primes,exponentials,factorials

by skellyrah, Apr 30, 2025, 6:31 PM

find all primes p,q such that $\frac{p^q+q^p-p-q}{p!-q!}$ is a prime number

This post has been edited 3 times. Last edited by skellyrah, 3 hours ago

primes

number theory

hard

Queue geo

by vincentwant, Apr 30, 2025, 3:54 PM

Let $ABC$ be an acute scalene triangle with circumcenter $O$ . Let $Y, Z$ be the feet of the altitudes from $B, C$ to $AC, AB$ respectively. Let $D$ be the midpoint of $BC$ . Let $\omega_1$ be the circle with diameter $AD$ . Let $Q\neq A$ be the intersection of $(ABC)$ and $\omega$ . Let $H$ be the orthocenter of $ABC$ . Let $K$ be the intersection of $AQ$ and $BC$ . Let $l_1,l_2$ be the lines through $Q$ tangent to $\omega,(AYZ)$ respectively. Let $I$ be the intersection of $l_1$ and $KH$ . Let $P$ be the intersection of $l_2$ and $YZ$ . Let $l$ be the line through $I$ parallel to $HD$ and let $O'$ be the reflection of $O$ across $l$ . Prove that $O'P$ is tangent to $(KPQ)$ .

This post has been edited 1 time. Last edited by vincentwant, Today at 3:55 PM

geometry

Very easy NT

by GreekIdiot, Apr 30, 2025, 2:49 PM

Prove that there exists no natural number $n>1$ such that $n \mid 2^n-1$ .

number theory

bezout

Elementary

Fermat s Little Theorem

Do not try to bash on beautiful geometry

by ItzsleepyXD, Apr 30, 2025, 9:30 AM

Let $ABC$ be triangle with point $D,E$ and $F$ on $BC,AB,CA$
such that $BE=CF$ and $E,F$ are on the same side of $BC$
Let $M$ be midpoint of segment $BC$ and $N$ be midpoint of segment $EF$
Let $G$ be intersection of $BF$ with $CE$ and $\dfrac{BD}{DC}=\dfrac{AC}{AB}$
Prove that $MN\parallel DG$

geometry

Functional Geometry

by GreekIdiot, Apr 27, 2025, 1:08 PM

Let $f: \pi \to \mathbb R$ be a function from the Euclidean plane to the real numbers such that $f(A)+f(B)+f(C)=f(O)+f(G)+f(H)$ for any acute triangle $\Delta ABC$ with circumcenter $O$ , centroid $G$ and orthocenter $H$ . Prove that $f$ is constant.

This post has been edited 1 time. Last edited by GreekIdiot, Apr 27, 2025, 1:08 PM

geometry

function

circumcircle

Can you construct the incenter of a triangle ABC?

by PennyLane_31, Oct 29, 2023, 1:53 AM

Given points $P$ and $Q$ , Jaqueline has a ruler that allows tracing the line $PQ$ . Jaqueline also has a special object that allows the construction of a circle of diameter $PQ$ . Also, always when two circles (or a circle and a line, or two lines) intersect, she can mark the points of the intersection with a pencil and trace more lines and circles using these dispositives by the points marked. Initially, she has an acute scalene triangle $ABC$ . Show that Jaqueline can construct the incenter of $ABC$ .

This post has been edited 1 time. Last edited by PennyLane_31, Oct 26, 2024, 3:18 PM

geometry

incenter

Right-angled triangle if circumcentre is on circle

by liberator, Jan 4, 2016, 9:41 PM

Let the excircle of triangle $ABC$ opposite the vertex $A$ be tangent to the side $BC$ at the point $A_1$ . Define the points $B_1$ on $CA$ and $C_1$ on $AB$ analogously, using the excircles opposite $B$ and $C$ , respectively. Suppose that the circumcentre of triangle $A_1B_1C_1$ lies on the circumcircle of triangle $ABC$ . Prove that triangle $ABC$ is right-angled.

Proposed by Alexander A. Polyansky, Russia

geometry

IMO

conditional geometry

Rectangle EFGH in incircle, prove that QIM = 90

by v_Enhance, Jul 18, 2014, 7:48 PM

Let $ABC$ be a triangle with incenter $I$ , and suppose the incircle is tangent to $CA$ and $AB$ at $E$ and $F$ . Denote by $G$ and $H$ the reflections of $E$ and $F$ over $I$ . Let $Q$ be the intersection of $BC$ with $GH$ , and let $M$ be the midpoint of $BC$ . Prove that $IQ$ and $IM$ are perpendicular.

geometry

rectangle

incenter

geometric transformation

reflection

geometry proposed

Another quadrilateral in a circle

by v_Enhance, May 3, 2013, 8:09 PM

Let $ABCD$ be a quadrilateral inscribed in a circle $\omega$ , and let $P$ be a point on the extension of $AC$ such that $PB$ and $PD$ are tangent to $\omega$ . The tangent at $C$ intersects $PD$ at $Q$ and the line $AD$ at $R$ . Let $E$ be the second point of intersection between $AQ$ and $\omega$ . Prove that $B$ , $E$ , $R$ are collinear.

af(a)+bf(b)+2ab=x^2 for all natural a, b - show that f(a)=a

by shoki, May 14, 2011, 3:53 PM

Suppose that $f : \mathbb{N} \rightarrow \mathbb{N}$ is a function for which the expression $af(a)+bf(b)+2ab$ for all $a,b \in \mathbb{N}$ is always a perfect square. Prove that $f(a)=a$ for all $a \in \mathbb{N}$ .

number theory proposed

number theory

Submit

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