A number theory problem from the British Math Olympiad

by Rainbow1971, Mar 28, 2025, 8:39 PM

I am a little surprised to find that I am (so far) unable to solve this little problem:

Quote:

Let $n$ be an integer. Show that, if $2 + 2 \sqrt{1+12n^2}$ is an integer, then it is a perfect square.

I set $k := \sqrt{1+12n^2}$ . If $2 + 2 \sqrt{1+12n^2}$ is an integer, then $k (=\sqrt{1+12n^2})$ is at least rational, so that $1 + 12n^2$ must be a perfect square then. Using Conway's topograph method, I have found out that the smallest non-negative pairs $(n, k)$ for which this happens are $(0,1), (2,7), (28,97)$ and $(390, 1351)$ , and that, for every such pair $(n,k)$ , the "next" such pair can be calculated as
$\begin{bmatrix} 7 & 2 \\ 24 & 7 \end{bmatrix} \begin{bmatrix} n \\ k \end{bmatrix} .$ The eigenvalues of that matrix are irrational, however, so that any calculation which uses powers of that matrix is a little cumbersome. There must be an easier way, but I cannot find it. Can you?

Thank you.

linear algebra

matrix

number theory

Monkeys have bananas

by nAalniaOMliO, Mar 28, 2025, 8:20 PM

Ten monkeys have 60 bananas. Each monkey has at least one banana and any two monkeys have different amounts of bananas.
Prove that any six monkeys can distribute their bananas between others such that all 4 remaining monkeys have the same amount of bananas.

combinatorics

A lot of numbers and statements

by nAalniaOMliO, Mar 28, 2025, 8:20 PM

101 numbers are written in a circle. Near the first number the statement "This number is bigger than the next one" is written, near the second "This number is bigger that the next two" and etc, near the 100th "This number is bigger than the next 100 numbers".
What is the maximum possible amount of the statements that can be true?

combinatorics

2025 Caucasus MO Seniors P2

by BR1F1SZ, Mar 26, 2025, 12:39 AM

Let $ABC$ be a triangle, and let $B_1$ and $B_2$ be points on segment $AC$ symmetric with respect to the midpoint of $AC$ . Let $\gamma_A$ denote the circle passing through $B_1$ and tangent to line $AB$ at $A$ . Similarly, let $\gamma_C$ denote the circle passing through $B_1$ and tangent to line $BC$ at $C$ . Let the circles $\gamma_A$ and $\gamma_C$ intersect again at point $B'$ ( $B' \neq B_1$ ). Prove that $\angle ABB' = \angle CBB_2$ .

geometry

Nordic 2025 P3

by anirbanbz, Mar 25, 2025, 12:36 PM

Let $ABC$ be an acute triangle with orthocenter $H$ and circumcenter $O$ . Let $E$ and $F$ be points on the line segments $AC$ and $AB$ respectively such that $AEHF$ is a parallelogram. Prove that $\vert OE \vert = \vert OF \vert$ .

NMC

geometry

Hard limits

by Snoop76, Mar 25, 2025, 9:13 AM

$a_n$ and $b_n$ satisfies the following recursion formulas: $a_{0}=1,$ $b_{0}=1$ , $a_{n+1}=a_{n}+b_{n}$ $\text{[math]}$ and $\text{[math]}$ $b_{n+1}=(2n+3)b_{n}+a_{n}$ . Find $\lim_{n \to \infty} \frac{a_n}{(2n-1)!!}$ $\text{[math]}$ and $\text{[math]}$ $\lim_{n \to \infty} \frac{b_n}{(2n+1)!!}.$

series

limit

double factorial

D1018 : Can you do that ?

by Dattier, Mar 24, 2025, 6:01 AM

We can find $A,B,C$ , such that $\gcd(A,B)=\gcd(C,A)=\gcd(A,2)=1$ and $\forall n \in \mathbb N^*, (C^n \times B \mod A) \mod 2=0$ .

For example :

$C=20$
$A=47650065401584409637777147310342834508082136874940478469495402430677786194142956609253842997905945723173497630499054266092849839$

$B=238877301561986449355077953728734922992395532218802882582141073061059783672634737309722816649187007910722185635031285098751698$

Can you find $A,B,C$ such that $A>3$ is prime, $C,B \in (\mathbb Z/A\mathbb Z)^*$ with $o(C)=(A-1)/2$ and $\forall n \in \mathbb N^*, (C^n \times B \mod A) \mod 2=0$ ?

arithmetic

number theory

A number theory about divisors which no one fully solved at the contest

by nAalniaOMliO, Jul 24, 2024, 7:40 PM

Let's call a pair of positive integers $(k,n)$ interesting if $n$ is composite and for every divisor $d<n$ of $n$ at least one of $d-k$ and $d+k$ is also a divisor of $n$
Find the number of interesting pairs $(k,n)$ with $k \leq 100$
M. Karpuk

This post has been edited 1 time. Last edited by nAalniaOMliO, Jul 27, 2024, 10:08 AM

number theory

f( - f (x) - f (y))= 1 -x - y , in Zxz

by parmenides51, Nov 22, 2020, 1:16 PM

Find all functions $f: Z \to Z$ that satisfy $f(-f (x) - f (y))= 1 -x - y$ for all $x, y \in Z$

This post has been edited 1 time. Last edited by parmenides51, Nov 22, 2020, 1:21 PM

functional equation

functional equation in Z

functional

algebra

USAMO 1981 #2

by Mrdavid445, Jul 26, 2011, 7:32 PM

Every pair of communities in a county are linked directly by one mode of transportation; bus, train, or airplane. All three methods of transportation are used in the county with no community being serviced by all three modes and no three communities being linked pairwise by the same mode. Determine the largest number of communities in this county.

AMC

USA(J)MO

USAMO

combinatorics unsolved

combinatorics

Submit

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