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

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0
votes
1answer
162 views

Suppose that a cube is inscribed in a sphere of radius one. What is the volume of the cube? my reasoning vs answer

Now my reasoning is that, s^2 + s^2 = 2^2, where s is the side of the cube, giving, s^3 = 2 sqrt 2. But the answer and explanation here is different: http://math.acadiau.ca/aumc/hints4.pdf how is the ...
7
votes
1answer
141 views

Bound on $|f(x)|^2 + |f'(x)|^2$

Let $f\in C^2(\mathbb{R})$ be a twice differentiable function satisfying $$|f(x)|^2\le a$$ and $$|f'(x)|^2 + |f''(x)|^2\le b$$ for all real $x$, where $a$ and $b$ are positive constants. Prove that ...
4
votes
1answer
119 views

What is the minimum number of locks on the cabinet that would satisfy these conditions?

Eleven scientists want to have a cabinet built where they will keep some top secret work. They want multiple locks installed, with keys distributed in such a way that if any six scientists are present ...
4
votes
1answer
77 views

Prove that $2^x < \prod_{p\le x} p < (13/4)^x$ for sufficiently large x

Prove that $2^x < \prod_{p\le x} p < (13/4)^x$ for sufficiently large x. Here $p$ is prime. So if we take logs we need to show for sufficiently large x, $x\log 2 < \sum_{p\le x}\log p < ...
0
votes
0answers
138 views

Iran Math Olympiad 2013 (Perfect Set)

Let $n$ be a natural number and suppose that $w_1,w_2,…,w_n$ are $n$ weights. We call the set of {$w_1,w_2,…,w_n$} to be a Perfect Set if we can achieve all of the 1, 2, …, W weights with sums of ...
1
vote
1answer
85 views

Diophantine Equation.

How many solutions are there in $\mathbb{N} \times \mathbb{N}$ to the equation $\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{1995}?$ How would you solve this? I have tried but am not sure how I should ...
2
votes
3answers
104 views

For any real numbers $a,b,c$ show that $\displaystyle \min\{(a-b)^2,(b-c)^2,(c-a)^2\} \leq \frac{a^2+b^2+c^2}{2}$

For any real numbers $a,b,c$ show that: $$ \min\{(a-b)^2,(b-c)^2,(c-a)^2\} \leq \frac{a^2+b^2+c^2}{2}$$ OK. So, here is my attempt to solve the problem: We can assume, Without Loss Of Generality, ...
4
votes
2answers
524 views

Prove that [0,1] is equivalent to (0,1) and give an explicit description of a 1-1 function from [0,1] onto (0,1)

The problem is stated as follows: Show that there is a one-to-one correspondence between the points of the closed interval $[0,1]$ and the points of the open interval $(0,1)$. Give an explicit ...
1
vote
1answer
27 views

Diophantine Approximation and Liouville Theorem

I'm reading the alternative proof of the MO problem: http://koopakoo.wordpress.com/2008/09/03/cgmo-2007-problem-7-and-liouvilles-theorem/ . However, I have a problem, namely that in the alternative ...
5
votes
1answer
125 views

Putnam Series Question

I'm studying for the Putnam Exam and am a bit confused about how to go about solving this problem. Sum the series $$ \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} \frac{m^2n}{3^m(n3^m + m3^n)}. $$ ...
0
votes
1answer
104 views

Prime factorization problem

The prime factorizations of $r + 1$ positive integers ($r \geq 1$) together involve only $r$ primes. Prove that there is a subset of these integers whose product is a perfect square. Now, I'm ...
7
votes
3answers
212 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 ...
19
votes
1answer
261 views

How to prove $\sum_{n=1}^\infty\operatorname{arccot}\frac{\sqrt[2^n]2+\cos\frac\pi{2^n}}{\sin\frac\pi{2^n}}=\operatorname{arccot}\frac{\ln2}\pi$?

How can I prove the following identity? $$\sum_{n=1}^\infty\operatorname{arccot}\frac{\sqrt[2^n]2+\cos\frac\pi{2^n}}{\sin\frac\pi{2^n}}=\operatorname{arccot}\frac{\ln2}\pi$$
6
votes
5answers
209 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 ...
3
votes
2answers
96 views

Interesting determinant: Let $A$ be an $n$ by $n$ matrix with entries $a_{i,j}$ given that $a_{i,j}=2$ if $i=j$

Let $A$ be an $n$ by $n$ matrix with entries $a_{i,j}$ given that $a_{i,j}=2$ if $i=j$, $a_{i,j}=1$ if $i-j\equiv\pm2\pmod n$, and $a_{i,j}=0$ otherwise. Find $\det A$. It seems that the ...
3
votes
1answer
126 views

Conjecture similar to Fermat's Theorem.

