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A Projection Theorem
buratinogigle   1
N an hour ago by aidan0626
Source: VN Math Olympiad For High School Students P1 - 2025
In triangle $ABC$, prove that
\[ a = b\cos C + c\cos B. \]
1 reply
buratinogigle
2 hours ago
aidan0626
an hour ago
J
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Interesting inequalities
sqing   4
N an hour ago by sqing
Source: Own
Let $ a,b $ be reals such that $ a^2+ab+b^2 =1$ . Prove that
$$ \frac{8}{ 5 }> \frac{1}{ a^2+1 }+ \frac{1}{ b^2+1 } \geq 1$$$$ \frac{9}{ 5 }\geq\frac{1}{ a^4+1 }+ \frac{1}{ b^4+1 } \geq 1$$$$ \frac{27}{ 14 }\geq \frac{1}{ a^6+1 }+ \frac{1}{ b^6+1 } \geq 1$$Let $ a,b $ be reals such that $ a^2+ab+b^2 =3$ . Prove that
$$ \frac{13}{ 10 }> \frac{1}{ a^2+1 }+ \frac{1}{ b^2+1 } \geq \frac{1}{ 2 }$$$$ \frac{6}{ 5 }>\frac{1}{ a^4+1 }+ \frac{1}{ b^4+1 } \geq \frac{1}{ 5 }$$$$ \frac{1}{ a^6+1 }+ \frac{1}{ b^6+1 } \geq \frac{1}{ 14 }$$
4 replies
sqing
Yesterday at 8:32 AM
sqing
an hour ago
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Ant wanna come to A
Rohit-2006   3
N an hour ago by Rohit-2006
An insect starts from $A$ and in $10$ steps and has to reach $A$ again. But in between one of the s steps and can't go $A$. Find probability. For example $ABCDCDEDEA$ is valid but $ABABCDEABA$ is not valid.

*Too many edits, my brain had gone to a trip
3 replies
Rohit-2006
Yesterday at 6:47 PM
Rohit-2006
an hour ago
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BMO Shortlist 2021 A5
Lukaluce   16
N an hour ago by Sedro
Source: BMO Shortlist 2021
Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that
$$f(xf(x + y)) = yf(x) + 1$$holds for all $x, y \in \mathbb{R}^{+}$.

Proposed by Nikola Velov, North Macedonia
16 replies
Lukaluce
May 8, 2022
Sedro
an hour ago
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Why is the old one deleted?
EeEeRUT   2
N an hour ago by ItzsleepyXD
Source: EGMO 2025 P1
For a positive integer $N$, let $c_1 < c_2 < \cdots < c_m$ be all positive integers smaller than $N$ that are coprime to $N$. Find all $N \geqslant 3$ such that $$\gcd( N, c_i + c_{i+1}) \neq 1$$for all $1 \leqslant i \leqslant m-1$

Here $\gcd(a, b)$ is the largest positive integer that divides both $a$ and $b$. Integers $a$ and $b$ are coprime if $\gcd(a, b) = 1$.
2 replies
EeEeRUT
2 hours ago
ItzsleepyXD
an hour ago
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Set summed with itself
Math-Problem-Solving   0
an hour ago
Source: Awesomemath Sample Problems
Let $A = \{1, 4, \ldots, n^2\}$ be the set of the first $n$ perfect squares of nonzero integers. Suppose that $A \subset B + B$ for some $B \subset \mathbb{Z}$. Here $B + B$ stands for the set $\{b_1 + b_2 : b_1, b_2 \in B\}$. Prove that $|B| \geq |A|^{2/3 - \epsilon}$ holds for every $\epsilon > 0$.
0 replies
Math-Problem-Solving
an hour ago
0 replies
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A Problem on a Rectangle
buratinogigle   0
2 hours ago
Source: VN Math Olympiad For High School Students P12 - 2025 - Bonus, MM Problem 2197
Let $ABCD$ be a rectangle and $P$ any point. Let $X, Y, Z, W, S, T$ be the foots of the perpendiculars from $P$ to the lines $AB, BC, CD, DA, AB, BD$, respectively. Let the perpendicular bisectors of $XY$ and $WZ$ intersect at $Q$, and those of $YZ$ and $XW$ intersect at $R$. Prove that the lines $QR$ and $ST$ are parallel.

