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USA GEO 2003
dreammath   21
N an hour ago by lpieleanu
Source: TST USA 2003
Let $ABC$ be a triangle and let $P$ be a point in its interior. Lines $PA$, $PB$, $PC$ intersect sides $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. Prove that
\[ [PAF]+[PBD]+[PCE]=\frac{1}{2}[ABC]  \]
if and only if $P$ lies on at least one of the medians of triangle $ABC$. (Here $[XYZ]$ denotes the area of triangle $XYZ$.)
21 replies
dreammath
Feb 16, 2004
lpieleanu
an hour ago
Nice "if and only if" function problem
ICE_CNME_4   10
N 2 hours ago by maromex
Let $f : [0, \infty) \to [0, \infty)$, $f(x) = \dfrac{ax + b}{cx + d}$, with $a, d \in (0, \infty)$, $b, c \in [0, \infty)$. Prove that there exists $n \in \mathbb{N}^*$ such that for every $x \geq 0$
\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\](For $n \in \mathbb{N}^*$ and $x \geq 0$, the notation $f_n(x)$ represents $\underbrace{(f \circ f \circ \dots \circ f)}_{n \text{ times}}(x)$. )

Please do it at 9th grade level. Thank you!
10 replies
ICE_CNME_4
Friday at 7:23 PM
maromex
2 hours ago
Inequality olympiad algebra
Foxellar   0
2 hours ago
Given that \( a, b, c \) are nonzero real numbers such that
\[
\frac{1}{abc} + \frac{1}{a} + \frac{1}{c} = \frac{1}{b},
\]let \( M \) be the maximum value of the expression
\[
\frac{4}{a^2 + 1} + \frac{4}{b^2 + 1} + \frac{7}{c^2 + 1}.
\]Determine the sum of the numerator and denominator of the simplified fraction representing \( M \).
0 replies
Foxellar
2 hours ago
0 replies
Integers on a cube
Rushil   6
N 2 hours ago by SomeonecoolLovesMaths
Source: Indian RMO 2004 Problem 2
Positive integers are written on all the faces of a cube, one on each. At each corner of the cube, the product of the numbers on the faces that meet at the vertex is written. The sum of the numbers written on the corners is 2004. If T denotes the sum of the numbers on all the faces, find the possible values of T.
6 replies
Rushil
Feb 28, 2006
SomeonecoolLovesMaths
2 hours ago
Tangents to circle concurrent on a line
Drytime   9
N 3 hours ago by Autistic_Turk
Source: Romania TST 3 2012, Problem 2
Let $\gamma$ be a circle and $l$ a line in its plane. Let $K$ be a point on $l$, located outside of $\gamma$. Let $KA$ and $KB$ be the tangents from $K$ to $\gamma$, where $A$ and $B$ are distinct points on $\gamma$. Let $P$ and $Q$ be two points on $\gamma$. Lines $PA$ and $PB$ intersect line $l$ in two points $R$ and respectively $S$. Lines $QR$ and $QS$ intersect the second time circle $\gamma$ in points $C$ and $D$. Prove that the tangents from $C$ and $D$ to $\gamma$ are concurrent on line $l$.
9 replies
Drytime
May 11, 2012
Autistic_Turk
3 hours ago
Quadratic
Rushil   8
N 3 hours ago by SomeonecoolLovesMaths
Source: Indian RMO 2004 Problem 3
Let $\alpha$ and $\beta$ be the roots of the equation $x^2 + mx -1 = 0$ where $m$ is an odd integer. Let $\lambda _n = \alpha ^n + \beta ^n , n \geq 0$
Prove that
(A) $\lambda _n$ is an integer
(B) gcd ( $\lambda _n , \lambda_{n+1}$) = 1 .
8 replies
Rushil
Feb 28, 2006
SomeonecoolLovesMaths
3 hours ago
$n^{22}-1$ and $n^{40}-1$
v_Enhance   6
N 3 hours ago by BossLu99
Source: OTIS Mock AIME 2024 #13
Let $S$ denote the sum of all integers $n$ such that $1 \leq n \leq 2024$ and exactly one of $n^{22}-1$ and $n^{40}-1$ is divisible by $2024$. Compute the remainder when $S$ is divided by $1000$.

