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Topic
First Poster
Last Poster
Geometry
MathsII-enjoy 1
N
by aidenkim119
Given triangle 
inscribed in 
, 
is the midpoint of arc 
of . The perpendicular bisector 
intersects 
at 
. 
intersects 
at 
different from 
. 
is the orthocenter of triangle . Prove that 
= 
inscribed in 
, 
is the midpoint of arc 
of . The perpendicular bisector 
intersects 
at 
. 
intersects 
at 
different from 
. 
is the orthocenter of triangle . Prove that 
= 
1 reply
Merlin's castle
gnoka 2
N
by Anzoteh
Source: 46th International Tournament of Towns, Senior A-Level P6, Fall 2024
Merlin's castle has 100 rooms and 1000 corridors. Each corridor links some two rooms. Each pair of rooms is linked by one corridor at most. Merlin has given out the plan of the castle to the wise men and declared the rules of the challenge. The wise men need to scatter across the rooms in a manner they wish. Each minute Merlin will choose a corridor and one of the wise men will have to pass along it from one of the rooms at its ends to the other one. Merlin wins when in both rooms on the ends of the chosen corridor there are no wise men. Let us call a number 
the magic number of the castle if wise men can pre-agree before the challenge and act in such a way that Merlin never wins, being the minimal possible number. What are the possible values of the magic number of the castle? (Merlin and all the wise men always know the location of all the wise men).
Timofey Vasilyev
the magic number of the castle if wise men can pre-agree before the challenge and act in such a way that Merlin never wins, being the minimal possible number. What are the possible values of the magic number of the castle? (Merlin and all the wise men always know the location of all the wise men).Timofey Vasilyev
2 replies
A drunk frog jumping ona grid in a weird way
Tintarn 4
N
by pi_quadrat_sechstel
Source: Baltic Way 2024, Problem 10
A frog is located on a unit square of an infinite grid oriented according to the cardinal directions. The frog makes moves consisting of jumping either one or two squares in the direction it is facing, and then turning according to the following rules:
i) If the frog jumps one square, it then turns
to the right;
ii) If the frog jumps two squares, it then turns to the left.
Is it possible for the frog to reach the square exactly
squares north of the initial square after some finite number of moves if it is initially facing:
a) North;
b) East?
i) If the frog jumps one square, it then turns
to the right;ii) If the frog jumps two squares, it then turns
to the left.Is it possible for the frog to reach the square exactly
squares north of the initial square after some finite number of moves if it is initially facing:a) North;
b) East?
4 replies
and 
. Prove that
Where 
and 
Prove that
Where 
Number Theory Chain!
JetFire008 57
N
by CHESSR1DER
I will post a question and someone has to answer it. Then they have to post a question and someone else will answer it and so on. We can only post questions related to Number Theory and each problem should be more difficult than the previous. Let's start!
?57 replies
Perpendicular bisector meets the circumcircle of another triangle
steppewolf 3
N
by Omerking
Source: 2023 Junior Macedonian Mathematical Olympiad P4
We are given an acute 
with circumcenter 
such that 
. The bisector of 
meets the circumcircle of at a second point 
. The perpendicular bisector of 
meets the circumcircle of 
for the second time at 
. The line 
meets the circumcircle of for the second time at 
. Prove that the lines 
, 
and are concurrent.
Authored by Petar Filipovski
with circumcenter 
such that 
. The bisector of 
meets the circumcircle of at a second point 
. The perpendicular bisector of 
meets the circumcircle of 
for the second time at 
. The line 
meets the circumcircle of for the second time at 
. Prove that the lines 
, 
and are concurrent.Authored by Petar Filipovski
3 replies
Quad formed by orthocenters has same area (all 7's!)
v_Enhance 35
N
by Wictro
Source: USA January TST for the 55th IMO 2014
Let 
be a cyclic quadrilateral, and let , , 
, and be the midpoints of , 
, 
, and 
respectively. Let 
, 
, 
and 
be the orthocenters of triangles 
, 
, 
and 
, respectively. Prove that the quadrilaterals and 
have the same area.
be a cyclic quadrilateral, and let , , 
, and be the midpoints of , 
, 
, and 
respectively. Let 
, 
, 
and 
be the orthocenters of triangles 
, 
, 
and 
, respectively. Prove that the quadrilaterals and 
have the same area.
35 replies
,![\[
A = \left\{ r \, \middle| \, r = \frac{a}{3}\left(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\right) + b\sqrt[3]{xyz}, \quad x, y, z \in \left[1, \frac{a}{b}\right] \right\},
\]](http://latex.artofproblemsolving.com/e/9/2/e92becb6264b8e5a9753f2288436a48523a25f29.png)
and![\[
B = \left[2\sqrt{ab}, a + b\right].
\]](http://latex.artofproblemsolving.com/2/d/2/2d25aa634389b768dc26408480bb71b316c0bf17.png)
.
A Segment Bisection Problem
buratinogigle 2
N
by aidenkim119
Source: VN Math Olympiad For High School Students P9 - 2025
In triangle , let the incircle 
touch sides 
at 
, respectively. Let 
lie on the line through perpendicular to . Let 
be the intersections of 
with 
, respectively. Let 
be the projections of 
onto line . Let 
be the second intersections of 
with the incircle . Let be the intersection of 
and 
. Prove that the line 
bisects segment 
.
, let the incircle 
touch sides 
at 
, respectively. Let 
lie on the line through perpendicular to . Let 
be the intersections of 
with 
, respectively. Let 
be the projections of 
onto line . Let 
be the second intersections of 
with the incircle . Let be the intersection of 
and 
. Prove that the line 
bisects segment 
.
2 replies
Typo in blog info
Craftybutterfly 3
N
by bpan2021
I found a typo in blog css. It is supposed to say Edit your blog's CSS in the text area below. not Edit your blog's CSS in the textarea below.
3 replies
