3
votes
2answers
33 views

partitions and their generating functions and Partitions of n

A partition of an integer, n, is one way of writing n as the sum of positive integers where the order of the addends (terms being added) does not matter. p(n, k) = number of partitions of n with k ...
2
votes
0answers
17 views

How prove that there are $a,b,c$ such that $a \in A, b \in B, c \in C$ and $a,b,c$ (with approriate order) is a arithmetic sequence?

Let $N=\{ 1,2,3,..., 3n \}$ with $n$ is a positive integer and $A,B,C$ are three arbitrary sets such that $A \cup B \cup C = N, A \cap B = B \cap C = C \cap A = \varnothing, |A| = |B| = |C| = n $. How ...
-4
votes
0answers
60 views

Prove that $\sqrt{n}$ is irrational [on hold]

Question: Using fundamental theorem of integers and the fact that every natural number that is not prime, prove that $\sqrt{n}$ is irrational unless $n=m^2$ for some $m\in\mathbb N$. Here is how I ...
6
votes
1answer
110 views

How prove $ y^2=x^3+x+1370^{1370}$ has at least 6 answers in $ \mathbb{Q}$?

How prove that $ y^2=x^3+x+1370^{1370}$ has at least 6 answers in $ \mathbb{Q}$?
-1
votes
1answer
49 views

Suppose f(x) + 2f(1/x) = x . Evaluate f(5) in simplest form. [closed]

If f(x) + 2(f(1/x)) = x, evaluate f(5). How can I go about solving this problem?
3
votes
0answers
92 views

After how many steps can compositions of $x\mapsto x+1$ and $x\mapsto x^2+1$ produce the same result starting from $1$ and $3$?

This problem is from a Turkish contest: Let $P(x)=x+1$ and $Q(x)=x^2+1$. Consider all sequences $(x_k,y_k)$ such that $(x_1,y_1)=(1,3)$ and $(x_{k+1},y_{k+1})$ is either $(P(x_k),Q(y_k))$ ...
4
votes
2answers
67 views

Find the 1005th digit after the decimal point expansion of the square root of N.

Let $N$ be the positive integer with $2008$ decimal digits, all of them $1$. That is, $N=1111...1111$, with $2008$ occurrences of the digit $1$. Find the $1005th$ digit after the decimal point ...
4
votes
1answer
266 views

Prove or disprove that there exists a unique positive integer sequence $\{a_{n}\}$ satisfying a condition

Question: Prove or disprove: there exists a unique positive integer sequence $\{a_{n}\}$ satisfying the following condition: $\forall m\in N^{+}$, there exists a unique integer sequence ...
4
votes
3answers
72 views

how find $\sum_{k \in A} \frac{1}{k-1} $ for $ A = \{ m^n| \text{ } m, n \in Z \text { and } m, n \ge 2 \} $

If $ A = \{ m^n| \text{ } m, n \in Z \text { and } m, n \ge 2 \} $, then how find $\sum_{k \in A} \frac{1}{k-1} $?
4
votes
3answers
104 views

$ 1987 \mid \left( n^n + (n+1)^n \right) $

Problem from the 1987 Leningrad Math Olympiad: Is there a positive integer $n$ such that $ n^n + \left( n + 1 \right)^n $ is divisible by $ 1987 $? The provided solution: The answer is ...
3
votes
1answer
94 views

Last 7 digits of 7th powers

Alice and Bob play the following game. They alternately select distinct nonzero digits from $1$ to $9$, until they have chosen seven such digits. Consider the resulting seven-digit number by joining ...
1
vote
0answers
97 views

Olympiad number theory question

Let $p,q$ and $r$ be prime numbers. It is given that $p$ divides $qr − 1$, $q$ divides $rp − 1$, and $r$ divides $pq − 1$. Determine all possible values of $pqr$. I think I'm missing something in ...
2
votes
3answers
108 views

Trouble with inequalities

I'm a 9th grade student, going into 10th grade. Math has always been a subject I enjoyed and excelled in. I'm writing a schoolboard-wide math contest next year in mid-February I believe. To prepare ...
0
votes
0answers
92 views

Number theory proofs relating to units

Moderator Note: This has been claimed to be a current contest question. It is being locked while we investigate. What is a counterexample for the proposition: If u ∈ Um has order n1 and u2 ∈ Um ...
2
votes
1answer
108 views

A term of the sequence

Let $a_0$ and $a_n$ be diff erent divisors of a natural number $m$, and $a_0, a_1, a_2,\cdots, a_n$ be a sequence of natural numbers such that it satisfies $$a_{i+1} = |a_i\pm a_{i-1}|\text{ for }0 ...
1
vote
0answers
30 views

Maximal number of inequivalent elements in $\mathbb{E}_p$

For a prime $p$ of the form $12k+1$ and $\mathbb{Z}_p=\{0,1,2,\cdots,p-1\}$, let $$ \displaystyle \mathbb{E}_p=\{(a,b) \mid a,b \in \mathbb{Z}_p,\quad p\nmid 4a^3+27b^2\}$$ For $(a,b), (a',b') \in ...
5
votes
3answers
101 views

Proving $n^2(n^2+16)$ is divisible by 720

Given that $n+1$ and $n-1$ are prime, we need to show that $n^2(n^2+16)$ is divisible by 720 for $n>6$. My attempt: We know that neither $n-1$ nor $n+1$ is divisible by $2$ or by $3$, therefore ...
2
votes
0answers
75 views

Find all pair(s) of positive integer $(a,b)$ such that $\frac{a^2}{2ab^2 -b^3+1}$ is also positive integer too?

