Concurrence
by LiamChen, Jun 5, 2025, 2:48 PM
Attachments:
Different scores possible in interview
by Rijul saini, Jun 4, 2025, 6:54 PM
In a job interview, the candidates are asked questions in a sequence. The initial score is 
. The candidate's score is calculated as follows:
after a correct answer, the score is increased by 
;
after a wrong answer, the score is divided by 
.
If the candidate is asked
questions and answers all of them, how many different scores are possible?
Note: Two different response sequences of the same length can result in the same score: the sequences
and 
with the same length, where 
denotes the correct answer and 
denotes the wrong answer, both result in the same score of 1.
Proposed by S. Muralidharan
. The candidate's score is calculated as follows:
after a correct answer, the score is increased by 
; after a wrong answer, the score is divided by 
.If the candidate is asked
questions and answers all of them, how many different scores are possible?Note: Two different response sequences of the same length can result in the same score: the sequences
and 
with the same length, where 
denotes the correct answer and 
denotes the wrong answer, both result in the same score of 1.Proposed by S. Muralidharan
This post has been edited 1 time. Last edited by Rijul saini, Yesterday at 7:26 PM
Calvin needs to cover all squares
by Rijul saini, Jun 4, 2025, 6:35 PM
Consider a 
board where we identify the squares with pairs 
where 
and 
denote the row and column number of that square, respectively.
Calvin picks two positive integers
and places a pawn at the bottom left corner (i.e. on 
) and makes the following moves. In his 
-th move, he moves the pawn from to either 
or 
if is odd and to either 
and 
if is even. Here all the numbers are taken modulo 
. Find the number of pairs 
that Calvin could have picked such that he can make moves so that the pawn covers all the squares on the board without being on any square twice.
Proposed by Tejaswi Navilarekallu
board where we identify the squares with pairs 
where 
and 
denote the row and column number of that square, respectively.Calvin picks two positive integers
and places a pawn at the bottom left corner (i.e. on 
) and makes the following moves. In his 
-th move, he moves the pawn from to either 
or 
if is odd and to either 
and 
if is even. Here all the numbers are taken modulo 
. Find the number of pairs 
that Calvin could have picked such that he can make moves so that the pawn covers all the squares on the board without being on any square twice.Proposed by Tejaswi Navilarekallu
This post has been edited 1 time. Last edited by Rijul saini, Yesterday at 7:21 PM
2-var inequality
by sqing, Jun 4, 2025, 12:55 PM
Let 
and 
Prove that
Let and 
Prove that
Let and 
Prove that
and 
Prove that
Let and 
Prove that
Let and 
Prove that
This post has been edited 2 times. Last edited by sqing, Yesterday at 1:19 PM
Super easy problem
by M11100111001Y1R, May 27, 2025, 7:17 AM
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?
XY is tangent to a fixed circle
by a_507_bc, Nov 12, 2022, 8:22 PM
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 .This post has been edited 2 times. Last edited by a_507_bc, Nov 13, 2022, 8:31 AM
set of points, there exist two lines containing n points
by jasperE3, Apr 5, 2021, 2:43 PM
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.You would not believe your eyes...
by willwin4sure, Jan 25, 2021, 5:00 PM
Ten million fireflies are glowing in 
at midnight. Some of the fireflies are friends, and friendship is always mutual. Every second, one firefly moves to a new position so that its distance from each one of its friends is the same as it was before moving. This is the only way that the fireflies ever change their positions. No two fireflies may ever occupy the same point.
Initially, no two fireflies, friends or not, are more than a meter away. Following some finite number of seconds, all fireflies find themselves at least ten million meters away from their original positions. Given this information, find the greatest possible number of friendships between the fireflies.
Nikolai Beluhov
at midnight. Some of the fireflies are friends, and friendship is always mutual. Every second, one firefly moves to a new position so that its distance from each one of its friends is the same as it was before moving. This is the only way that the fireflies ever change their positions. No two fireflies may ever occupy the same point.Initially, no two fireflies, friends or not, are more than a meter away. Following some finite number of seconds, all fireflies find themselves at least ten million meters away from their original positions. Given this information, find the greatest possible number of friendships between the fireflies.
Nikolai Beluhov
This post has been edited 2 times. Last edited by v_Enhance, Jun 21, 2022, 3:14 AM
Number theory
by falantrng, Feb 25, 2018, 10:55 AM
Let 
be positive integers such that 
and 
. Let 
be the set of values attained by 
as runs through the positive integers. Show that is the set of all positive divisors of some positive integer.
be positive integers such that 
and 
. Let 
be the set of values attained by 
as runs through the positive integers. Show that is the set of all positive divisors of some positive integer.This post has been edited 2 times. Last edited by v_Enhance, Feb 25, 2018, 11:06 AM
Reason: LaTeX copy edits
Reason: LaTeX copy edits
IMO 2015 Problem 3
by liberator, Jul 14, 2015, 2:48 PM
Problem: Let be an acute triangle with 
. Let 
be its cirumcircle, 
its orthocenter, and 
the foot of the altitude from . Let 
be the midpoint of 
. Let 
be the point on such that 
and let 
be the point on such that 
. Assume that the points , , , and are all different and lie on in this order.
