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set of points, there exist two lines containing n points
jasperE3 1
N
by ririgggg
Source: 2004 Brazil TST Test 2 P1
Find the smallest positive integer 
that satisfies the following condition: For every finite set of points on the plane, if for any points from this set there exist two lines containing all the points, then there exist two lines containing all points from the set.
that satisfies the following condition: For every finite set of points on the plane, if for any points from this set there exist two lines containing all the points, then there exist two lines containing all points from the set.
1 reply
XY is tangent to a fixed circle
a_507_bc 2
N
by math-olympiad-clown
Source: Baltic Way 2022/15
Let 
be a circle, and 
are two fixed points on . Given a third point 
on , let 
and 
denote the feet of the altitudes from 
and 
, respectively, in the triangle 
. Prove that there exists a fixed circle 
such that 
is tangent to regardless of the choice of the point .
be a circle, and 
are two fixed points on . Given a third point 
on , let 
and 
denote the feet of the altitudes from 
and 
, respectively, in the triangle 
. Prove that there exists a fixed circle 
such that 
is tangent to regardless of the choice of the point .
2 replies
Super easy problem
M11100111001Y1R 6
N
by sami1618
Source: Iran TST 2025 Test 2 Problem 1
The numbers from 2 to 99 are written on a board. At each step, one of the following operations is performed:
Choose a natural number 
such that 
. If both numbers and 
are on the board, erase both.
Choose a natural number such that 
. If both numbers and 
are on the board, erase both.
By performing these operations, what is the maximum number of numbers that can be erased from the board?
Choose a natural number 
such that 
. If both numbers and 
are on the board, erase both.
Choose a natural number such that 
. If both numbers and 
are on the board, erase both.By performing these operations, what is the maximum number of numbers that can be erased from the board?
6 replies
Beware the degeneracies!
Rijul saini 7
N
by Adywastaken
Source: India IMOTC 2025 Day 1 Problem 1
Let 
be real numbers satisfying 
Prove that 
and determine all cases of equality.
Proposed by Shantanu Nene
be real numbers satisfying 
Prove that 
and determine all cases of equality.Proposed by Shantanu Nene
7 replies
13th PMO Area Part 1 #17
scarlet128 1
N
by scarlet128
Source: https://pmo.ph/wp-content/uploads/2014/08/13thPMO-Area_ver5.pdf
The number x is chosen randomly from the interval (0, 1]. Define y = floor of (log base 4(x)). Find the sum of the lengths of all subintervals of (0, 1] for which y is odd.
1 reply
Romanian Geo
oVlad 3
N
by NuMBeRaToRiC
Source: Romania TST 2025 Day 1 P2
Let be a scalene acute triangle with incentre 
and circumcentre 
. Let 
cross 
at 
. On circle , let and be the mid-arc points of and 
, respectively. Let 
cross 
at 
and let 
cross 
at 
. Prove that the lines 
and 
are concurrent on the external bisector of 
.
David-Andrei Anghel
be a scalene acute triangle with incentre 
and circumcentre 
. Let 
cross 
at 
. On circle , let and be the mid-arc points of and 
, respectively. Let 
cross 
at 
and let 
cross 
at 
. Prove that the lines 
and 
are concurrent on the external bisector of 
.David-Andrei Anghel
3 replies
1 viewing
IMO 2011 Problem 5
orl 86
N
by bjump
Let 
be a function from the set of integers to the set of positive integers. Suppose that, for any two integers 
and , the difference 
is divisible by 
. Prove that, for all integers and with 
, the number 
is divisible by 
.
Proposed by Mahyar Sefidgaran, Iran
be a function from the set of integers to the set of positive integers. Suppose that, for any two integers 
and , the difference 
is divisible by 
. Prove that, for all integers and with 
, the number 
is divisible by 
.Proposed by Mahyar Sefidgaran, Iran
86 replies
11th PMO Nationals, Easy #5
scarlet128 1
N
by Mathzeus1024
Source: https://pmo.ph/wp-content/uploads/2020/12/11th-PMO-Questions.pdf
Solve for x : 2(floor of x) = x + 2{x}
1 reply
Cute Geometry
EthanWYX2009 0
In triangle 
, let 
and 
be a pair of isogonal conjugate points. The line 
intersects 
at 
, and the line 
intersects at 
. Let the circumcircle of 
and the circumcircle of 
intersect again at 
(other than ). Prove that the line 
bisects 
.
IMAGE
, let 
and 
be a pair of isogonal conjugate points. The line 
intersects 
at 
, and the line 
intersects at 
. Let the circumcircle of 
and the circumcircle of 
intersect again at 
(other than ). Prove that the line 
bisects 
.IMAGE
0 replies
Interior point of ABC
Jackson0423 0
Let D be an interior point of the acute triangle ABC with AB > AC so that ∠DAB = ∠CAD. The point E on the segment AC satisfies ∠ADE = ∠BCD, the point F on the segment AB satisfies ∠F DA = ∠DBC, and the point X on the line AC satisfies CX = BX. Let O1 and O2 be the circumcenters of the triangles ADC and EXD, respectively. Prove that the lines BC, EF, and O1O2 are concurrent
0 replies
