0
votes
1answer
83 views

lifting the exponent lemma for $p=2$.

I am trying to understand the proof of theorem 3 (in p.4) of http://www.artofproblemsolving.com/Resources/Papers/LTE.pdf However, I dont understand the last sentence "This means the power of $2$ in ...
7
votes
1answer
103 views

Mediteranean Mathematics Olympiad 2014 number theory problem.

I paraphrase it slightly to make it shorter. Prove for every integer $S\geq100$ there exists a positive integer $P$ such that there are at least two different solutions in positive integers(up to ...
-2
votes
3answers
160 views

Equation $a^{n}+b^{n}=2008$ has no integers solutions. [closed]

Prove that the equation $a^{n}+b^{n}=2008$ has no solutions for $a,b,n\in\mathbb{Z}, n\geq2.$
6
votes
1answer
92 views

Finding all such polynomials under a gcd condition

Find all such polynomial $f(x)\in \mathbb{Z}[x]$ such that $$ \forall n\in \mathbb{N} \quad \gcd(f(n),f(2^n))=1$$ This is a problem from the Indian Team Selection Test. Can someone give me a solution ...
10
votes
3answers
176 views

If $a^4+b^4\in\mathbb Q$ and $a^3+b^3\in\mathbb Q$ and $a^2+b^2\in\mathbb Q$, prove that $a+b\in\mathbb Q$ and $ab\in\mathbb Q$.

If $\begin{cases}a^4+b^4\in\mathbb Q\\ a^3+b^3\in\mathbb Q\\ a^2+b^2\in\mathbb Q\end{cases}$, prove that $a+b\in\mathbb Q$ and $ab\in\mathbb Q$. It is given that $a,b\in\mathbb R$. The proof of ...
6
votes
4answers
156 views

Prove that there is an integer $n$ such that $n^{1992}$ starts with $1992$ one's.

This was taken from an old Brazilian Mathematical Olympiad (1992). As the title says, we're supposed to prove that there is an integer $n$ such that $n^{1992}$ starts with $1992$ one's (in the ...
1
vote
1answer
45 views

How many different right triangles are possible with the shorter side of odd length?

I was trying to solve this problem but unable to figure it out completely. I thing number of was odd integer $n$ can be the side of right triangle is number of factor of $\frac{n^2}{2}$. Can some one ...
1
vote
2answers
104 views

Given perimeter of triangle and one side, find other two sides

In triangle ABC, all three sides have integer lengths. If AB = 21, the perimeter is 54, and the area is a positive integer, what are the lengths of BC and AC? I tried using Heron's Formula, but I ...
1
vote
1answer
45 views

Number theory problem, trigonometry

Suppose $p$ and $q$ are relatively prime positive integers, and that $x$ is a positive rational number. Given that $x \in [-\frac{1}{2}, \frac{1}{2}]$ and $$q\sin{\pi x} = p$$ how can we compute $p, ...
1
vote
0answers
45 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
174 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 $5$ ...
11
votes
1answer
217 views

Curious number theory problem

$k,m,n\in\mathbb{N}$ satisfy $k^{m+n}=nm^n$. How can I show that $m=k$ and $n=k^k\,?$
7
votes
2answers
126 views

$a^3+3a^2+a$ is never a perfect square.

Prove that no number of the form $ a^3+3a^2+a $, for a positive integer $a$, is a perfect square. This problem was published in the Italian national competition (Cesenatico 1991). I've been ...
2
votes
3answers
106 views

For any positive integers $a$ and $b$, the number $(36a+b)(a+36b)$ can never be a power of $2$.

APMO 1998: Show that for any positive integers $a$ and $b$, the number $(36a+b)(a+36b)$ can never be a power of $2$. The solution I've read substitutes $a=2^Ap,b=2^Bq$ where $p$ and $q$ are ...
4
votes
1answer
188 views

Math Olympiads: GCD of terms in a sequence equals GCD of terms in other sequence

Recently, someone asked for a proof of a problem from the Russian Mathematical Olympiad, 1995. Math Olympiads: GCD of terms in a sequence equals GCD of their indices. The problem was to show that if ...
5
votes
2answers
161 views

Math Olympiads: GCD of terms in a sequence equals GCD of their indices.

The sequence $a_1 ,a_2 ,a_3 ,...$ of positive integers satisfies $\text{gcd}(a_i ,a_j ) = \text{gcd} (i, j)$ for $i \neq j$. Prove that $a_i = i$ for all $i$. Source: Russian Mathematical Olympiad, ...
2
votes
3answers
54 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.
-1
votes
2answers
84 views

Show the sum is equal to a product of six primes

On a set of math challenges, one of them is to prove that 145678+456781+567814+678145+781456+814567 is the product of six different primes. This sounds like number theory to me, but I have no ...
6
votes
2answers
275 views

A generalization of IMO 1977 problem 2

Here is the IMO 1977 problem 2: In a finite sequence of real numbers the sum of any seven successive terms is negative, and the sum of any eleven successive terms is positive. Determine the ...
1
vote
0answers
45 views

Prove that for every prime $p > 100$ and every integer $r$ $\exists a, b$ such that $p \mid a^2 + b^5 - r$

Prove that for every prime $p > 100$ and every integer $r$ $\exists a, b \in \mathbb{Z}$ such that $p \mid a^2 + b^5 - r$ Preferably without using Jacobi sums, I have already seen a solution using ...
1
vote
1answer
46 views

chain of divisibility relation

Let $a$ and $b$ be positive integers such that $a | b^2, b^2 | a^3, a^3 | b^4, b^4 | a^5, \cdots $ Prove that $a = b$. My way is as follows: Let $A=v_p(a), B=v_p(b)$ be the exact power of a prime $p$ ...
8
votes
2answers
244 views

Computing the last non-zero digit of ${1027 \choose 41}$?

