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PQ = r and 6 more conditions
avisioner   40
N 22 minutes ago by wu2481632
Source: 2023 ISL G2
Let $ABC$ be a triangle with $AC > BC,$ let $\omega$ be the circumcircle of $\triangle ABC,$ and let $r$ be its radius. Point $P$ is chosen on $\overline{AC}$ such taht $BC=CP,$ and point $S$ is the foot of the perpendicular from $P$ to $\overline{AB}$. Ray $BP$ mets $\omega$ again at $D$. Point $Q$ is chosen on line $SP$ such that $PQ = r$ and $S,P,Q$ lie on a line in that order. Finally, let $E$ be a point satisfying $\overline{AE} \perp \overline{CQ}$ and $\overline{BE} \perp \overline{DQ}$. Prove that $E$ lies on $\omega$.
40 replies
avisioner
Jul 17, 2024
wu2481632
22 minutes ago
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this hAOpefully shoudn't BE weird
popop614   47
N an hour ago by Tsikaloudakis
Source: 2023 IMO Shortlist G1
Let $ABCDE$ be a convex pentagon such that $\angle ABC = \angle AED = 90^\circ$. Suppose that the midpoint of $CD$ is the circumcenter of triangle $ABE$. Let $O$ be the circumcenter of triangle $ACD$.

Prove that line $AO$ passes through the midpoint of segment $BE$.
47 replies
popop614
Jul 17, 2024
Tsikaloudakis
an hour ago
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Symmetric FE
Phorphyrion   9
N an hour ago by MuradSafarli
Source: 2023 Israel TST Test 7 P1
Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that for all $x, y\in \mathbb{R}$ the following holds:
\[f(x)+f(y)=f(xy)+f(f(x)+f(y))\]
9 replies
Phorphyrion
May 9, 2023
MuradSafarli
an hour ago
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Diagonals BD,CE concurrent with diameter AO in cyclic ABCDE
WakeUp   11
N an hour ago by zhoujef000
Source: Romanian TST 2002
Let $ABCDE$ be a cyclic pentagon inscribed in a circle of centre $O$ which has angles $\angle B=120^{\circ},\angle C=120^{\circ},$ $\angle D=130^{\circ},\angle E=100^{\circ}$. Show that the diagonals $BD$ and $CE$ meet at a point belonging to the diameter $AO$.

Dinu Șerbănescu
11 replies
WakeUp
Feb 5, 2011
zhoujef000
an hour ago
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Easy Geometry Problem in Taiwan TST
chengbilly   5
N 2 hours ago by Tamam
Source: 2025 Taiwan TST Round 1 Independent Study 2-G
Suppose $I$ and $I_A$ are the incenter and the $A$-excenter of triangle $ABC$, respectively.
Let $M$ be the midpoint of arc $BAC$ on the circumcircle, and $D$ be the foot of the
perpendicular from $I_A$ to $BC$. The line $MI$ intersects the circumcircle again at $T$ . For
any point $X$ on the circumcircle of triangle $ABC$, let $XT$ intersect $BC$ at $Y$ . Prove
that $A, D, X, Y$ are concyclic.
5 replies
chengbilly
Mar 6, 2025
Tamam
2 hours ago
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one cyclic formed by two cyclic
CrazyInMath   32
N 2 hours ago by kamatadu
Source: EGMO 2025/3
Let $ABC$ be an acute triangle. Points $B, D, E$, and $C$ lie on a line in this order and satisfy $BD = DE = EC$. Let $M$ and $N$ be the midpoints of $AD$ and $AE$, respectively. Suppose triangle $ADE$ is acute, and let $H$ be its orthocentre. Points $P$ and $Q$ lie on lines $BM$ and $CN$, respectively, such that $D, H, M,$ and $P$ are concyclic and pairwise different, and $E, H, N,$ and $Q$ are concyclic and pairwise different. Prove that $P, Q, N,$ and $M$ are concyclic.
32 replies
1 viewing
CrazyInMath
Apr 13, 2025
kamatadu
2 hours ago
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powers sums and triangular numbers
gaussious   3
N 2 hours ago by RagvaloD
prove 1^k+2^k+3^k + \cdots + n^k \text{is divisible by } \frac{n(n+1)}{2} \text{when} k \text{is odd}
3 replies
gaussious
6 hours ago
RagvaloD
2 hours ago
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Beautiful geometry
m4thbl3nd3r   0
2 hours ago
Let $\omega$ be the circumcircle of triangle $ABC$, $M$ is the midpoint of $BC$ and $E$ be the second intersection of $AM$ and $\omega$. Tangent line of $\omega$ at $E$ intersects $BC$ at $P$, let $PKL$ be a transversal of $\omega$ and $X,Y$ be intersections of $AK,AL$ with $BC$. Let $PF$ be a tangent line of $\omega$. Prove that $LYFP$ is cyclic
0 replies
m4thbl3nd3r
2 hours ago
0 replies
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prove |a-b| is a square, given a-b=a^2c-b^2d
Alpha314159   4
N 2 hours ago by Leman_Nabiyeva
Source: Macau Inter High School Competition
Let $a, b$ be integers such that there are consecutive integers $c,d$ satisfy $$a-b=a^2 c-b^2 d$$.
Prove : $|a-b|$ is a perfect square.
4 replies
1 viewing
Alpha314159
Mar 7, 2020
Leman_Nabiyeva
2 hours ago
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Inspired by pennypc123456789
sqing   0
2 hours ago
Source: Own
Let $ a,b,c\geq 0 $and $ab+bc+ca\neq 0.$ Prove that
$$ \frac{9-8\sqrt 2+5\sqrt 5}{6}\leq\frac{a + b}{a + 2b + c} + \dfrac{b + c}{b + 2c + a}+\dfrac{c + a}{c + 2a + b}\leq \frac{9+8\sqrt 2-5\sqrt 5}{6}$$
0 replies
sqing
2 hours ago
0 replies
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The Bank of Bath
TelMarin   100
N 2 hours ago by Ihatecombin
Source: IMO 2019, problem 5
The Bank of Bath issues coins with an $H$ on one side and a $T$ on the other. Harry has $n$ of these coins arranged in a line from left to right. He repeatedly performs the following operation: if there are exactly $k>0$ coins showing $H$, then he turns over the $k$th coin from the left; otherwise, all coins show $T$ and he stops. For example, if $n=3$ the process starting with the configuration $THT$ would be $THT \to HHT \to HTT \to TTT$, which stops after three operations.

