Inspired by A_E_R
by sqing, Apr 13, 2025, 12:18 PM
Let 
and 
Prove that
Let and 
Prove that
and 
Prove that
Let and 
Prove that
and 
Prove that
Interesting inequality
by A_E_R, Apr 13, 2025, 9:41 AM
This post has been edited 2 times. Last edited by A_E_R, an hour ago
Reason: size
Reason: size
Number Theory Chain!
by JetFire008, Apr 7, 2025, 7:14 AM
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!
?This post has been edited 1 time. Last edited by JetFire008, Apr 7, 2025, 7:14 AM
Abelkonkurransen 2025 1b
by Lil_flip38, Mar 20, 2025, 11:07 AM
In Duckville there is a perpetual trophy with the words “Best child of Duckville” engraved on it. Each inhabitant of Duckville has a non-empty list (which never changes) of other inhabitants of Duckville. Whoever receives the trophy
gets to keep it for one day, and then passes it on to someone on their list the next day. Gregers has previously received the trophy. It turns out that each time he does receive it, he is guaranteed to receive it again exactly
days later (but perhaps earlier, as well). Hedvig received the trophy today. Determine all integers 
for which we can be absolutely certain that she cannot receive the trophy again in 
days, given the above information.
gets to keep it for one day, and then passes it on to someone on their list the next day. Gregers has previously received the trophy. It turns out that each time he does receive it, he is guaranteed to receive it again exactly
days later (but perhaps earlier, as well). Hedvig received the trophy today. Determine all integers 
for which we can be absolutely certain that she cannot receive the trophy again in 
days, given the above information.Concurrent lines
by BR1F1SZ, Mar 7, 2025, 8:23 PM
Let 
be a trapezoid with parallel sides 
and 
, where 
. A circle passing through 
and 
intersects 
again at 
respectively. Prove that 
are concurrent.
be a trapezoid with parallel sides 
and 
, where 
. A circle passing through 
and 
intersects 
again at 
respectively. Prove that 
are concurrent.This post has been edited 1 time. Last edited by BR1F1SZ, Mar 7, 2025, 8:35 PM
Arithmetic progression
by BR1F1SZ, Mar 7, 2025, 8:20 PM
Suppose an infinite non-constant arithmetic progression of integers contains 
in it. Prove that there are an infinite number of perfect cubes in this progression. (A perfect cube is an integer of the form 
, where 
is an integer. For example, 
, 
and are perfect cubes.)
in it. Prove that there are an infinite number of perfect cubes in this progression. (A perfect cube is an integer of the form 
, where 
is an integer. For example, 
, 
and are perfect cubes.)IMO Shortlist 2017 Geometry
by liberator, Oct 4, 2018, 9:40 PM
Here are my solutions to some of the geometry problems from this year's IMO Shortlist.
G1. Let
be a convex pentagon such that 
, 
, and 
. Prove that the perpendicular line from 
to 
and the line segments 
and 
are concurrent.
G3. Let
be the circumcenter of an acute triangle 
. Line 
intersects the altitudes of through 
and at 
and 
, respectively. The altitudes meet at 
. Prove that the circumcenter of triangle 
lies on a median of triangle .
G4. In triangle, let 
be the excircle opposite to 
. Let 
and 
be the points where is tangent to 
, and , respectively. The circle 
intersects line at and . Let 
be the midpoint of 
. Prove that the circle 
is tangent to .
G1. Let
be a convex pentagon such that 
, 
, and 
. Prove that the perpendicular line from 
to 
and the line segments 
and 
are concurrent.![[asy]
size(6cm);
pointpen=black; pathpen=rgb(0.4,0.6,0.8); pointfontpen=fontsize(10);
pair X=dir(270), W=dir(215), Y=dir(10), V=W*X/Y, Z=X/W*Y, A=2*V*W/(V+W), B=2*W*X/(W+X), C=2*X*Y/(X+Y), D=2*Y*Z/(Y+Z), E=2*Z*V/(Z+V), P=extension(E,A,C,D), Q=extension(A,B,D,E), T=foot(E,B,C);
D(E--A--B--C--D--E--T);
DPA(E--P--D^^A--Q--E,pathpen+dashed);
D(unitcircle,heavygreen);
D(rightanglemark(E,T,C,5));
DPA(anglemark(B,A,E)^^anglemark(D,C,B),red);
DPA(anglemark(E,D,C)^^anglemark(C,B,A),purple);
D("A",A,SW);
D("B",B,SW);
D("C",C,SE);
D("D",D,ENE);
D("E",E,N);
D("P",P,NE);
D("Q",Q,WNW);
D("T",T);
[/asy]](http://latex.artofproblemsolving.com/c/c/9/cc9c07b357e53df571029bb533dfe85be2b84966.png)
;
is a kite, so has an incircle. Similarly, if 
,
has an incircle. Hence 
be the foot from 
.