I am working on the following problem: Let $x_n$ be a sequence of positive odd numbers. If $N$ is the number of ordered pairs $(x_1, x_2, x_3, \dots, x_{42})$ such that $$x_1 + x_2 + x_3 + \dots + ...
1
vote
1answer
38 views

Number theory problem - powers

Find the smallest prime $p$ such that for any $1 \leq k \leq 10$ relatively prime to $p$, one of $k, k^2,\ldots k^{p - 2}$ is congruent to $1$ modulo $p$. I am honestly not sure how to approach this ...
-2
votes
2answers
171 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 ...
20
votes
4answers
2k views

Sum of four squares not a prime

Let $ a, b, c, d $ be natural numbers such that $ ab=cd $. Prove that $ a^2+b^2+c^2+d^2 $ is not a prime. I am clueless on this one. I tried contradiction, but didn't get anywhere. Can you help? ...
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
0answers
87 views

Proving that $\sqrt{4ab-1}=m^2$ is equivalent to $a=b$. where $a$ and $b$ are non zero integers

So the original question was to prove that if $4ab-1$ divides $4a^2-1)^2$, then $a=b$ where $a$ and $b$ are non zero integers. (IMO 2007) I proceed this way: $(4a²-1)²/(4ab-1)=q$ where $q$ is ...
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 ...
3
votes
1answer
142 views

Answering a flawed Mathletes question (finding $x^2 + y^2 = p$ given $p$ for large $p$)

There was a mathletes meet today (high school) and this was one of the questions: "-Some background on Fermat's 4k+1 sum of square theorem- One such prime is $367369$. What integers $x, y$ satisfy ...
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
69 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 ...
0
votes
1answer
62 views

Brazil 2002 first problem neater result?

Brazil's 2002 first problem basically asks to prove that for any positive integer n, there are n integers $m_1,m_2\dots m_n$ where $1\leq m_i\leq9$ such that $m_1^2+m_2^2+\ldots+m_n^2=a^2$ for some ...
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
385 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 ...
1
vote
0answers
76 views

(AIME) number theory question [duplicate]

How many integers less than 1000 can be expressed in the form $$\frac{(x + y + z)^2}{xyz} $$ where $x, y, z$ are integers? So far, I've attempted substituting certain values of $x, y, z$. For ...
2
votes
2answers
67 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. ...
4
votes
1answer
312 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?
2
votes
1answer
76 views

How to prove $\binom{2n}{n}\frac{1}{n+1} = \prod \limits_{i = 2}^n \frac{2i-1}{i+1} $?

How to prove this closed form involving Catalan numbers? $$\binom{2n}{n}\frac{1}{n+1} = \prod \limits_{i = 2}^n \frac{2 \times (2i-1)}{i+1} $$ I have seen this being used here. Not sure how to derive ...
3
votes
2answers
140 views

If $x=123456789101112131415161718$, then $x\equiv 6\pmod{16}$ and $x\equiv 0\pmod 6$

BdMO 2013 Rajshahi Build a number by writing down consecutive natural numbers starting from $1$ which is divisible by $6$ and gives a reminder of $6$ upon division by $16$. Such a number is ...
1
vote
1answer
52 views

Number theory problem, parity

Let $f(x)$ denote the number of (not necessarily distinct) prime factors of $x$. Let $n > 1$ be the smallest positive integer for which there are more $i$ with with $f(i)$ even than $f(i)$ odd in ...
1
vote
3answers
82 views

There exists an integer with alternating digits $1$ and $2$ which is divisible by $2013$

Could someone give me hints in how to solve the following (rather interesting) problem? Prove that there exists an integer consisting of an alternance of $1$s and $2$s with as many $1$s as $2$s ...
6
votes
2answers
135 views

Prove that $\sqrt[2012]{2012!}<\sqrt[2013]{2013!}$

I need to prove that $\sqrt[2012]{2012!}<\sqrt[2013]{2013!}$ My attempt: Let $a=\sqrt[2012]{2012!}$ and $b=\sqrt[2013]{2013!}$ Then $\displaystyle\frac{b^{2012}}{a^{2012}}=\frac{2013}{b}$ ...
5
votes
3answers
222 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 ... ...
3
votes
0answers
154 views

How many times is the digit $3$ repeated in $9^{666}$? [closed]

How many times is the digit $3$ repeated in the number $9^{666}$ ? Thanks.
3
votes
1answer
237 views

Find a number leaving a particular remainder with 3 different numbers

I have the following question: Let $N$ be the greatest number that will divide $1305, 4665$ and $6905$, leaving the same remainder in each case. What is the sum of digits of $N$. My approach ...
4
votes
1answer
168 views

IMO Hong Kong TST 2014

Let $m,n$ be distinct positive integer not exceeding 2013 and $d$ be their gcd. Suppose $d^2|3(m-n)$. Find the greatest possible value of $d(m+n)$. I only know $m-n$ should be a perfect square, but ...
12
votes
2answers
489 views

If 1 boy knows r girls and 1 girl knows r boys ,then number of boys=girls

Yet another question from BdMO 2013 Nationals: In a class,every boy knows $r$ number of girls and every girl knows $r$ number of boys.Show that there are equal number of boys and girls[Assume that ...
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
100 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 ...