Raymond Zhu

6 replies
v_Enhance
Jan 16, 2024
BossLu99
3 hours ago
Parallelogram in the Plane
Taco12   8
N 3 hours ago by lpieleanu
Source: 2023 Canada EGMO TST/2
Parallelogram $ABCD$ is given in the plane. The incircle of triangle $ABC$ has center $I$ and is tangent to diagonal $AC$ at $X$. Let $Y$ be the center of parallelogram $ABCD$. Show that $DX$ and $IY$ are parallel.
8 replies
Taco12
Feb 10, 2023
lpieleanu
3 hours ago
Combinatorial
|nSan|ty   7
N 3 hours ago by SomeonecoolLovesMaths
Source: RMO 2007 problem
How many 6-digit numbers are there such that-:
a)The digits of each number are all from the set $ \{1,2,3,4,5\}$
b)any digit that appears in the number appears at least twice ?
(Example: $ 225252$ is valid while $ 222133$ is not)
[weightage 17/100]
7 replies
|nSan|ty
Oct 10, 2007
SomeonecoolLovesMaths
3 hours ago
pairs (m, n) such that a fractional expression is an integer
cielblue   0
4 hours ago
Find all pairs $(m,\ n)$ of positive integers such that $\frac{m^3-mn+1}{m^2+mn+2}$ is an integer.
0 replies
cielblue
4 hours ago
0 replies
the same prime factors
andria   6
N 4 hours ago by MathLuis
Source: Iranian third round number theory P4
$a,b,c,d,k,l$ are positive integers such that for every natural number $n$ the set of prime factors of $n^k+a^n+c,n^l+b^n+d$ are same. prove that $k=l,a=b,c=d$.
6 replies
andria
Sep 6, 2015
MathLuis
4 hours ago
4 lines concurrent
Zavyk09   7
N May 2, 2025 by bin_sherlo
Source: Homework
Let $ABC$ be triangle with circumcenter $(O)$ and orthocenter $H$. $BH, CH$ intersect $(O)$ again at $K, L$ respectively. Lines through $H$ parallel to $AB, AC$ intersects $AC, AB$ at $E, F$ respectively. Point $D$ such that $HKDL$ is a parallelogram. Prove that lines $KE, LF$ and $AD$ are concurrent at a point on $OH$.
7 replies
Zavyk09
Apr 9, 2025
bin_sherlo
May 2, 2025
4 lines concurrent
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Source: Homework
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Zavyk09
16 posts
#1 • 1 Y
Y by PikaPika999
Let $ABC$ be triangle with circumcenter $(O)$ and orthocenter $H$. $BH, CH$ intersect $(O)$ again at $K, L$ respectively. Lines through $H$ parallel to $AB, AC$ intersects $AC, AB$ at $E, F$ respectively. Point $D$ such that $HKDL$ is a parallelogram. Prove that lines $KE, LF$ and $AD$ are concurrent at a point on $OH$.
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aidenkim119
34 posts
#2 • 1 Y
Y by PikaPika999
............
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aidenkim119
34 posts
#3 • 1 Y
Y by PikaPika999
First three are trivial by pascal, but AD looks a bit hard / '
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ItzsleepyXD
149 posts
#4 • 1 Y
Y by PikaPika999
Redefine
Let $ABC$ be triangle with circumcenter $(O)$ and orthocenter $H$. $A'$ is antipode of $A$ . $O'$ is circumcenter of $(BOC)$ . Point $E,F$ satisfied $OE // A'F // AB , OF // A'E // AC$ then prove $OH,O'A',BE,CF$ concurrent .

Let $B',C'$ be antipode of $B,C$ respectively.
MMP: Fix $(O),B,C$ . Move $A$ on $(O)$ deg 2.
Since $A',E,C'$ collinear and $A',F,B'$ collinear
By $\angle C'EO = \angle BAC = \angle BC'C = \angle C'BO$ so $C',B,O,E$ concyclic.
implies that $E$ deg 2. Also $F$ deg 2.
So line $BE,CF$ deg 1.
$H=$ reflection of $A \infty_{\perp BC} \cap (O)$ across $BC$
Since $H,A'$ deg 2. implies that line $O'A'.OH$ deg 2.
We want to prove $BE,CF,OH$ concurrent first and $BE,CF,O'A'$ concurrent.
but both have deg 1+1+2+1 = 5 .

choose $A= B,C,B',C'$ and midpoint of arc $BC$
the rest of problem is easy. $\square$
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pingupignu
49 posts
#5 • 1 Y
Y by PikaPika999
My solution may not be elegant but here's some DDIT spam for you guys to enjoy :blush:.
Let $X = KE \cap LF$. I will show that $X, A, D$ and $X, O, H$ are collinear.

