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AT // BC wanted
parmenides51   110
N an hour ago by YaoAOPS
Source: IMO 2019 SL G1
Let $ABC$ be a triangle. Circle $\Gamma$ passes through $A$, meets segments $AB$ and $AC$ again at points $D$ and $E$ respectively, and intersects segment $BC$ at $F$ and $G$ such that $F$ lies between $B$ and $G$. The tangent to circle $BDF$ at $F$ and the tangent to circle $CEG$ at $G$ meet at point $T$. Suppose that points $A$ and $T$ are distinct. Prove that line $AT$ is parallel to $BC$.

(Nigeria)
110 replies
parmenides51
Sep 22, 2020
YaoAOPS
an hour ago
Inspired by old results
sqing   1
N an hour ago by sqing
Source: Own
Let $ a,\,b,\,c>0 .$ Prove that
$$\frac{a^3+b^3+c^3+5abc}{(a+2b)(b+2c)(c+2a)}\geq \frac{8}{27}$$$$\frac{a^3+b^3+c^3+\frac{59}{10}abc}{(a+2b)(b+2c)(c+2a)}\geq \frac{89}{270}$$

1 reply
sqing
an hour ago
sqing
an hour ago
areas of rectangles by n x k lines on a rectangle
parmenides51   1
N an hour ago by Gggvds1
Source: (2021-) 2022 XV 15th Dürer r Math Competition Finals Day 1 E5 E+2
Annie drew a rectangle and partitioned it into $n$ rows and $k$ columns with horizontal and vertical lines. Annie knows the area of the resulting $n \cdot k$ little rectangles while Benny does not. Annie reveals the area of some of these small rectangles to Benny. Given $n$ and $k$ at least how many of the small rectangle’s areas did Annie have to reveal, if from the given information Benny can determine the areas of all the $n \cdot k$ little rectangles?
For example in the case $n = 3$ and $k = 4$ revealing the areas of the $10$ small rectangles if enough information to find the areas of the remaining two little rectangles.
IMAGE
1 reply
parmenides51
Nov 29, 2022
Gggvds1
an hour ago
k-triangular sets
navi_09220114   1
N an hour ago by flower417477
Source: TASIMO 2025 Day 2 Problem 6
For an integer $k\geq 1$, we call a set $\mathcal{S}$ of $n\geq k$ points in a plane $k$-triangular if no three of them lie on the same line and whenever at most $k$ (possibly zero) points are removed from $\mathcal{S}$, the convex hull of the resulting set is a non-degenerate triangle. For given positive integer $k$, find all integers $n\geq k$ such that there exists a $k$-triangular set consisting of $n$ points.

Note. A set of points in a Euclidean plane is defined to be convex if it contains the line segments connecting each pair of its points. The convex hull of a shape is the smallest convex set that contains it.
1 reply
1 viewing
navi_09220114
May 19, 2025
flower417477
an hour ago
Game theory in 2025???
Iveela   1
N an hour ago by flower417477
Source: IMSC 2025 P3
Alice and Bob play a game on a $K_{2026}$. They take turns with Alice playing first. Initially all edges are uncolored. On Alice's turn she chooses any uncolored edge and colors it red. On Bob's turn, he chooses 1, 2, or 3 uncolored edges and colors them blue. The game ends once all edges have been colored. Let $r$ and $b$ be the number of vertices of the largest red and blue clique, respectively. Bob wins if at the end of the game $b > r$. Show that Bob has a winning strategy.

Note: $K_n$ denotes the complete graph with $n$ vertices and where there is an edge between any pair of vertices. A red (or blue, respectively) clique refers to a complete subgraph of which all the edges are red (or blue, respectively).
1 reply
Iveela
Jul 5, 2025
flower417477
an hour ago
A number theory problem
MathMaxGreat   1
N an hour ago by Assassino9931
Source: 2025 Summer NSMO
Prove there exists finite numbers of $(a,b),s.t.$ $a^{[\sqrt{a}]}-1=b!$
1 reply
MathMaxGreat
5 hours ago
Assassino9931
an hour ago
Application of contraction mapping to the limit of a sequence
gia_russell   0
2 hours ago
Consider the sequence $(x_n)$ defined by:

$$
\begin{cases}
x_1 = 3 \\
x_{n+1} = \dfrac{1}{3}\left(2x_n + \dfrac{3}{x_n^2} \right) \quad \forall n \ge 1
\end{cases}
$$
Prove that the sequence $(x_n)$ converges.
Find $\lim x_n$.
0 replies
gia_russell
2 hours ago
0 replies
A geometry problem
MathMaxGreat   0
2 hours ago
Source: 2025 Summer NSMO
In $\triangle ABC$, there’s two points $X,Y$ in $\angle BAC$, $Y$ is on $\odot ABC$, $\angle BAX= \angle CAY$.
$F$ is on $AY$, s.t. $AC$ bisect the external angle of $\angle BCF$. $E$ is on $AC$, s.t. $AC=CE$.
$\odot (BFY),\odot (ACY)$ intersect at $T$. Prove: $d_{T\rightarrow AX}=| d_{B\rightarrow AX} -d_{C\rightarrow AX}|$
0 replies
MathMaxGreat
2 hours ago
0 replies
another geo in IMSC
ZeroHero   5
N 2 hours ago by Assassino9931
Given acute angled triangle $ABC$ with orthocenter $H$. Let $M$ be midpoint of arc $BC$ that containing $A$. Assume that $BH$ meet $MC$ at $F$ similarly $CH$ meet $MB$ at $E$, and $EF$ meet $BC$ at $K$. If $L$ is foot of perpendicular dropped from $M$ to line $KH$, prove that $AH=AL$.
5 replies
ZeroHero
Jul 6, 2025
Assassino9931
2 hours ago
Number theory
ZeroHero   2
N 2 hours ago by Assassino9931
Source: Interesting
Let $a$ and $b$ be integers with $a>b\geq 5$. Prove that there exist a positive integer $k$ and positive integers $c_{1},c_{2},... ,c_{k}$ with $c_{1}=a$ and $c_{k}=b$, such that $c_{i}^2+c_{i+1}^2$ is a perfect square for all $1\leq i<k$
2 replies
ZeroHero
Jun 29, 2025
Assassino9931
2 hours ago
a