Another number theory problem. I can find the small value of $b$ such that 0,1,2. But, I cannot find the upper limit of $b$, such that the value of $b$ is limited. How can I find the solution ...
4
votes
3answers
97 views

Find all positive integer solutions $(x,y,z)$ that satisfy $5^x \cdot 7^y +4= 3^z$?

This is another contest math-problem. The only problem that I cannot find the way to tackle this problem. Can anybody try to provide the solution to solve this problem? Thanks
5
votes
4answers
107 views

Integer solutions to $a^{2014} +2015\cdot b! = 2014^{2015}$

How many solutions are there for $a^{2014} +2015\cdot b! = 2014^{2015}$, with $a,b$ positive integers? This is another contest problem that I got from my friend. Can anybody help me find the ...
35
votes
4answers
968 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 ...
1
vote
1answer
99 views

What are some of the more efficient ways of studying for an Olympiad?

This September I am participating in a competition called the Australian Intermediate Mathematics olympiad, and you may not have heard of it but it's very similar to the AIME. Could you please tell me ...
3
votes
3answers
151 views

Find the maximum value of $abc$

$a,b,c$ are three positive real numbers such that $ab+bc+ca=12$. Then find the maximum value of $abc$
2
votes
3answers
118 views

Maximize $\sqrt{2x + 13} + \sqrt[3]{3y+5} + \sqrt[4]{8z+12}$

Given three non-negative (as pointed out by Calvin Lin) real numbers $x+y+z = 3$, find the maximum value of $\sqrt{2x + 13} + \sqrt[3]{3y+5} + \sqrt[4]{8z+12}$. (Source : Singapore Math Olympiad ...
3
votes
2answers
64 views

Almost perfect numbers

A positive integer $n$ is called almost perfect if the sum of its divisors smaller than $n$ is $n-1$. What are all almost perfect numbers $n$ such that some power $n^k$ is also almost perfect for at ...
6
votes
4answers
184 views

Finding all primes $(p,q)$ for perfect squares.

Find all prime pairs $(p,q)$ such that $2p-1, 2q-1, 2pq-1$ are all perfect squares. Source: St.Petersburg Olympiad 2011 I have only found the pair $(5,5)$ so I am thinking that maybe a modulo $5$ ...
2
votes
4answers
73 views

Prove that $(2m+1)^2 - 4(2n+1)$ can never be a perfect square where m, n are integers

I could prove it hit and trial method. But I was thinking to come up with a general and a more 'mathematically' correct method, but I did not reach anywhere. Thanks a lot for any help.
13
votes
4answers
439 views

prove Diophantine equation has no solution $\prod_{i=1}^{2014}(x+i)=\prod_{i=1}^{4028}(y+i)$

show that this equation $$(x+1)(x+2)(x+3)\cdots(x+2014)=(y+1)(y+2)(y+3)\cdots(y+4028)$$ have no positive integer solution. This problem is china TST (2014),I remember a famous result? maybe ...
4
votes
2answers
116 views

Finding the value of $(bc-ad)(ac-bd)(ab-cd)$

Let $a,b,c,d$ be $4$ distinct non-zero integers such that $a+b+c+d = 0$. It is know that the number $$M = (bc - ad)(ac - bd)(ab-cd)$$ lies strictly between $96100$ and $98000$. Determine the value ...
-2
votes
2answers
199 views

Even or Odd for factorial

Moderator Note: This is a current contest question on codechef.com. Given $N$ and $M$ I need to tell whether $\left\lfloor \large\frac{N!}{M} \right\rfloor$ is even or odd.How to do this ...
2
votes
1answer
150 views

2012 USAJMO Problem 5

For distinct positive integers $a, b < 2012$, define $f(a, b)$ to be the number of integers $k$ with $1 \le k<2012$ such that the remainder when $ak$ divided by 2012 is greater than that of $bk$ ...
3
votes
2answers
117 views

Eliminate numbers from $1,2,3. . .30$ such that the remaining sequence does not contain both $x$ and $2x$

BdMO 2014 nationals From the sequence 1,2,3. . . .30,pick another sequence of numbers such that if x is in our new sequence,then 2x is not there(or vice versa).What is the maximum number of terms ...
1
vote
1answer
45 views

For a prime $p$, $6p\mid a^p+1$ for no $a$ or infinitely many $a$

BdMO Nationals Secondary: Show that for any prime $p$, there are either infinitely many or no positive integer $a$, so that $6p$ divides $a^p+1$ . Find all those primes for which there exists no ...
5
votes
1answer
177 views