bisects 
,G3. Let
be the circumcenter of an acute triangle 
. Line 
intersects the altitudes of through 
and at 
and 
, respectively. The altitudes meet at 
. Prove that the circumcenter of triangle 
lies on a median of triangle .![[asy]
unitsize(3cm);
pointpen = black; pathpen = rgb(0.4,0.6,0.8); pointfontpen = fontsize(10);
pair A=dir(125), B=dir(210), C=dir(-30), E=foot(B,C,A), F=foot(C,A,B), O=origin, P=extension(B,E,O,A), Q=extension(C,F,O,A), H=A+B+C, M=(B+C)/2, X=foot(H,A,M), G=extension(E,F,B,C);
D(A--B--C--cycle);
DPA(Q--A^^B--E^^C--F);
D(A--M,red+dashed);
DPA(circumcircle(P,Q,H)^^A--H,heavygreen);
DPA(X--G--E^^G--B,dashed+pathpen);
D("A",A,dir(A));
D("B",B,dir(B));
D("C",C,dir(C));
D("E",E,NE);
D("F",F,NW);
D("H",H,SE);
D("O",O,dir(0));
D("P",P);
D("Q",Q,NE);
D("X",X,dir(0));
D(G); D(M);
[/asy]](http://latex.artofproblemsolving.com/5/1/e/51ee6eed1502f3913ff1dcf03722b90c1b30032b.png)
be the foot from 
;
.
,
is tangent to 
.
is the polar of G4. In triangle
, let 
be the excircle opposite to 
. Let 
and 
be the points where is tangent to 
, and , respectively. The circle 
intersects line at and . Let 
be the midpoint of 
. Prove that the circle 
is tangent to .![[asy]
size(10cm);
pointpen=black; pathpen=rgb(0.4,0.6,0.8); pointfontpen=fontsize(10);
pair A=dir(130), B=dir(210), C=dir(-30), I_A=2*dir(-90)-incenter(A,B,C), D=foot(I_A,B,C), E=foot(I_A,A,C), F=foot(I_A,A,B);
path w=D(circumcircle(A,E,F),red), l=(100*B-99*C)--(100*C-99*B);
pair P=IP(l, w), Q=OP(l, w), R=extension(B,C,E,F), N=foot(I_A, A, D), M=(A+D)/2, T=2*N-D;
DPA(E--A--F--R^^P--R--T--A);
D(CP(I_A,D),heavygreen);
D(circumcircle(M,P,Q),purple+dashed);
D("A",A,dir(A));
D("B",B,SW);
D("C",C,NE);
D("D",D,SSW);
D("E",E,dir(0));
D("F",F,SW);
D("P",P,W);
D("Q",Q,NE);
D("R",R,dir(0));
D("T",T,SE);
D("M",M,SSW);
D("N",N,W);
D("I_A",I_A);
[/asy]](http://latex.artofproblemsolving.com/f/6/e/f6e50d745586db5e9317c53d85eff9941f7e1ea4.png)
be the pole of 
be the midpoint of 
;
,
.
,
is cyclic. Hence 
,Maximum area of the triangle
by adityaguharoy, Jan 19, 2017, 11:42 AM
If in some triangle 
we are given :
and one side length of the triangle equals 
, then under these conditions find the maximum area of the triangle .
we are given :
and one side length of the triangle equals 
, then under these conditions find the maximum area of the triangle .Injective arithmetic comparison
by adityaguharoy, Jan 16, 2017, 3:41 AM
Show or refute :
For every injective function
there are elements 
in an arithmetic progression in the order 
such that 
.
For every injective function
there are elements 
in an arithmetic progression in the order 
such that 
.Convex quad
by MithsApprentice, Oct 27, 2005, 7:54 AM
Let 
be a convex quadrilateral such that diagonals 
and 
intersect at right angles, and let 
be their intersection. Prove that the reflections of across 
are concyclic.
be a convex quadrilateral such that diagonals 
and 
intersect at right angles, and let 
be their intersection. Prove that the reflections of across 
are concyclic.
October 2018
April 2018
April 2016
December 2014
October 2014
September 2014
August 2014