Algebraic Manipulation

Let $p,q,r,s$ be real numbers that satisfy
$p^2+q^2=r^2+s^2$ $p^2+s^2-ps=q^2+r^2+qr$ Hence find the value of $\frac{pq+rs}{ps+qr}$

Click to reveal hidden text

Hint : Use Sine/Cosine Rule

1 reply

Darealzolt

Yesterday at 3:40 AM

vanstraelen

an hour ago

Ducks can play games now apparently

MortemEtInteritum 35

N 2 hours ago by pi271828

Source: USA TST(ST) 2020 #1

Let $a$ , $b$ , $c$ be fixed positive integers. There are $a+b+c$ ducks sitting in a
circle, one behind the other. Each duck picks either rock, paper, or scissors, with $a$ ducks
picking rock, $b$ ducks picking paper, and $c$ ducks picking scissors.
A move consists of an operation of one of the following three forms:

[list]
[*] If a duck picking rock sits behind a duck picking scissors, they switch places.
[*] If a duck picking paper sits behind a duck picking rock, they switch places.
[*] If a duck picking scissors sits behind a duck picking paper, they switch places.
[/list]
Determine, in terms of $a$ , $b$ , and $c$ , the maximum number of moves which could take
place, over all possible initial configurations.

35 replies

1 viewing

MortemEtInteritum

Nov 16, 2020

pi271828

2 hours ago

2017 IGO Advanced P3

bgn 18

N 2 hours ago by Circumcircle

Source: 4th Iranian Geometry Olympiad (Advanced) P3

Let $O$ be the circumcenter of triangle $ABC$ . Line $CO$ intersects the altitude from $A$ at point $K$ . Let $P,M$ be the midpoints of $AK$ , $AC$ respectively. If $PO$ intersects $BC$ at $Y$ , and the circumcircle of triangle $BCM$ meets $AB$ at $X$ , prove that $BXOY$ is cyclic.

Proposed by Ali Daeinabi - Hamid Pardazi

18 replies

bgn

Sep 15, 2017

Circumcircle

2 hours ago

Own made functional equation

JARP091 1

N 3 hours ago by JARP091

Source: Own (Maybe?)

$\text{Find all functions } f : \mathbb{R} \to \mathbb{R} \text{ such that:} \\ f(a^4 + a^2b^2 + b^4) = f\left((a^2 - f(ab) + b^2)(a^2 + f(ab) + b^2)\right)$

1 reply

JARP091

May 31, 2025

JARP091

3 hours ago

Euler line of incircle touching points /Reposted/

Eagle116 6

N 3 hours ago by pigeon123

Let $ABC$ be a triangle with incentre $I$ and circumcentre $O$ . Let $D,E,F$ be the touchpoints of the incircle with $BC$ , $CA$ , $AB$ respectively. Prove that $OI$ is the Euler line of $\vartriangle DEF$ .

6 replies

Eagle116

Apr 19, 2025

pigeon123

3 hours ago

Parallel lines on a rhombus

buratinogigle 1

N 3 hours ago by Giabach298

Source: Own, Entrance Exam for Grade 10 Admission, HSGS 2025

Given the rhombus $ABCD$ with its incircle $\omega$ . Let $E$ and $F$ be the points of tangency of $\omega$ with $AB$ and $AC$ respectively. On the edges $CB$ and $CD$ , take points $G$ and $H$ such that $GH$ is tangent to $\omega$ at $P$ . Suppose $Q$ is the intersection point of the lines $EG$ and $FH$ . Prove that two lines $AP$ and $CQ$ are parallel or coincide.

1 reply

buratinogigle

5 hours ago

Giabach298

3 hours ago

Orthocenter lies on circumcircle

whatshisbucket 90

N 3 hours ago by bjump

Source: 2017 ELMO #2

Let $ABC$ be a triangle with orthocenter $H,$ and let $M$ be the midpoint of $\overline{BC}.$ Suppose that $P$ and $Q$ are distinct points on the circle with diameter $\overline{AH},$ different from $A,$ such that $M$ lies on line $PQ.$ Prove that the orthocenter of $\triangle APQ$ lies on the circumcircle of $\triangle ABC.$

Proposed by Michael Ren

90 replies

whatshisbucket

Jun 26, 2017

bjump

3 hours ago

Polish MO Finals 2014, Problem 4

j___d 3

N 4 hours ago by ariopro1387

Source: Polish MO Finals 2014

Denote the set of positive rational numbers by $\mathbb{Q}_{+}$ . Find all functions $f: \mathbb{Q}_{+}\rightarrow \mathbb{Q}_{+}$ that satisfy
$\underbrace{f(f(f(\dots f(f}_{n}(q))\dots )))=f(nq)$ for all integers $n\ge 1$ and rational numbers $q>0$ .