I was wondering about a problem which i could reduce to asking the following Does there exist a set $a,b,c$ of prime numbers such that $$a^a+b^b=c^c$$ Is it really a tough problem or do you think ...
2
votes
1answer
323 views

Korean Math Olympiad 1993 (geometry)

Consider a triangle ABC with BC = $a$, CA = $b$, AB = $c$. Let D be the midpoint of BC and E be the intersection of the bisector of A with BC. The circle through A, D, E meets AC, AB again at F, G ...
1
vote
2answers
60 views

Putnam-Style Sequences Problem

Let $S_1$ denote the sequence of positive integers $1,2,3,4,5,6,\ldots,$ and define the sequence $S_{n+1}$ in terms of $S_n$ by adding $1$ to those integers in $S_n$ which are divisible by $n$. Thus, ...
1
vote
1answer
57 views

Prove that for $n\ge1$, $\xi-\frac{h_n}{k_n}=(-1)^nk_n^{-2}\left(\xi_{n+1}+\langle 0,a_n,a_{n-1},…,a_2,a_1\rangle\right)^{-1}$

Prove that for $n\ge1$, $$\xi-\frac{h_n}{k_n}=(-1)^nk_n^{-2}\left(\xi_{n+1}+\langle 0,a_n,a_{n-1},...,a_2,a_1\rangle\right)^{-1}$$ In addition, show that ...
1
vote
4answers
974 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.
0
votes
1answer
70 views

Numbers with integer multiples using only digits $2$ and $6$ (Austria Mathematical Olympiad 2006)

Let $N$ be a positive integer. How many non-negative integers $n ≤ N$ are there that have an integer multiple, that only uses the digits $2$ and $6$ in decimal representation? Obviously, $n$ can't be ...
3
votes
1answer
49 views

Prove $\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)^{2}\geq (a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)$

Prove that $$\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)^{2}\geq (a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)$$ for $a,b,c>0$ Any hints/solutions?
1
vote
1answer
23 views

Interesting continued fraction problem $|r_i-u_0/u_1|\le\frac1{k_ik_{i+1}}$

Let $u_0/u_1$ be a rational number in lowest terms, and write $u_0/u_1=\langle a_0, a_1,...,a_n\rangle$ in standard continued fraction notation. Show that if $0\le i<n$, then ...
2
votes
1answer
68 views

Is $\sum_{n=1}^{\infty}n/(b_1 + b_2 + \cdots+ b_n)$ convergent?

Let $\sum_{n=1}^{\infty}a_n$ be a convergent series of positive terms (so $a_i > 0$ for all $i$) and set $b_n = 1/(na_n^2)$ for $n\ge1$. Is $\sum_{n=1}^{\infty}n/(b_1 + b_2 + \cdots + b_n)$ ...
7
votes
1answer
140 views

Define a sequence by $a_1 = 1, a_2 = 1/2$, and $a_{n+2} = a_{n+1} - a_na_{n+1}/2$ for $n$ a positive integer.

Define a sequence by $a_1 = 1, a_2 = 1/2$, and $$a_{n+2} = a_{n+1} - a_na_{n+1}/2$$ for $n$ a positive integer. Find $$\lim_{n\to\infty}na_n$$ if it exists. Well, we can deduce that $\lim a_n=0$ by ...
5
votes
1answer
191 views

Korea Math Olympiad 1993

An integer which is the area of a right-angled triangle with integer sides is called Pythagorean. Prove that for every positive integer n > 12 there exists a Pythagorean number between n and 2n.
1
vote
1answer
235 views

Rigorous Statements: “It suffices to show that […]” and Variations

Mathematical proofs display a variety of proof styles and language used. One of the common statements I have seen are "It suffices to show that [...]" and "We want to show that [...]". Of course, ...
3
votes
2answers
129 views

The number 3211000 is 7-special

Define a positive integer $k$ to be $n$-special if it satisfies the following properties: It has $n$ digits (0, 1, ..., 9) The 1st digit is equal to the number of 0's in the decimal representation ...
6
votes
2answers
114 views

$a_1=1,a_{n+1}=\frac{n}{a_n}+\frac{a_n}{n}$. Prove that for $n\ge4$, $\lfloor{a_n^2}\rfloor=n$

Define a sequence $\left\lbrace a_{n}\right\rbrace$ by $\displaystyle{a_{1} = 1\,,\ a_{n + 1} = {n \over a_n} + {a_n \over n}.\quad}$ Prove that for $n \geq 4,\,\,\left\lfloor ...
4
votes
3answers
157 views

Find all polynomials $P(x)$ such that $2xP(x)=(x+1)P(x-1)+(x-1)P(x+1)$.