(a) Show that, for each initial configuration, Harry stops after a finite number of operations.

(b) For each initial configuration $C$, let $L(C)$ be the number of operations before Harry stops. For example, $L(THT) = 3$ and $L(TTT) = 0$. Determine the average value of $L(C)$ over all $2^n$ possible initial configurations $C$.

Proposed by David Altizio, USA
100 replies
TelMarin
Jul 17, 2019
Ihatecombin
2 hours ago
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pretty well known
dotscom26   2
N Apr 4, 2025 by Giant_PT
Let $\triangle ABC$ be a scalene triangle such that $\Omega$ is its incircle. $AB$ is tangent to $\Omega$ at $D$. A point $E$ ($E \notin \Omega$) is located on $BC$.

Let $\omega_1$, $\omega_2$, and $\omega_3$ be the incircles of the triangles $BED$, $ADE$, and $AEC$, respectively.

Show that the common tangent to $\omega_1$ and $\omega_3$ is also tangent to $\omega_2$.

2 replies
dotscom26
Apr 3, 2025
Giant_PT
Apr 4, 2025
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pretty well known
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Reveal topic tags
geometry
Inversion
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dotscom26
36 posts
dotscom26
#1 h  Apr 3, 2025, 2:03 AM
Y by
Let $\triangle ABC$ be a scalene triangle such that $\Omega$ is its incircle. $AB$ is tangent to $\Omega$ at $D$. A point $E$ ($E \notin \Omega$) is located on $BC$.

Let $\omega_1$, $\omega_2$, and $\omega_3$ be the incircles of the triangles $BED$, $ADE$, and $AEC$, respectively.

Show that the common tangent to $\omega_1$ and $\omega_3$ is also tangent to $\omega_2$.
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dotscom26
36 posts
dotscom26
#2 h  Apr 3, 2025, 11:37 PM
Y by
bump....
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Giant_PT
25 posts
Giant_PT
#3 h  Apr 4, 2025, 2:13 AM • 1 Y
Y by waterbottle432
Definitely not the intended solution. Define points as shown in the diagram, and line $FG$ is the common tangent between $\omega_1$ and $\omega_3.$ Now, it suffices to prove that quadrilateral $ADMO$ has an incircle. Through length calculations, we have
$$AD+BC=AC+BD\implies AD+IJ=AK+DH\implies AD+FG=AN+DL\implies AD+MO=AO+MD.$$Clearly this proves that quadrilateral $ADMO$ has an incircle due to Pitot's theorem.
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This post has been edited 1 time. Last edited by Giant_PT, Apr 4, 2025, 2:14 AM
Reason: Typo
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