3 replies

j___d

Jul 27, 2016

ariopro1387

4 hours ago

S(an) greater than S(n)

ilovemath0402 1

N 4 hours ago by ilovemath0402

Source: Inspired by an old result

Find all positive integer $n$ such that $S(an)\ge S(n) \quad \forall a \in \mathbb{Z}^{+}$ ( $S(n)$ is sum of digit of $n$ in base 10)
P/s: Original problem

The original one was prove that $10^t-1$ are one of such $n$ then i wonder can we find out all such $n$ satisfied the inequality

1 reply

ilovemath0402

5 hours ago

ilovemath0402

4 hours ago

Hagge-like circles, Jerabek hyperbola, Lemoine cubic

kosmonauten3114 0

4 hours ago

Source: My own

Let $\triangle{ABC}$ be a scalene oblique triangle with circumcenter $O$ and orthocenter $H$ , and $P$ ( $\neq \text{X(3), X(4)}$ , $\notin \odot(ABC)$ ) a point in the plane.
Let $\triangle{A_1B_1C_1}$ , $\triangle{A_2B_2C_2}$ be the circumcevian triangles of $O$ , $P$ , respectively.
Let $\triangle{P_AP_BP_C}$ be the pedal triangle of $P$ with respect to $\triangle{ABC}$ .
Let $A_1'$ be the reflection in $P_A$ of $A_1$ . Define $B_1'$ , $C_1'$ cyclically.
Let $A_2'$ be the reflection in $P_A$ of $A_2$ . Define $B_2'$ , $C_2'$ cyclically.
Let $O_1$ , $O_2$ be the circumcenters of $\triangle{A_1'B_1'C_1'}$ , $\triangle{A_2'B_2'C_2'}$ , respectively.

Prove that:
1) $P$ , $O_1$ , $O_2$ are collinear if and only if $P$ lies on the Jerabek hyperbola of $\triangle{ABC}$ .
2) $H$ , $O_1$ , $O_2$ are collinear if and only if $P$ lies on the Lemoine cubic (= $\text{K009}$ ) of $\triangle{ABC}$ .

0 replies

kosmonauten3114

4 hours ago

0 replies

Incenter perpendiculars and angle congruences

math154 84

N 4 hours ago by zuat.e

Source: ELMO Shortlist 2012, G3

$ABC$ is a triangle with incenter $I$ . The foot of the perpendicular from $I$ to $BC$ is $D$ , and the foot of the perpendicular from $I$ to $AD$ is $P$ . Prove that $\angle BPD = \angle DPC$ .

Alex Zhu.

84 replies

math154

Jul 2, 2012

zuat.e

4 hours ago

Algebraic Manipulation

Darealzolt 1

N Apr 30, 2025 by Soupboy0

Find the number of pairs of real numbers $a, b, c$ that satisfy the equation $a^4 + b^4 + c^4 + 1 = 4abc$ .

1 reply

Darealzolt

Apr 30, 2025

Soupboy0

Apr 30, 2025

Algebraic Manipulation

G H J

Reveal topic tags

algebra

G H BBookmark kLocked kLocked NReply

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

Darealzolt

22 posts

Darealzolt

#1 h Apr 30, 2025, 1:25 PM

Y by

Find the number of pairs of real numbers $a, b, c$ that satisfy the equation $a^4 + b^4 + c^4 + 1 = 4abc$ .

Z K Y

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

Soupboy0

499 posts

Soupboy0

#2 h Apr 30, 2025, 2:36 PM

Y by

$(a, b, c) = (1, 1, 1), (-1, -1, 1)$ and permutations. Note that if $a, b, c > 0$ , then $a^4+b^4+c^4+1 \ge 4abc$ by AM-GM, with equality at $a=b=c=1$ , which is the only solution for $a, b, c > 0$ . Also note that if $1$ or all of $a, b, c$ are negative, then there are no solutions by the Trivial inequality. Also note that when $2$ of $a, b, c$ are negative, then this case is symmetric to $a, b, c > 0$ . Therefore, there are $4$ solutions

Z K Y

N Quick Reply