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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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0 replies
jlacosta
Apr 2, 2025
0 replies
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Geo with unnecessary condition
egxa   8
N 22 minutes ago by ErTeeEs06
Source: Turkey Olympic Revenge 2024 P4
Let the circumcircle of a triangle $ABC$ be $\Gamma$. The tangents to $\Gamma$ at $B,C$ meet at point $E$. For a point $F$ on line $BC$ which is not on the segment $BC$, let the midpoint of $EF$ be $G$. Lines $GB,GC$ meet $\Gamma$ again at points $I,H$ respectively. Let $M$ be the midpoint of $BC$. Prove that the points $F,I,H,M$ lie on a circle.

Proposed by Mehmet Can Baştemir
8 replies
egxa
Aug 6, 2024
ErTeeEs06
22 minutes ago
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USAMO 2000 Problem 3
MithsApprentice   9
N 2 hours ago by Anto0110
A game of solitaire is played with $R$ red cards, $W$ white cards, and $B$ blue cards. A player plays all the cards one at a time. With each play he accumulates a penalty. If he plays a blue card, then he is charged a penalty which is the number of white cards still in his hand. If he plays a white card, then he is charged a penalty which is twice the number of red cards still in his hand. If he plays a red card, then he is charged a penalty which is three times the number of blue cards still in his hand. Find, as a function of $R, W,$ and $B,$ the minimal total penalty a player can amass and all the ways in which this minimum can be achieved.
9 replies
MithsApprentice
Oct 1, 2005
Anto0110
2 hours ago
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Problem 4
blug   1
N 2 hours ago by Filipjack
Source: Polish Math Olympiad 2025 Finals P4
A positive integer $n\geq 2$ and a set $S$ consisting of $2n$ disting positive integers smaller than $n^2$ are given. Prove that there exists a positive integer $r\in \{1, 2, ..., n\}$ that can be written in the form $r=a-b$, for $a, b\in \mathbb{S}$ in at least $3$ different ways.
1 reply
blug
Today at 11:59 AM
Filipjack
2 hours ago
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Strange angle condition and concyclic points
lminsl   126
N 3 hours ago by cj13609517288
Source: IMO 2019 Problem 2
In triangle $ABC$, point $A_1$ lies on side $BC$ and point $B_1$ lies on side $AC$. Let $P$ and $Q$ be points on segments $AA_1$ and $BB_1$, respectively, such that $PQ$ is parallel to $AB$. Let $P_1$ be a point on line $PB_1$, such that $B_1$ lies strictly between $P$ and $P_1$, and $\angle PP_1C=\angle BAC$. Similarly, let $Q_1$ be the point on line $QA_1$, such that $A_1$ lies strictly between $Q$ and $Q_1$, and $\angle CQ_1Q=\angle CBA$.

Prove that points $P,Q,P_1$, and $Q_1$ are concyclic.

Proposed by Anton Trygub, Ukraine
126 replies
lminsl
Jul 16, 2019
cj13609517288
3 hours ago
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Geo Mock #9
Bluesoul   1
N Today at 3:19 PM by vanstraelen
Consider $\triangle{ABC}$ with $AB=12, AC=22$. The points $D,E$ lie on $AB,AC$ respectively, such that $\frac{AD}{BD}=\frac{AE}{CE}=3$. Extend $CD,BE$ to meet the circumcircle of $\triangle{ABC}$ at $P,Q$ respectively. Let the circumcircles of $\triangle{ADP}, \triangle{AEQ}$ meet at points $A,T$. Extend $AT$ to $BC$ at $R$, given $AR=16$, find $[ABC]$.
1 reply
Bluesoul
Apr 1, 2025
vanstraelen
Today at 3:19 PM
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Regarding IMO prepartion
omega2007   0
Today at 3:14 PM
<Hey Everyone'>
I'm 10 grader student and Im starting prepration for maths olympiad..>>> From scratch (not 2+2=4 )

Do you haves compilled resources of Handouts,
PDF,
Links,
List of books topic wise
which are shared on AOPS (and from your prespective) for maths olympiad and any useful thing, which will help me in boosting Maths olympiad prepration.
0 replies
omega2007
Today at 3:14 PM
0 replies
J
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Geo Mock #6
Bluesoul   1
N Today at 1:59 PM by vanstraelen
Consider triangle $ABC$ with $AB=5, BC=8, AC=7$, denote the incenter of the triangle as $I$. Extend $BI$ to meet the circumcircle of $\triangle{AIC}$ at $Q\neq I$, find the length of $QC$.
1 reply
Bluesoul
Apr 1, 2025
vanstraelen
Today at 1:59 PM
J
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Congruence
Ecrin_eren   1
N Today at 1:39 PM by Ecrin_eren
Find the number of integer pairs (x, y) satisfying the congruence equation:

3y² + 3x²y + y³ ≡ 3x² (mod 41)

for 0 ≤ x, y < 41.