MM Problem
0 replies
buratinogigle
2 hours ago
0 replies
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The difference of the two angles is 180 degrees
buratinogigle   0
2 hours ago
Source: VN Math Olympiad For High School Students P11 - 2025
In triangle $ABC$, let $D$ be the midpoint of $AB$, and $E$ the midpoint of $CD$. Suppose $\angle ACD = 2\angle DEB$. Prove that
\[ 2\angle AED-\angle DCB =180^\circ. \]
0 replies
buratinogigle
2 hours ago
0 replies
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A Segment Bisection Problem
buratinogigle   0
2 hours ago
Source: VN Math Olympiad For High School Students P9 - 2025
In triangle $ABC$, let the incircle $\omega$ touch sides $BC, CA, AB$ at $D, E, F$, respectively. Let $P$ lie on the line through $D$ perpendicular to $BC$. Let $Q, R$ be the intersections of $PC, PB$ with $EF$, respectively. Let $K, L$ be the projections of $R, Q$ onto line $BC$. Let $M, N$ be the second intersections of $DQ, DR$ with the incircle $\omega$. Let $S$ be the intersection of $KM$ and $LN$. Prove that the line $DS$ bisects segment $QR$.
0 replies
buratinogigle
2 hours ago
0 replies
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A Characterization of Rectangles
buratinogigle   0
2 hours ago
Source: VN Math Olympiad For High School Students P8 - 2025
Prove that if a convex quadrilateral $ABCD$ satisfies the equation
\[ (AB + CD)^2 + (AD + BC)^2 = (AC + BD)^2, \]then $ABCD$ must be a rectangle.
0 replies
buratinogigle
2 hours ago
0 replies
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A Generalization of Ptolemy's Theorem
buratinogigle   0
2 hours ago
Source: VN Math Olympiad For High School Students P7 - 2025
Given a convex quadrilateral $ABCD$, define
\[ \alpha = |\angle ADB - \angle ACB| = |\angle DAC - \angle DBC| \quad\text{and}\quad \beta = |\angle ABD - \angle ACD| = |\angle BAC - \angle BDC|. \]Prove that
\[ AC \cdot BD = AD \cdot BC \cos\alpha + AB \cdot CD \cos\beta. \]
0 replies
buratinogigle
2 hours ago
0 replies
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parallel
phuong   2
N Sep 17, 2020 by jayme
Source: unsolved geometry
Let ABC be a triangle and its incircle touches BC,CA,AB at D,E, F, reps. AD cuts (I) again at G. H lies on line EF such that $GH \perp AD$. Prove that $AH \parallel BC$.
2 replies
phuong
Sep 26, 2016
jayme
Sep 17, 2020
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parallel
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Reveal topic tags
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G H BBookmark kLocked kLocked NReply
Source: unsolved geometry
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phuong
210 posts
phuong
#1 h  Sep 26, 2016, 7:11 AM • 2 Y
Y by Adventure10, Mango247
Let ABC be a triangle and its incircle touches BC,CA,AB at D,E, F, reps. AD cuts (I) again at G. H lies on line EF such that $GH \perp AD$. Prove that $AH \parallel BC$.
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jayme
9775 posts
jayme
#2 h  Sep 26, 2016, 11:46 AM • 1 Y
Y by Adventure10
Dear Mathlinkers,
have a look at
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=111468

Sincerely
Jean-Louis
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jayme
9775 posts
jayme
#3 h  Sep 17, 2020, 12:53 PM
Y by
Dear Mathlinkers,

http://jl.ayme.pagesperso-orange.fr/Docs/01.%201.%20Parallele%20a%20un%20cote%20du%20triangle%20passant%20par%20un%20sommet%20du%20triangle.pdf

p. 6-7.

Sincerely
Jean-Louis
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