elementary combinatorial game theory? By combinatorial game theory I mean chiefly the study of zero-sum, deterministic two-player games (perhaps even more ...
8
votes
3answers
413 views

Proving $a^ab^b + a^bb^a \le 1$, given $a + b = 1$

Given $a + b = 1$, Prove that $a^ab^b + a^bb^a \le 1$; $a$ and $b$ are positive real numbers.
5
votes
1answer
246 views

Smallest possible value on Fibonacci Function

Suppose $f$ is a polynomial with integer coefficients, such that for all non-negative integers $n$ the $n$-th Fibonacci number $u_n$ divides $f(u_{n+1})$. Find the smallest possible positive value of ...
9
votes
4answers
566 views

Irrational painting device

Part a) of the following problem appeared in one of the Putnam Exams (sorry, don't know which year exactly). If you want to solve Part a) don't read Part b). You have a painting device, which given ...
5
votes
1answer
82 views

Find all such functions defined on the space

$f:\mathbb{R}^3\to \mathbb{R}^{\ast}$ is such that for any non-degenerate tetrahedron $ABCD$ with $O$ the center of the inscribed sphere, we have : $$f(O)=f(A)f(B)f(C)f(D) $$ Prove that $f(X)=1$ for ...
1
vote
2answers
143 views

$7$ points inside a circle at equal distances

BdMO 2014 There are $7$ points on a circle.Any 2 consecutive points are at equal distance from one another.How many acute angled triangles can you form taking any 3 of these points? I believe ...
6
votes
3answers
478 views

Expected value uniform decreasing function

We are given a function $f(n,k)$ as for(i=0;i < k;i++) n = rand(n); return n; rand is defined as a random number generator that uniformly generates values ...
5
votes
1answer
258 views

Mathematical problem with square numbers in the decimal system

Moderator Note: this is a question from the Federal Mathematics Competition 2013. Good morning, here's another (pretty difficult) mathematical problem... The task may sound a little strange (I'm ...
5
votes
1answer
297 views

Is it possible for the number created by ordering $1$ to $n$ where $n > 1$ be a palindrome?

Is it possible for the number created by the consecutive numbers $1$ to $n$ where $n > 1$ be a palindrome eg. $1234567\ldots n$? I believe this is a contest problem, but how would one solve ...
0
votes
1answer
293 views

Select elements from $N$ sets

$N$ sets are given which can have any number of elements from $1-100$ each. Now we need to count arrangements in which we select $1$ element from each set under the condition that we can not choose ...
0
votes
1answer
255 views

Applying a Function to Square Matrices

Moderator Note: This question is from a contest which ended 1 Dec 2012. Consider a polynomial $f$ with complex coefficients. Call such $f$ broken if we can find a square matrix $M$ such that $M ...
12
votes
3answers
304 views

Prove that $\exists a,g,h\in G\colon h=aga^{-1}, g\neq h ,gh=hg$ in a finite non-abelian group $G$.

Let $G$ be a finite and non-abelian group. How do I prove the following statement? $$\exists a,g,h\in G \colon\quad h=aga^{-1},\ g\neq h ,\ gh=hg.$$ Thanks in advance.
9
votes
2answers
352 views

Functional Equation: a little tricky

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $f[f(x)^2+f(y)]=xf(x)+y$ for all real numbers $x$ and $y$. Clearly $f(x)=x$ is a solution, check by substitution. I'm at a loss as ...
4
votes
2answers
140 views

I have used Cauchy and Jensen. It is not helping me very much. Advice on solving this problem.

Let $a$, $b$ and $c$ be positive real numbers with $abc=1$. Prove that $$ \frac{a^{n+2}}{a^n+(n-1)b^n}+\frac{b^{n+2}}{b^n+(n-1)c^n}+\frac{c^{n+2}}{c^n+(n-1)a^n} \geq \frac{3}{n} $$ for each ...
4
votes
1answer
153 views

Prove that $f$ is a linear combination of $f_1,f_2,\dots,f_n$.

Let $V$ be a vector space and let $f, f_1,f_2,\dots,f_n$ be linear maps from $V$ to $\mathbb{R}$. Suppose that $f(x)=0$ whenever $f_1(x)=f_2(x)=\cdots=f_n(x)=0$. Prove that $f$ is a linear combination ...
3
votes
1answer
207 views

Uniformly distributed probability problem

May you have an idea for the following exercise I found from some olympiad. Each day you have to bring home one full can of water. To do so you go to the next well and make the can full. On the way ...