Sequence where the sum of digits of all numbers is 7

BdMO 2014 We define a sequence starting with $a_1=7,a_2=16,\ldots,\,$ such that the sum of digits of all numbers of the sequence is $7$ and if $m>n$,then $a_m>a_n$ i.e. all such numbers are ...
1
vote
1answer
71 views

Eliminating numbers from the sequence $1,2,3,4,5,6,7…400$

BdMO 2014 Let us take the sequence $1,2,3,4,5,6,7....400$ .We are going to remove numbers from the sequence such that the sum of any 2 numbers of the remaining sequence is not divisible by 7.What ...
3
votes
2answers
314 views

Finding all positive integers $x,y,z$ that satisfy $3^x - 5^y = z^2$

Find all positive integers $x,y,z$ that satisfy: $$3^x - 5^y = z^2.$$ I think that $(x,y,z)= (2,1,2)$ will be the only solution. But how to prove that?
1
vote
2answers
88 views

If $7$ is the first digit of $2^n$, what is the first digit of $5^n$?

Let $2^n = 7\cdot 10^x + p$ and $5^n = a\cdot 10^y + r$ And now what? (We're in base $10$)
7
votes
1answer
126 views

South Africa National Olympiad 2000 (Tile 4xn rectangle using 2x1 tiles)

Let $A_n$ be the number of ways to tile a $4×n$ rectangle using $2×1$ tiles. Prove that $A_n$ is divisible by 2 if and only if $A_n$ is divisible by 3. My attempt: Define basic shapes A, B and C, ...
1
vote
0answers
44 views

Two Perfect Squares--$(3n+1) \& (4n+1)$. [duplicate]

Assume $n$ is a Natural Number which satisfies the following 2 properties simultaneously: $01$ . $(3n+1)$=$a$12 for some Natural Number $a$1. $02$ . $(4n+1)$=$a$22 for some Natural Number $a$2. ...
8
votes
1answer
397 views

IMO 1979 problem

The question is $$\text{If }\, p, \ q\in \mathbb{N}, \;1-\frac12+\frac13-\frac14-\dotsb-\frac{1}{1318}+\frac{1}{1319}=\frac{p}{q}.\qquad \text{Prove that } 1979\mid p.$$ So my solution went like ...
2
votes
2answers
68 views

Divisibility Of Positve Integers [closed]

Suppose a,b and c are three positive integers which satisfy the condition that ($a$2+$b$2+$c$2) is divisible by $(a+b+c)$. Prove that there exists infinitely many positive integers $n$ for which ...
40
votes
7answers
9k views

There exists a power of 2 such that the last five digits are all 3's or 6's. Find the last 5 digits of this number

I just took an olympiad and I'm wondering how this problem is solved. Problem: There exists a power of 2 such that the last five digits are all 3's or 6's. Find the last 5 digits of this number. ...
5
votes
3answers
238 views

What is the smallest natural number n?

What is the smallest natural number n for which there is a natural k, such that, the lasts 2012 digit in the representation decimal of $n^k$ are equal to 1? I don't even know how to start with it ... ...
2
votes
2answers
156 views

AIME number theory problem (unique factorization domains)

I'd greatly appreciate some help with the following problem, from a mock AIME I took. Compute the largest squarefree positive integer $n$ such that $\mathbb{Q}(\sqrt{-n})\cap \overline{\mathbb{Z}}$ ...
3
votes
2answers
166 views

How find this $a^3+b^3+c^3-20(a+3)(b+3)(c+3)=2013$ equation integer solution

if $a,b,c\in Z$,and $a\le b\le c$ and such $$\begin{cases} a+b+c=-3\\ a^3+b^3+c^3-20(a+3)(b+3)(c+3)=2013 \end{cases}$$ Find the value $3a+b+2c=?$ my try $$a+b+c=-3\Longrightarrow ...
2
votes
1answer
81 views

Highest $n$ such that $2^n|a^{2012}+a^{2013}+a^{2014}+\cdots +a^{3012}$,$a=4k+2$

A question from BdMO 2013 Nationals: Let $a$ be an integer divisible by 2 but not divisible by 4. What is the largest positive integer n such that ...
3
votes
2answers
101 views

2 is a primitive root mod $3^h$ for any positive integer $h$

It's easy to verify that 2 is a primitive root mod $3^2$. But then why does it follow that 2 is a primitive root mod $3^h$ for any positive integer $h$? This was used in the solution of 2009 Putnam ...
0
votes
0answers
49 views

Find all rational solutions to $x^3 - y^2 = 2$. [duplicate]

Find all rational solutions to $x^3 - y^2 = 2$. The only integers solutions are $(3,\pm5)$: http://mathforum.org/library/drmath/view/51569.html
7
votes
3answers
388 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.
3
votes
1answer
127 views

Korean Math Olympiad 2000 (floor function, quadratic mod) [closed]

Let $p$ be a prime such that $p ≡ 1\ (\mathrm{mod}\ 4)$. Evaluate ...