Part 1:
I first claim that $\angle \infty_{CH}XL = \angle \infty_{BH}XK$. This follows from
$$\angle \infty_{CH}XL = \angle FLH = \angle FHL = 90^\circ - \angle BHL = 90^\circ - \angle CHK = \angle EHK = \angle XKH = \angle \infty_{BH}XK.$$Then, applying DDIT on $X \cup LDKH$ we see that $$(XK, XL), (XH, XD), (X\infty_{BH}, X\infty_{CH})$$are reciprocal pairs under some involution on $\mathcal{P}_X$. This involution must be a reflection in the angle bisector of $\angle KXL$. Hence $XH, XD$ are isogonal in $\angle KXL$.

Next, since $AK=AL$ (well-known), $LF=FH=AE$, $AF=EH=EK$, we yield $\triangle LAF \cong \triangle AKE$.
I claim that $X\infty_{AB}, X\infty_{AC}$ are isogonal in $\angle XKL$. This is because $$\angle \infty_{AB}XF = 180^\circ - \angle AFL = 180^\circ - \angle AEK = \angle AEX = \angle \infty_{AC}XE.$$Applying DDIT on $X \cup AEHF$ would then give $XA, XH$ are isogonal in $\angle EXF$. Since $XH, XD$, $XH, XA$ are isogonal in $\angle KXL = \angle EXF$ we conclude that $XA \equiv XD$, or $X \in AD$.

Part 2:
I first prove that $X\infty_{CH}, OL, AK$ concur at a point $S$ on $(XLK)$. For this, let $S = X\infty_{CH} \cap OL$, where from a short angle chase we get $$\angle XSL = \angle OLH = B-A = (90^\circ - A) - (90^\circ - B) = \angle AKL - \angle LCB$$$$= \angle AKL - \angle LAF = \angle AKL - \angle  AKE = \angle EKL = \angle XKL$$Hence $XSKL$ are cyclic, and from
$$\angle SKX = \angle SLX = \angle FLO = \angle ALO - \angle ALF = A - \angle EAK = A - (90^\circ - C)$$which equals
$$= A+C-90^\circ = 90^\circ - B = \angle LAB = \angle AKE = \angle AKX$$$\implies S \in AK$. The claim is proven.

From DDIT in $X \cup AKOL$ we have the reciprocal pairs $$(XA, XO), (XK, XL), (XS, XT)$$where $T = AL \cap KO \cap X \infty_{BH}$ (similarly).
since $(XS, XT) = (X\infty_{CH}, X\infty_{BH})$ and we have established $(XA, XH), (X\infty_{CH}, X\infty_{BH})$ are isogonal in $\angle KXL$ we get $XH \equiv XO \implies X \in OH$. The problem is solved. $\blacksquare$
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tomsuhapbia
6 posts
#6 • 1 Y
Y by Amkan2022
We need two well-known lemmas about the isogonal line:
1. Given a $\triangle ABC$ and a point $P$ satisfied $\angle ABP=\angle ACP$. Let $Q$ be the reflection of $P$ in the midpoint of $BC$. Then $AP$ and $AQ$ are isogonals wrt $\angle BAC$.
2. Given a trapezoid $ABCD$ $(AB\parallel CD)$ is inscribed $(O)$. Let $E,F$ be the intersections of $BC$ and $AD$; $AC$ and $BD$. Let $S$ be a abitary point on $(O)$. Then $SE,SG$ are isogonals wrt $\angle ASB$.

Back to the problem: Let $X,Y$ be the intersections of $OL$ and $AK$; $AL$ and $OK$. By symmetric, we have $AK=AH=AL$ so $AO$ is the perpendicular bisector of $KL$ then $XKLY$ is a trapezoid. Let $LK$ intersects $KE$ at $Z$.

We have
$$\angle FZK=\angle ZEH-\angle ZFC=180^\circ-\angle KEH-\angle HFC=180^\circ-3\angle BAC$$and
$$\angle LYK=\angle AOK-\angle OAL=2(90^\circ-\angle OAL)-\angle OAL=180^\circ-3\angle OAL=180^\circ-3\angle BAC=\angle FZK\,(2)$$since
$$\angle OAL=\dfrac{1}{2}\angle KAL=\dfrac{1}{2}(\angle KAB+\angle BAH+\angle HAC+\angle CAL)=\angle BAC$$$(2)$ leads to $Z$ lies on the circumcircle of $XKLY$. From the lemma 2, we obtain that $ZA,ZO$ are isogonals wrt $\angle LZK$ $(1)$. We also have $\angle ZFE=\angle LFE=\angle KEH=180^\circ-\angle ZEH$ and $\angle HEZ=\angle HKE=\angle FLH=180^\circ-\angle HLZ$, using lemma 1 and get $(XD,XH)$ and $(XH,XA)$ are two isogonal pairs wrt $\angle LZK\equiv\angle FZE$, so $A,D,Z$ are collinear. Combine with $(1)$ and we conclude $Z,O,H$ are collinear or $AD,LF,KE,OH$ are concurrent at $Z$.

