chat gpt

by fuv870, Mar 16, 2025, 9:51 PM

The chat gpt alreadly knows how to solve the problem of IMO USAMO and AMC?

Perfect Squares, Infinite Integers and Integers

by steven_zhang123, Mar 16, 2025, 12:06 PM

For which integer \( h \), are there infinitely many positive integers \( n \) such that \( \lfloor \sqrt{h^2 + 1} \cdot n \rfloor \) is a perfect square? (Here \( \lfloor x \rfloor \) denotes the integer part of the real number \( x \)?

Unsolved Diophantine(I think)

by Nuran2010, Mar 14, 2025, 4:41 PM

Find all solutions for the equation $2^n=p+3^p$ where $n$ is a positive integer and $p$ is a prime.(Don't get mad at me,I've used the search function and did not see a correct and complete solution anywhere.)

Another NT FE

by nukelauncher, Sep 22, 2020, 11:58 PM

Find all functions $f:\mathbb Z_{>0}\to \mathbb Z_{>0}$ such that $a+f(b)$ divides $a^2+bf(a)$ for all positive integers $a$ and $b$ with $a+b>2019$.

Variable point on the median

by MarkBcc168, Jun 11, 2019, 12:23 AM

Let $ABC$ be a scalene triangle with circumcircle $\Gamma$. Let $M$ be the midpoint of $BC$. A variable point $P$ is selected in the line segment $AM$. The circumcircles of triangles $BPM$ and $CPM$ intersect $\Gamma$ again at points $D$ and $E$, respectively. The lines $DP$ and $EP$ intersect (a second time) the circumcircles to triangles $CPM$ and $BPM$ at $X$ and $Y$, respectively. Prove that as $P$ varies, the circumcircle of $\triangle AXY$ passes through a fixed point $T$ distinct from $A$.

Whoever wrote this... doesn't know what concise means :\

by shiningsunnyday, Apr 11, 2017, 3:46 PM

2004 ISL G2 wrote:
Let $\Gamma$ be a circle and let $d$ be a line such that $\Gamma$ and $d$ have no common points. Further, let $AB$ be a diameter of the circle $\Gamma$; assume that this diameter $AB$ is perpendicular to the line $d$, and the point $B$ is nearer to the line $d$ than the point $A$. Let $C$ be an arbitrary point on the circle $\Gamma$, different from the points $A$ and $B$. Let $D$ be the point of intersection of the lines $AC$ and $d$. One of the two tangents from the point $D$ to the circle $\Gamma$ touches this circle $\Gamma$ at a point $E$; hereby, we assume that the points $B$ and $E$ lie in the same halfplane with respect to the line $AC$. Denote by $F$ the point of intersection of the lines $BE$ and $d$. Let the line $AF$ intersect the circle $\Gamma$ at a point $G$, different from $A$. Prove that the reflection of the point $G$ in the line $AB$ lies on the line $CF$.

Solution

Can anyone spot the projective solution (since I found this in a projective chapter in Lemmas).

IMO 2014 Problem 1

by Amir Hossein, Jul 8, 2014, 12:17 PM

Let $a_0 < a_1 < a_2 < \dots$ be an infinite sequence of positive integers. Prove that there exists a unique integer $n\geq 1$ such that
\[a_n < \frac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.\]
Proposed by Gerhard Wöginger, Austria.
This post has been edited 1 time. Last edited by v_Enhance, Nov 5, 2023, 5:17 PM
Reason: missing < sign

Sequences and limit

by lehungvietbao, Jan 3, 2014, 10:32 AM

Let $({{x}_{n}}),({{y}_{n}})$ be two positive sequences defined by ${{x}_{1}}=1,{{y}_{1}}=\sqrt{3}$ and
\[ \begin{cases}  {{x}_{n+1}}{{y}_{n+1}}-{{x}_{n}}=0 \\   x_{n+1}^{2}+{{y}_{n}}=2 \end{cases} \] for all $n=1,2,3,\ldots$.
Prove that they are converges and find their limits.

IMO 2012 P5

by mathmdmb, Jul 11, 2012, 7:03 PM

Let $ABC$ be a triangle with $\angle BCA=90^{\circ}$, and let $D$ be the foot of the altitude from $C$. Let $X$ be a point in the interior of the segment $CD$. Let $K$ be the point on the segment $AX$ such that $BK=BC$. Similarly, let $L$ be the point on the segment $BX$ such that $AL=AC$. Let $M$ be the point of intersection of $AL$ and $BK$.

Show that $MK=ML$.

Proposed by Josef Tkadlec, Czech Republic
This post has been edited 3 times. Last edited by Eternica, Jun 19, 2024, 10:03 AM

One secuence satisfying condition

by hatchguy, Sep 4, 2011, 12:18 AM

Prove that there exists only one infinite secuence of positive integers $a_1,a_2,...$ with $a_1=1$, $a_2>1$ and $a_{n+1}^3 + 1 = a_na_{n+2}$ for all positive integers $n$.

Can this sequence be bounded?

by darij grinberg, Jan 19, 2005, 11:00 AM

Let $a_0$, $a_1$, $a_2$, ... be an infinite sequence of real numbers satisfying the equation $a_n=\left|a_{n+1}-a_{n+2}\right|$ for all $n\geq 0$, where $a_0$ and $a_1$ are two different positive reals.

Can this sequence $a_0$, $a_1$, $a_2$, ... be bounded?

Proposed by Mihai Bălună, Romania
This post has been edited 1 time. Last edited by djmathman, Sep 27, 2015, 2:12 PM

To share with readers my favorite problem I came across today :) (Shout for contrib.)

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  • Let $ ABC$ be an equilateral triangle of side length $ 1$. Let $ D$ be the point such that $ C$ is the midpoint of $ BD$, and let $ I$ be the incenter of triangle $ ACD$. Let $ E$ be the point on line $ AB$ such that $ DE$ and $ BI$ are perpendicular. $ \

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