Inequality
by lgx57, May 28, 2025, 7:21 AM
,
. Prove that
,
3^n + 61 is a square
by VideoCake, May 26, 2025, 5:14 PM
Determine all positive integers 
such that 
is the square of an integer.
such that 
is the square of an integer.Serbian selection contest for the IMO 2025 - P6
by OgnjenTesic, May 22, 2025, 4:07 PM
For an 
table filled with natural numbers, we say it is a divisor table if:
- the numbers in the
-th row are exactly all the divisors of some natural number 
,
- the numbers in the
-th column are exactly all the divisors of some natural number 
,
-
for every 
.
A prime number
is given. Determine the smallest natural number 
, divisible by , such that there exists an divisor table, or prove that such does not exist.
Proposed by Pavle Martinović
table filled with natural numbers, we say it is a divisor table if:- the numbers in the
-th row are exactly all the divisors of some natural number 
,- the numbers in the
-th column are exactly all the divisors of some natural number 
,-
for every 
.A prime number
is given. Determine the smallest natural number 
, divisible by , such that there exists an divisor table, or prove that such does not exist.Proposed by Pavle Martinović
prove that
D1033 : A problem of probability for dominoes 3*1
by Dattier, May 15, 2025, 12:29 PM
Let 
a grid of 9*9, we choose a little square in of this grid three times, we can choose three times the same.
What the probability of cover with 3*1 dominoes this grid removed by theses little squares (one, two or three) ?
a grid of 9*9, we choose a little square in of this grid three times, we can choose three times the same.What the probability of cover with 3*1 dominoes this grid removed by theses little squares (one, two or three) ?
Problem 7
by SlovEcience, May 14, 2025, 11:03 AM
Consider the sequence 
defined by 
and
![\[
u_{n+1} = \frac{1}{2}u_n^2 - 4 \quad \text{for all } n \in \mathbb{N}.
\]](//latex.artofproblemsolving.com/9/9/4/994aa754cc1288ce4f28a95a0276e64282fb5f66.png)
a) Prove that there exist infinitely many positive integers such that 
.
b) Compute
![\[
\lim_{n \to \infty} \frac{2u_{n+1}}{u_0u_1\cdots u_n}.
\]](//latex.artofproblemsolving.com/f/f/1/ff174e7431cbfbc17c650d109651241286756a1a.png)
defined by 
and![\[
u_{n+1} = \frac{1}{2}u_n^2 - 4 \quad \text{for all } n \in \mathbb{N}.
\]](http://latex.artofproblemsolving.com/9/9/4/994aa754cc1288ce4f28a95a0276e64282fb5f66.png)
a) Prove that there exist infinitely many positive integers such that 
.b) Compute
![\[
\lim_{n \to \infty} \frac{2u_{n+1}}{u_0u_1\cdots u_n}.
\]](http://latex.artofproblemsolving.com/f/f/1/ff174e7431cbfbc17c650d109651241286756a1a.png)
Easy but Nice 12
by TelvCohl, Mar 8, 2025, 3:49 PM
Given a 
with orthocenter 
and a point 
lying on the Euler line of 
Prove that the midpoint of 
lies on the Thomson cubic of the pedal triangle of WRT
with orthocenter 
and a point 
lying on the Euler line of 
Prove that the midpoint of 
lies on the Thomson cubic of the pedal triangle of WRT Inequality with xy+yz+zx=1
by Kimchiks926, Nov 12, 2022, 5:34 PM
The positive real numbers 
satisfy 
. Prove that:
satisfy 
. Prove that:
This post has been edited 1 time. Last edited by Kimchiks926, Nov 13, 2022, 2:27 PM
Reason: typo
Reason: typo
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