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

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8
votes
5answers
784 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 ...
9
votes
5answers
404 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 ...
0
votes
1answer
923 views

How many permutations of $N$ number that sum of every two adjacent is at most $M$

Moderator Note: This is a current contest question on codechef.com. This codechef.com contest ended on $17$ June. —Brian M. Scott Given two integers $N$ and $M$, find how many permutations ...
5
votes
4answers
7k 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?
2
votes
2answers
601 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 ...
67
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 ...
11
votes
2answers
370 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
5answers
1k 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 ...
6
votes
1answer
322 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 $a_k>0,k=1,2,\dots,n$, $$\sum_{k=1}^n \frac{2k+1}{a_1+a_2+\ldots+a_k}<4\sum_{k=1}^n\frac1{a_k}\;.$$
14
votes
2answers
741 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
187 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 ...
4
votes
1answer
282 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?
7
votes
2answers
210 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?
6
votes
4answers
631 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: ...
1
vote
0answers
241 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 ...
46
votes
6answers
5k 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 ...
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 $P(x^2+1)=P(x)^2+1$
13
votes
5answers
1k 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
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 ...
20
votes
2answers
1k 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 ...
5
votes
3answers
213 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 ...
11
votes
3answers
396 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 ...
7
votes
3answers
386 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
234 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
545 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 ...
1
vote
2answers
104 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
464 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
246 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
290 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 ...
2
votes
1answer
245 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
285 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
267 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
1answer
145 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
202 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 ...
1
vote
0answers
193 views

Modular arithematic Equation

We have an equation: $a^x+b^x+c^x \equiv m \pmod n $ also given $a,b,c < y $ what are the total number of solutions of this equation?
1
vote
1answer
292 views

What is shortcut to this contest algebra problem about polynomial?

The polynomial $P(x)=x^4 + ax^3 + bx^2 +cx + d$ has the property that $p(k)=11k$ for $k=1,2,3,4$. Compute $c$. The answer is $-39$.
0
votes
2answers
226 views

Prove that there is a $\delta$ such that $\int_{0}^{1} (f(x))^2dx\leq \delta$$\int_{0}^{1} (f'(x))^2dx$ for all $f$ with these conditions

Let $S=\{f:\mathbb{R} \to \mathbb{R}\}$ that satisfies: $\forall f\in S$, $f'$ exists and $f'$ is continuous and $f(0)=f(1)=0$. Please prove that $\exists \delta :\forall f\in S$ s.t. $\int_{0}^{1} ...
4
votes
1answer
613 views

Pennies on a checkerboard.

Here is a question on pennies on checkerboard. It isnt a homework question. I saw it in a book. ...
4
votes
1answer
376 views

How to find all rational numbers satisfy this equation?

Find all rational number $a,b,c$ satisfy: $$a+b+c=abc$$ I try to change this in different forms like $(ab-1)c = a+b$, $(ac-1)b = a+c$, $(cb-1)a = b+c$ etc but it won't help...
3
votes
1answer
247 views

block matrices problem

Let $A,B,C$ and $D$ be n by n matrics such that $AC=CA$. Prove that $\det \begin{pmatrix} A & B\\ C & D \end{pmatrix}=\det(AD-CB)$. The solution is to first assume that $A$ is invertible and ...
3
votes
1answer
302 views

Game Theory Matching a Deck of Cards

Moderator Note: This question is from a contest which ended 1 Dec 2012. Suppose we have a deck of cards labeled from $1$ to $52$. Let them be shuffled in a random configuration, then made ...
1
vote
2answers
47 views

Connecting square vertexes with minimal road

I have four cities in $A=(0,0),B=(1,0),C=(1,1),D=(0,1)$. I am asked to build the shortest motorway to connect the cities. How can I do that? I was thinking that first I need some compactness argument ...
1
vote
3answers
562 views

Find the four digit number?

Find a four digit number which is an exact square such that the first two digits are the same and also its last two digits are also the same.
1
vote
2answers
196 views

Understanding Less Frequent Form of Induction? (Putnam and Beyond)

I won't paste the question here since my problem is not a technical one but a conceptual one. Book is here: (Page 22 of the pdf) I do not understand why it is necessarily to induct $2^{k}$ to show ...
35
votes
4answers
895 views

How to prove $k!+(2k)!+\cdots+(nk)!$ has a prime divisor greater than $k!$

Question: Let $k$ be a positive integer. Show that there exist $n$ such that $$I=k!+(2k)!+(3k)!+\cdots+(nk)!$$ has a prime divisor $P$ such that $P>k!$. My idea: Let us denote by ...
23
votes
2answers
632 views

Does there exist a sequence $\{a_n\}_{n\ge1}$ with $a_n < a_{n+1}+a_{n^2}$ such that $\sum_{n=1}^{\infty}a_n$ converges?

Does there exist a sequence $\{a_n\}_{n\ge1}$ with $a_n < a_{n+1}+a_{n^2}$ such that $\sum_{n=1}^{\infty}a_n$ converges? Does there exist a sequence with the same property but with each term ...
26
votes
3answers
548 views

Square matrices satisfying certain relations must have dimension divisible by $3$

I saw this tucked away in a MathOverflow comment and am asking this question to preserve (and advertise?) it. It's a nice problem! Problem: Suppose $A$ and $B$ are real $n\times n$ matrices with ...
32
votes
3answers
1k views

A generalization of IMO 1983 problem 6

Note: This question has a bounty that will expire in just a few days. Let $a,b,c$ and $d$ be the lengths of the sides of a quadrilateral. Show that $$ab^2(b-c)+bc^2(c-d)+cd^2(d-a)+da^2(a-b)\ge 0$$ ...
8
votes
5answers
814 views

Favorite Math Competition Problems

I'm running a weekly math contest for a summer camp and would like to compile a list of interesting problems. The problems may presuppose mathematical knowledge up to but not including Calculus. For ...
19
votes
3answers
947 views

Finding all integer solutions of $5^x+7^y=2^z$

Find all integers $x,y,z$ such that $5^x+7^y=2^z$. This one comes from an online contest that I arranged some years ago, and I can assure that a completely elementary solution exists.