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

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18
votes
6answers
18k 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?
10
votes
4answers
4k 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.
14
votes
4answers
727 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 ...
7
votes
2answers
269 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?
8
votes
5answers
834 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 ...
64
votes
6answers
10k 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 ...
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 ...
5
votes
3answers
433 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 ...
4
votes
1answer
419 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?
74
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 ...
18
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 ...
15
votes
2answers
741 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 ...
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 ...
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 ...
7
votes
2answers
221 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. ...
8
votes
1answer
381 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 $a_k>0,k=1,2,\dots,n$, $$\sum_{k=1}^n \frac{2k+1}{a_1+a_2+\ldots+a_k}<4\sum_{k=1}^n\frac1{a_k}\;.$$
2
votes
2answers
689 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
568 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 ...
0
votes
2answers
75 views

Putnam and Beyond AM-GM help

From Putnam and Beyond: The Solution is: The only part I do NOT understand is how: $a_k + b_k = 1$ for every $k$? The problem just specifies nonnegative numbers?
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
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
561 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 ...
9
votes
4answers
471 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 ...
10
votes
2answers
230 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 ...
5
votes
4answers
391 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: ...
16
votes
2answers
858 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 ...
6
votes
5answers
237 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 ...
7
votes
1answer
234 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
698 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: ...
6
votes
1answer
719 views

Product of three consecutive positive integers is never a perfect power

I am trying to prove that the product of three consecutive positive integers is never a perfect power. Can anyone point to gaps in my proof and/or post an alternate solution? Let the three ...
5
votes
3answers
565 views

prove that $\text{rank}(AB)\ge\text{rank}(A)+\text{rank}(B)-n.$

If $A$ is a $m \times n$ matrix and $B$ a $n \times k$ matrix, prove that $$\text{rank}(AB)\ge\text{rank}(A)+\text{rank}(B)-n.$$ Also show when equality occurs.
1
vote
2answers
84 views

Placing $9$ black and $9$ white rooks on $ 6\times6$ board without any attacks between different colours

In how many ways can $9$ black and $9$ white rooks be placed on a $6 × 6$ chess board, so that no white rook can capture a black one? A rook can capture another piece if it is in the same rank (row) ...
91
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: ...
49
votes
10answers
1k views

Arc length contest! Minimize the arc length of $f(x)$ when given three conditions.

Contest: Give an example of a continuous function $f$ that satisfies three conditions: $f(x) \geq 0$ on the interval $0\leq x\leq 1$; $f(0)=0$ and $f(1)=0$; the area bounded by the graph of $f$ and ...
25
votes
8answers
3k views

Reasoning that $ \sin2x=2 \sin x \cos x$

In mathcounts teacher told us to use the formula $ \sin2x=2 \sin x \cos x$. What's the math behind this formula that made it true? Can someone explain?
24
votes
6answers
2k views

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

Find all polynomials $P$ such that $P(x^2+1)=P(x)^2+1$
11
votes
2answers
2k views

Preparing for Mathematics Olympiad

Friends, Please don't take it as an off-topic question. I actually want to learn different concepts of maths and physics as well as of chemistry. I am preparing for Mathematics Olympiad , can any ...
16
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
718 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, ...
9
votes
3answers
722 views

How many 0's are in the end of this expansion?

How many $0's$ are in the end of: $$1^1 \cdot 2^2 \cdot 3^3 \cdot 4^4.... 99^{99}$$ The answer is supposed to be $1100$ but I have absolutely NO clue how to get there. Any advice?
12
votes
1answer
610 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
594 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
424 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
248 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 ...
10
votes
4answers
574 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 ...
7
votes
3answers
175 views

When is $2^x+3^y$ a perfect square?

If $x$ and $y$ are positive integers, then when is $2^x+3^y$ a perfect square? I tried this question a lot but failed. I tried dividing cases into when $x,y$ are even/odd, but still have no idea ...
7
votes
3answers
301 views

All roots of the quartic equation $a x^4 + b x^3 + x^2 + x + 1 = 0$ cannot be real

Problem Prove that all roots of $a x^4 + b x^3 + x^2 + x + 1 = 0$ cannot be real. Here $a,b \in \mathbb R$, and $a \neq 0$. Source This is one of the previous year problem of Regional ...
5
votes
1answer
85 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 ...
3
votes
3answers
295 views

Putnam 1967 problem, integration

I have some problems understanding this solution of this problem: Given $f,g : \mathbb{R} \rightarrow \mathbb{R}$ - continuous functions with period $1$, prove that $$\lim _{n \rightarrow \infty} ...
1
vote
1answer
72 views

Find minimum of $P=\frac{\sqrt{3(2x^2+2x+1)}}{3}+\frac{1}{\sqrt{2x^2+(3-\sqrt{3})x +3}}+\frac{1}{\sqrt{2x^2+(3+\sqrt{3})x +3}}$

For $x\in\mathbb{R}$ find minimum of $P$. $P=\dfrac{\sqrt{3(2x^2+2x+1)}}{3}+\dfrac{1}{\sqrt{2x^2+(3-\sqrt{3})x +3}}+\dfrac{1}{\sqrt{2x^2+(3+\sqrt{3})x +3}}$ Source : Viet Nam national test for high ...