Find all polynomials $P(x)$ such that $2xP(x)=(x+1)P(x-1)+(x-1)P(x+1)$. Well, if $\deg P\le 3$ this is easy since we can deduce $P(0)=P(1)=P(-1)$ by letting $x=0,1,-1$
4
votes
1answer
202 views

No primes in this sequence

Here's a fun little problem: Prove that the sequence $$10001, 100010001, 1000100010001, \cdots$$ contains no prime numbers.
0
votes
1answer
101 views

In a party with 2000 persons, determine # of people who know everyone

In a party with 2000 persons, among any set of four there is at least one person who knows each of the other three. There are three people who are not mutually acquainted with each other. How many ...
0
votes
1answer
86 views

Czech Republic Math Olympiad 2008 Problem

In decimal representation, we call an integer k-carboxylic if and only if it can be represented as a sum of k distinct integers, all of them greater than 9, whose digits are the same. For instance, ...
7
votes
2answers
160 views

Continuous functions on $\mathbb{R}^2$ with special property

The following problem is from Miklos Schweitzer competition (Year 1983, Problem 7): Prove that if the function $f: \mathbb{R}^{2}\to [0, 1]$ is continuous, and its average on every circle of ...
1
vote
2answers
51 views

Recursive formular and closed-form questions

Follow the question the $f(n)=4n-1$ and $F(n)=\sum_{k=0}^nf(k)$. And it ask you to write the recursive of $F(n)$. But I only know the recursive of $f(n)$ is $$f(n)=\begin{cases} -1,&\text{if ...
0
votes
1answer
69 views

Conditions for convergence of a geometric series [duplicate]

This question concerns the infinite geometric series formula. It turns out there is a nice formula for the sum of an infinite geometric series. Consider the infinite geometric series ...
5
votes
1answer
94 views

Macedonia National Olympiad 2010

Problem The point O is the center of the circumscribed circle of the acute-angled triangle ABC. The line AO cuts the side BC in point N, and the line BO cuts the side AC at point M. Prove that if ...
1
vote
2answers
63 views

Contest Math Possible Triangles

In the xy-plane, how many triangles have each of their vertices at points (a,b) where a,b are integers satisfying 1 ≤ a ≤ 5 and 1≤b≤5? I got twenty-five, but something tells me this isn't right. I ...
2
votes
1answer
60 views

Calculating number of pages from sum of page numbers

A novel has 6 chapters. As usual, starting from the first page of the first chapter, the pages of the novel are numbered $1, 2, 3, 4, \ldots$. Also, each chapter begins on a new page. The last ...
1
vote
1answer
56 views

Young Tableaux Generalizing

The entries in a array include all the digits from 1 through 9, arranged so that the entries in every row and column are in increasing order. How many such arrays are there? (2010 AMC12 B) The ...
1
vote
2answers
179 views

France Olympiad Team Selection Test 2005

In an international meeting of n ≥ 3 participants, 14 languages are spoken. We know that: - Any 3 participants speak a common language. - No language is spoken by more than half of the participants. ...
1
vote
0answers
59 views

Who made now part of the problem?

Who came up with the meme of putting the current year as a four digit number into exercise problems? Is there a known first historical account?
3
votes
1answer
105 views

AMC Problem Help 12B 2010

A geometric sequence $(a_n)$ has $a_1=\sin x$, $a_2=\cos x$ , and $a_3=\tan x$ for some real number $x$. For what value of $n$ does $a_n=1+\cos x$? The AMC website has a solution to this, and ...
3
votes
2answers
137 views

Combinatorial Proof Of A Number Theory Theorem--Confusion

I came across a combinatorial proof of the Fermat's Little Theorem which states that If $p$ is a prime number then the number ($a$$p$-$a$) is a multiple of $p$ for any natural number $a$. Let me ...
9
votes
4answers
2k views

How to do well on Math Olympiads

I'm a high school student who really likes maths and I'm quite good at school. I want to start training maths by myself but I think I need some guidelines. I want to do well on IMO but I don't know ...
0
votes
1answer
111 views

Polynomials as sum of squares

Sometimes I have seen some math's competition problem solutions made by completing the expression as sum of squares. What is the intuition/computer program behind these solutions? For example, Prove ...
2
votes
1answer
164 views

Number of ways of expressing $n$ as a sum of positive integers

a) Let $s_n$ denote the number of ways of expressing $n$ as a sum of positive integers. Thus $s_1=1$, $s_2=2$, and $s_3=4$ (the four ways are $3$, $2+1$, $1+2$, and $1+1+1$). Prove that ...
4
votes
2answers
245 views

Logic Puzzle of Diamonds and sons

I came across a math problem and I need a solution for this. An old man has 49 diamonds. Each one has a different worth as $1, $2, $3, ….. $49. He has 7 sons and he ...
4
votes
2answers
232 views

Algebra question from Australia national olympiad 2013

Find all positive integers $n$ for which there are real numbers $x_1, \; x_2, \cdots,\; x_n$ satisfying $$(i) \; \; -1<x_i<1 \; for \; i=1,2, \cdots n$$ $$(ii) \; \; x_1+x_2+ \cdots +x_n=0 \; ...
1
vote
1answer
138 views

China Girls Math Olympiad (CGMO) 2002

There are 3n girl students who took part in a summer camp. There were three girl students to be on duty every day. When the summer camp ended, it was found that any two of the 3n students had been on ...