a nice prob for number theory

by Jackson0423, Apr 30, 2025, 4:14 PM

Let $n$ be a positive integer, and let its positive divisors be
$d_1 < d_2 < \cdots < d_k.$ Define $f(n)$ to be the number of ordered pairs $(i, j)$ with $1 \le i, j \le k$ such that $\gcd(d_i, d_j) = 1$ .

Find $f(3431 \times 2999)$ .

Also, find a general formula for $f(n)$ when
$n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k},$ where the $p_i$ are distinct primes and the $e_i$ are positive integers.

number theory

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, 6 hours ago

geometry

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

1 line solution to Inequality

by ItzsleepyXD, Apr 30, 2025, 9:27 AM

Let $x_1,x_2,\dots,x_n$ be positive real integer such that $x_1^2+x_2^2+\cdots+x_n^2=2$ Prove that
$\sum_{i=1}^{n}\frac{1}{x_i^3(x_{i-1}+x_{i+1})}\geqslant \left(\sum_{i=1}^{n}\frac{x_i}{x_{i-1}+x_{i+1}}\right)^3$ such that $x_{n+1}=x_1$ and $x_0=x_n$

inequalities

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.

C-B=60 <degrees>

by Sasha, Apr 10, 2005, 1:25 PM

Let $O$ be the circumcenter of an acute-angled triangle $ABC$ with ${\angle B<\angle C}$ . The line $AO$ meets the side $BC$ at $D$ . The circumcenters of the triangles $ABD$ and $ACD$ are $E$ and $F$ , respectively. Extend the sides $BA$ and $CA$ beyond $A$ , and choose on the respective extensions points $G$ and $H$ such that ${AG=AC}$ and ${AH=AB}$ . Prove that the quadrilateral $EFGH$ is a rectangle if and only if ${\angle ACB-\angle ABC=60^{\circ }}$ .

Proposed by Hojoo Lee, Korea

This post has been edited 1 time. Last edited by djmathman, Aug 1, 2015, 2:52 AM
Reason: Official version is better than non-official one

Art of Problem Solving Blog

by Jackson0423, Apr 30, 2025, 4:14 PM

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

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

by ItzsleepyXD, Apr 30, 2025, 9:27 AM

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

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

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

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

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

by Sasha, Apr 10, 2005, 1:25 PM