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AT // BC wanted
parmenides51 110
N
an hour ago
by YaoAOPS
Source: IMO 2019 SL G1
Let
be a triangle. Circle
passes through
, meets segments
and
again at points
and
respectively, and intersects segment
at
and
such that
lies between
and
. The tangent to circle
at
and the tangent to circle
at
meet at point
. Suppose that points
and
are distinct. Prove that line
is parallel to
.
(Nigeria)






















(Nigeria)
110 replies
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
rows and
columns with horizontal and vertical lines. Annie knows the area of the resulting
little rectangles while Benny does not. Annie reveals the area of some of these small rectangles to Benny. Given
and
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
little rectangles?
For example in the case
and
revealing the areas of the
small rectangles if enough information to find the areas of the remaining two little rectangles.
IMAGE






For example in the case



IMAGE
1 reply

k-triangular sets
navi_09220114 1
N
an hour ago
by flower417477
Source: TASIMO 2025 Day 2 Problem 6
For an integer
, we call a set
of
points in a plane
-triangular if no three of them lie on the same line and whenever at most
(possibly zero) points are removed from
, the convex hull of the resulting set is a non-degenerate triangle. For given positive integer
, find all integers
such that there exists a
-triangular set consisting of
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.










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
Game theory in 2025???
Iveela 1
N
an hour ago
by flower417477
Source: IMSC 2025 P3
Alice and Bob play a game on a
. 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
and
be the number of vertices of the largest red and blue clique, respectively. Bob wins if at the end of the game
. Show that Bob has a winning strategy.
Note:
denotes the complete graph with
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).




Note:


1 reply
A number theory problem
MathMaxGreat 1
N
an hour ago
by Assassino9931
Source: 2025 Summer NSMO
Prove there exists finite numbers of

![$a^{[\sqrt{a}]}-1=b!$](http://latex.artofproblemsolving.com/f/7/e/f7e9c99c1c6be70ab50714e8fbabf7a02227d2d4.png)
1 reply
Application of contraction mapping to the limit of a sequence
gia_russell 0
2 hours ago
Consider the sequence
defined by:

Prove that the sequence
converges.
Find
.


Prove that the sequence

Find

0 replies
A geometry problem
MathMaxGreat 0
2 hours ago
Source: 2025 Summer NSMO
In
, there’s two points
in
,
is on
,
.
is on
, s.t.
bisect the external angle of
.
is on
, s.t.
.
intersect at
. Prove:
















0 replies
another geo in IMSC
ZeroHero 5
N
2 hours ago
by Assassino9931
Given acute angled triangle
with orthocenter
. Let
be midpoint of arc
that containing
. Assume that
meet
at
similarly
meet
at
, and
meet
at
. If
is foot of perpendicular dropped from
to line
, prove that
.


















5 replies
Number theory
ZeroHero 2
N
2 hours ago
by Assassino9931
Source: Interesting
Let
and
be integers with
. Prove that there exist a positive integer
and positive integers
with
and
, such that
is a perfect square for all









2 replies