https://i.postimg.cc/hJQdTNdW/image.png
This post has been edited 2 times. Last edited by tomsuhapbia, May 1, 2025, 5:25 PM
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hectorleo123
347 posts
#7
Y by
I apologize for the complex bash, but I couldn't find another way.

Let \( B' \) and \( C' \) be the antipodes of \( B \) and \( C \).
Since \( L \) is the reflection of \( H \) over \( AB \), we have \( \angle FLB = \angle FHB = 90^\circ \) (since \( FH \parallel AC \perp BH \)).
Analogously, \( \angle EKC = 90^\circ \).
\(\Rightarrow B', F, L \) are collinear and \( C', E, K \) are collinear.
By Pascal's Theorem on
\[ \binom{B, L, C'}{C, K, B'} \]we get that \( KE, LF \), and \( OH \) are concurrent.
Now it suffices to prove that \( AD, KE \), and \( LF \) are concurrent.

We use complex numbers, where \( (ABC) \) is the unit circle and \( a = 1 \).
Let
\[
k = -\frac{c}{b}, \quad l = -\frac{b}{c}, \quad c' = -c, \quad b' = -b, \quad h = b + c + 1, \quad o = 0
\]\[
d + b + c + 1 = d + h = l + k = -\frac{b}{c} - \frac{c}{b}
\]\[
\Rightarrow d = -\frac{b^2 + c^2 + b^2c + bc^2 + bc}{bc}
\]
Let \( X = KC' \cap LB' \).
We have:
\[
\frac{x + c}{\overline{x + c}} = \frac{c - \frac{c}{b}}{\overline{c - \frac{c}{b}}} = -\frac{c^2}{b}
\Rightarrow \overline{x} = -\frac{xb + bc + c}{c^2}
\]Analogously,
\[
\overline{x} = -\frac{xc + bc + b}{b^2}
\]Equating both expressions:
\[
(b - c)(b^2c + bc^2 + bc - x(b^2 + bc + c^2)) = 0
\Rightarrow x = -\frac{bc(b + c + 1)}{b^2 + bc + c^2}
\]
Points \( A, D, X \) are collinear if and only if
\[
\frac{d - 1}{x - 1} \in \mathbb{R}
\]Substituting:
\[
\frac{(b^2 + c^2 + b^2c + bc^2 + 2bc)/bc}{(b^2c + bc^2 + 2bc + b^2 + c^2)/(b^2 + bc + c^2)} = \frac{b^2 + bc + c^2}{bc}\in \mathbb{R}_\blacksquare
\]
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bin_sherlo
733 posts
#8
Y by
Let $B',C'$ be the antipodes of $B,C$ on $(ABC)$. Also let $LC'\cap KB'=W,KC'\cap LB'=P$.
Claim: $B',F,L$ and $C',E,K$ are collinear.
Proof: Pascal at $KB'LCAB$ yields $AC_{\infty},B'L\cap AB,H$ are collinear thus, $B'L\cap AB=F$. Similarily $C',E,K$ are collinear.
Claim: $P$ lies on $OH$.
Proof: Pascal at $BKC'CLB'$ gives $H,P,O$ are collinear.
Claim: $A,D,P$ are collinear.
Proof: Notice that $W,L,H,K$ lie on the circle with diameter $WH$ and since $AH=AK=AL$, $A$ must be the circumcenter of $(KLHW)$. Hence $W,A,H$ are collinear.
DDIT at $DLHK$ implies $(\overline{AD},\overline{AH}),(\overline{AK},\overline{AL}),(\overline{AC'},\overline{AB'})$ is an involution. DDIT at $B'KC'L$ gives $(\overline{AP},\overline{AH}),(\overline{AK},\overline{AL}),(\overline{AC'},\overline{AB'})$ is an involution. Combining these implies $AD\equiv AP$ hence $A,D,P$ are collinear as desired.$\blacksquare$
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