Prove that the circumcircles of triangles
and 
are tangent to each other.
Proposed by Ukraine
be an acute triangle with 
. Let 
be its cirumcircle, 
its orthocenter, and 
the foot of the altitude from . Let 
be the midpoint of 
. Let 
be the point on such that 
and let 
be the point on such that 
. Assume that the points , , , and are all different and lie on in this order.Prove that the circumcircles of triangles
and 
are tangent to each other.Proposed by Ukraine
![[asy]
/* IMO 2015 Problem 3, free script by liberator, 14 July 2015 */
import olympiad;
import cse5;
unitsize(3.2cm);
pen ddd = rgb(0.4,0.6,0.8);
pointpen=black;
defaultpen(fontsize(10pt));
pair A = dir(75);
pair B = dir(205);
pair C = dir(-25);
pair O = origin;
pair H = orthocenter(A,B,C);
pair F = foot(A,B,C);
pair D = IP(L(H,F,0,2015),unitcircle);
pair M = midpoint(B--C);
pair Ap = B+C-H;
pair Q = IP(L(M,H,0,2015), unitcircle);
pair Qp = A+Ap-Q;
pair K = IP(L(Qp,H,-0.1,2015), unitcircle);
pair P = circumcenter(D,H,K);
D(unitcircle, heavygreen);
D(circumcircle(K,Q,H), red+linewidth(1));
D(circumcircle(F,K,M), heavymagenta);
D(circumcircle(D,H,K), grey+linetype("0 2")+linewidth(1));
D(A--B--C--cycle, ddd+linewidth(1));
D(A--D, ddd);
D(A--Qp, ddd);
D(Q--Ap, ddd+dashed);
D(K--Qp, ddd+dashed);
D(H--P--K, ddd);
D(rightanglemark(C,F,A,2), ddd);
D("A",A,dir(A)*2);
D("B",B,dir(B));
D("C",C,dir(C));
D("D",D,dir(-110));
D("F",F,dir(F));
D("H",H,dir(-65)*2);
D("K",K,dir(-15)*1.6);
D("M",M,dir(-70));
D("P",P,dir(P));
D("Q",Q,dir(Q));
D("A'",Ap,dir(Ap));
D("Q'",Qp,dir(Qp));
D("\Gamma",O,dir(160)*28);
[/asy]](http://latex.artofproblemsolving.com/a/a/6/aa6ae4ba390a833b56195faffc62aaf837297443.png)
be the antipodes of 
respectively w.r.t 
are collinear. Also observe that 
are collinear, since 
be the circumcenter of 
,
:
.
to 
at 
;
from rectangle 
,
,
,
is tangent to 
.
,
is tangent to 
:
,This post has been edited 2 times. Last edited by liberator, Jul 14, 2015, 8:58 PM
Coaxal circles in incenter/excenter configuration
by liberator, Apr 15, 2015, 7:18 PM
Problem: Let be a triangle, whose excircle (opposite ) touches 
at 
respectively. Denote 
as the intersection of the lines 
and 
, and 
as the intersection of the lines 
and 
. If 
is the excenter of 
, opposite , prove that the circumcircles of triangles 
and 
are coaxal.
Commentary: We may replace "excircle" with "incircle", "excenter" with "incenter", and the result still holds.
See interactive diagram here.
be a triangle, whose excircle (opposite ) touches 
at 
respectively. Denote 
as the intersection of the lines 
and 
, and 
as the intersection of the lines 
and 
. If 
is the excenter of 
, opposite , prove that the circumcircles of triangles 
and 
are coaxal.Commentary: We may replace "excircle" with "incircle", "excenter" with "incenter", and the result still holds.
See interactive diagram here.
and 
.
.
is cyclic, from perpendicularity of tangents.
is tangent to 
and 
,
.
and 
,
.
at 
,
.
,Attachments:
Rectangle EFGH in incircle, prove that QIM = 90
by v_Enhance, Jul 18, 2014, 7:48 PM
Let be a triangle with incenter 
, and suppose the incircle is tangent to and at and . Denote by 
and the reflections of and over . Let be the intersection of with 
, and let be the midpoint of . Prove that 
and 
are perpendicular.
be a triangle with incenter 
, and suppose the incircle is tangent to and at and . Denote by 
and the reflections of and over . Let be the intersection of with 
, and let be the midpoint of . Prove that 
and 
are perpendicular.
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