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Topics in undergraduate and graduate studies

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College Math Topics in undergraduate and graduate studies

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Arc Midpoints Form Cyclic Quadrilateral

ike.chen 57

N an hour ago by cj13609517288

Source: ISL 2022/G2

In the acute-angled triangle $ABC$ , the point $F$ is the foot of the altitude from $A$ , and $P$ is a point on the segment $AF$ . The lines through $P$ parallel to $AC$ and $AB$ meet $BC$ at $D$ and $E$ , respectively. Points $X \ne A$ and $Y \ne A$ lie on the circles $ABD$ and $ACE$ , respectively, such that $DA = DX$ and $EA = EY$ .
Prove that $B, C, X,$ and $Y$ are concyclic.

57 replies

1 viewing

ike.chen

Jul 9, 2023

cj13609517288

an hour ago

Complex number

ronitdeb 0

an hour ago

Let $z_1, ... ,z_5$ be vertices of regular pentagon inscribed in a circle whose radius is $2$ and center is at $6+i8$ . Find all possible values of $z_1^2+z_2^2+...+z_5^2$

0 replies

ronitdeb

an hour ago

0 replies

Elementary Problems Compilation

Saucepan_man02 29

N an hour ago by Electrodynamix777

Could anyone send some elementary problems, which have tricky and short elegant methods to solve?

For example like this one:
Solve over reals: $a^2 + b^2 + c^2 + d^2 -ab-bc-cd-d +2/5=0$

29 replies

Saucepan_man02

May 26, 2025

Electrodynamix777

an hour ago

Generic Real-valued FE

lucas3617 4

N 2 hours ago by GreekIdiot

$f: \mathbb{R} -> \mathbb{R}$ , find all functions where $f(2x+f(2y-x))+f(-x)+f(y)=2f(x)+f(y-2x)+f(2y)$ for all $x$ , $y \in \mathbb{R}$

4 replies

lucas3617

Apr 25, 2025

GreekIdiot

2 hours ago

Find all possible values of BT/BM

va2010 54

N 2 hours ago by lpieleanu

Source: 2015 ISL G4

Let $ABC$ be an acute triangle and let $M$ be the midpoint of $AC$ . A circle $\omega$ passing through $B$ and $M$ meets the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $T$ be the point such that $BPTQ$ is a parallelogram. Suppose that $T$ lies on the circumcircle of $ABC$ . Determine all possible values of $\frac{BT}{BM}$ .

54 replies

va2010

Jul 7, 2016

lpieleanu

2 hours ago

A Familiar Point

v4913 52

N 2 hours ago by SimplisticFormulas

Source: EGMO 2023/6

Let $ABC$ be a triangle with circumcircle $\Omega$ . Let $S_b$ and $S_c$ respectively denote the midpoints of the arcs $AC$ and $AB$ that do not contain the third vertex. Let $N_a$ denote the midpoint of arc $BAC$ (the arc $BC$ including $A$ ). Let $I$ be the incenter of $ABC$ . Let $\omega_b$ be the circle that is tangent to $AB$ and internally tangent to $\Omega$ at $S_b$ , and let $\omega_c$ be the circle that is tangent to $AC$ and internally tangent to $\Omega$ at $S_c$ . Show that the line $IN_a$ , and the lines through the intersections of $\omega_b$ and $\omega_c$ , meet on $\Omega$ .

52 replies

v4913

Apr 16, 2023

SimplisticFormulas

2 hours ago

Tangential quadrilateral and 8 lengths

popcorn1 72

N 2 hours ago by cj13609517288

Source: IMO 2021 P4

Let $\Gamma$ be a circle with centre $I$ , and $A B C D$ a convex quadrilateral such that each of the segments $A B, B C, C D$ and $D A$ is tangent to $\Gamma$ . Let $\Omega$ be the circumcircle of the triangle $A I C$ . The extension of $B A$ beyond $A$ meets $\Omega$ at $X$ , and the extension of $B C$ beyond $C$ meets $\Omega$ at $Z$ . The extensions of $A D$ and $C D$ beyond $D$ meet $\Omega$ at $Y$ and $T$ , respectively. Prove that $A D+D T+T X+X A=C D+D Y+Y Z+Z C.$
Proposed by Dominik Burek, Poland and Tomasz Ciesla, Poland

72 replies

popcorn1

Jul 20, 2021

cj13609517288

2 hours ago

An algorithm for discovering prime numbers?

Lukaluce 3

N 2 hours ago by TopGbulliedU

Source: 2025 Junior Macedonian Mathematical Olympiad P3

Is there an infinite sequence of prime numbers $p_1, p_2, ..., p_n, ...,$ such that for every $i \in \mathbb{N}, p_{i + 1} \in \{2p_i - 1, 2p_i + 1\}$ is satisfied? Explain the answer.

3 replies

Lukaluce

May 18, 2025

TopGbulliedU

2 hours ago

Random concyclicity in a square config

Maths_VC 5

N 2 hours ago by Royal_mhyasd

Source: Serbia JBMO TST 2025, Problem 1

Let $M$ be a random point on the smaller arc $AB$ of the circumcircle of square $ABCD$ , and let $N$ be the intersection point of segments $AC$ and $DM$ . The feet of the tangents from point $D$ to the circumcircle of the triangle $OMN$ are $P$ and $Q$ , where $O$ is the center of the square. Prove that points $A$ , $C$ , $P$ and $Q$ lie on a single circle.

5 replies

Maths_VC

Tuesday at 7:38 PM

Royal_mhyasd

2 hours ago

Basic ideas in junior diophantine equations

Maths_VC 3

N 3 hours ago by Royal_mhyasd

Source: Serbia JBMO TST 2025, Problem 3

Determine all positive integers $a, b$ and $c$ such that
$2$ $\cdot$ $10^a + 5^b = 2025^c$

3 replies

Maths_VC

Tuesday at 7:54 PM

Royal_mhyasd

3 hours ago

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