1 reply
Ecrin_eren
Yesterday at 10:34 AM
Ecrin_eren
Today at 1:39 PM
J
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Probability
Ecrin_eren   1
N Today at 1:38 PM by Ecrin_eren
In a board, James randomly writes A , B or C in each cell. What is the probability that, for every row and every column, the number of A 's modulo 3 is equal to the number of B's modulo 3?

1 reply
Ecrin_eren
Yesterday at 11:21 AM
Ecrin_eren
Today at 1:38 PM
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Excalibur Identity
jjsunpu   9
N Today at 12:21 PM by fruitmonster97
proof is below
9 replies
jjsunpu
Yesterday at 3:27 PM
fruitmonster97
Today at 12:21 PM
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New geometry problem
titaniumfalcon   2
N Today at 12:06 PM by fruitmonster97
Post any solutions you have, with explanation or proof if possible, good luck!
2 replies
titaniumfalcon
Yesterday at 10:40 PM
fruitmonster97
Today at 12:06 PM
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.problem.
Cobedangiu   2
N Today at 12:06 PM by Lankou
Find the integer coefficients after expanding Newton's binomial:
$$(\frac{3}{2}-\frac{2}{3}x^2)^n (n \in Z)$$
2 replies
Cobedangiu
Today at 6:20 AM
Lankou
Today at 12:06 PM
J
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Inequalities
sqing   23
N Today at 11:43 AM by sqing
Let $ a,b,c\geq 0 $ and $a+b+c=1$. Prove that
$$a^2+b^2+ ab +21abc\leq\frac{512}{441}$$Equality holds when $a=b=\frac{38}{21},c=\frac{5}{214}.$
$$a^2+b^2+ ab +19abc\leq\frac{10648}{9747}$$Equality holds when $a=b=\frac{22}{57},c=\frac{13}{57}.$
$$a^2+b^2+ ab +22abc\leq\frac{15625}{13068}$$Equality holds when $a=b=\frac{25}{66},c=\frac{8}{33}.$
23 replies
sqing
Mar 26, 2025
sqing
Today at 11:43 AM
J
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Puzzling p&c question
Hunter87   0
Today at 8:51 AM
There are 9 numbered tickets (distinct) to be distributed (1-9), and 7 contestants. Each contestant must get atleast one ticket. Two PARTICULAR contestants (say, A and B) can never get tickets with adjacent numbers and any contestant can get more than 1 ticket. All tickets are to be distributed.
In how many ways can this be done?
0 replies
Hunter87
Today at 8:51 AM
0 replies
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Problem 2
Mualpha7   1
N Jul 22, 2015 by hOLObESUTO
Source: Paraguayan Mathematical Olympiad 2006
Consider all right triangles with integer sides such that the length of the hypotenuse and one of the two sides are consecutive. How many such triangles exist?
1 reply
Mualpha7
Jul 22, 2015
hOLObESUTO
Jul 22, 2015
J
Problem 2
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Reveal topic tags
geometry
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G H BBookmark kLocked kLocked NReply
Source: Paraguayan Mathematical Olympiad 2006
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Mualpha7
70 posts
Mualpha7
#1 h  Jul 22, 2015, 6:14 AM • 2 Y
Y by Adventure10, Mango247
Consider all right triangles with integer sides such that the length of the hypotenuse and one of the two sides are consecutive. How many such triangles exist?
Z K Y
The post below has been deleted. Click to close.
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hOLObESUTO
17 posts
hOLObESUTO
#2 h  Jul 22, 2015, 6:48 AM • 2 Y
Y by Adventure10, Mango247
Infinitely many.
Let one leg be k where k is odd.
The hypotneuse is of length $\frac{k^2+1}{2}$ and the other leg of length $\frac{k^2-1}{2}$.
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