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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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[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
Apr 2, 2025
0 replies
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Problems for v_p(n)
xytunghoanh   1
N 13 minutes ago by whwlqkd
Hello everyone. I need some easy problems for $v_p(n)$ (use in a problem for junior) to practice. Can anyone share to me?
Thanks :>
1 reply
1 viewing
xytunghoanh
an hour ago
whwlqkd
13 minutes ago
J
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Parallel Lines and Q Point
taptya17   13
N 18 minutes ago by everythingpi3141592
Source: India EGMO TST 2025 Day 1 P3
Let $\Delta ABC$ be an acute angled scalene triangle with circumcircle $\omega$. Let $O$ and $H$ be the circumcenter and orthocenter of $\Delta ABC,$ respectively. Let $E,F$ and $Q$ be points on segments $AB,AC$ and $\omega$, respectively, such that
$$\angle BHE=\angle CHF=\angle AQH=90^\circ.$$Prove that $OQ$ and $AH$ intersect on the circumcircle of $\Delta AEF$.

Proposed by Antareep Nath
13 replies
taptya17
Dec 13, 2024
everythingpi3141592
18 minutes ago
J
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Cool combinatorial problem (grid)
Anto0110   0
18 minutes ago
Suppose you have an $m \cdot n$ grid with $m$ rows and $n$ columns, and each square of the grid is filled with a non-negative integer. Let $a$ be the average of all the numbers in the grid. Prove that if $m >(10a+10)^n$ the there exist two identical rows (meaning same numbers in the same order).
0 replies
Anto0110
18 minutes ago
0 replies
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Cards and combi
AlephG_64   0
25 minutes ago
Source: 2025 Finals Portuguese Mathematical Olympiad P6
Maria wants to solve a challenge with a deck of cards, each with a different figure. Initially, the cards are distributed randomly into two piles, not necessarily in equal parts. Maria's goal is to get all the cards into the same pile.

On each turn, Maria takes the top card from each pile and compares them. In the rule book, there's a table that indicates, for each card match, which of the two wins. Both cards are then placed on the bottom of the winning card in the order Maria chooses. The challenge ends when all the cards are in one pile.

Show that it is always possible for Maria to solve the challenge. Regardless of the initial distribution of the cards and the table in the rule book.
0 replies
AlephG_64
25 minutes ago
0 replies
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that statement is true
pennypc123456789   3
N 2 hours ago by sqing
we have $a^3+b^3 = 2$ and $3(a^4+b^4)+2a^4b^4 \le 8 $ , then we can deduce $a^2+b^2$ \le 2 $ ?
3 replies
pennypc123456789
Mar 23, 2025
sqing
2 hours ago
J
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Distance vs time swimming problem
smalkaram_3549   1
N 2 hours ago by Lankou
How should I approach a problem where we deal with velocities becoming negative and stuff. I know that they both travel 3 Lengths of the pool before meeting a second time.
1 reply
smalkaram_3549
Today at 2:57 AM
Lankou
2 hours ago
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.problem.
Cobedangiu   4
N 3 hours ago by Lankou
Find the integer coefficients after expanding Newton's binomial:
$$(\frac{3}{2}-\frac{2}{3}x^2)^n (n \in Z)$$
4 replies
Cobedangiu
Yesterday at 6:20 AM
Lankou
3 hours ago
J
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inequalities - 5/4
pennypc123456789   2
N 3 hours ago by sqing
Given real numbers $x, y$ satisfying $|x| \le 3, |y| \le 3$. Prove that:
\[ 0 \le (x^2 + 1)(y^2 + 1) + 4(x - 1)(y - 1) \le 164. \]
2 replies
pennypc123456789
5 hours ago
sqing
3 hours ago
J
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Source of a combinatorics problem
isodynamicappolonius2903   0
Today at 6:47 AM
Does anyone know exactly the source of this problem?? I just remember that it from a combinatorics book of Russia.
0 replies
isodynamicappolonius2903
Today at 6:47 AM
0 replies
J
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KSEA NMSC Mock Contest Group B (Last Problem)
Shiyul   5
N Today at 3:26 AM by Shiyul
Let $a_n$ be a sequence defined by $a_n = a^2 + 1$. Then the product of four consecutive terms in $a_n$ can be written as the product of two terms in $a_n$. Find $p + q$ if $(a_(11))(a_(12))(a_(13))(a_(14)) = (a_p)(a_q)$.
5 replies
Shiyul
Yesterday at 3:09 PM
Shiyul
Today at 3:26 AM
J
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Regarding IMO prepartion
omega2007   1
N Today at 2:49 AM by omega2007
<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.
1 reply
omega2007
Yesterday at 3:14 PM
omega2007
Today at 2:49 AM
J
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School Math Problem
math_cool123   6
N Today at 2:37 AM by anduran
Find all ordered pairs of nonzero integers $(a, b)$ that satisfy $$(a^2+b)(a+b^2)=(a-b)^3.$$
6 replies
math_cool123
Apr 2, 2025
anduran
Today at 2:37 AM
J
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New geometry problem
titaniumfalcon   3
N Yesterday at 11:16 PM by mathprodigy2011
Post any solutions you have, with explanation or proof if possible, good luck!
3 replies
titaniumfalcon
Thursday at 10:40 PM
mathprodigy2011
Yesterday at 11:16 PM
J
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Geo Mock #10
Bluesoul   2
N Yesterday at 8:26 PM by Bluesoul
Consider acute $\triangle{ABC}$ with $AB=10$, $AC<BC$ and area $135$. The circle $\omega$ with diameter $AB$ meets $BC$ at $E$. Let the orthocenter of the triangle be $H$, connect $CH$ and extend to meet $\omega$ at $N$ such that $NC>HC$ and $NE$ is the diameter of $\omega$. Draw the circumcircle $\Gamma$ of $\triangle{AHB}$, chord $XY$ of $\Gamma$ is tangent to $\omega$ and it passes through $N$, compute $XY$.
2 replies
Bluesoul
Apr 1, 2025
Bluesoul
Yesterday at 8:26 PM
J
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St.Peterburg, P2 Grade 10, 2013
mathuz   1
N Apr 27, 2014 by MariusBocanu
in a convex quadrilateral $ABCD$ , $M,N$ are midpoints of $BC,AD$ respectively. If $AM=BN$ and $DM=CN$ then prove that $AC=BD$.



S. Berlov
1 reply
mathuz
Apr 27, 2014
MariusBocanu
Apr 27, 2014
J
St.Peterburg, P2 Grade 10, 2013
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Reveal topic tags
geometry proposed
geometry
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mathuz
1512 posts
mathuz
#1 h  Apr 27, 2014, 4:27 AM • 1 Y
Y by Adventure10
in a convex quadrilateral $ABCD$ , $M,N$ are midpoints of $BC,AD$ respectively. If $AM=BN$ and $DM=CN$ then prove that $AC=BD$.



S. Berlov
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MariusBocanu
429 posts
MariusBocanu
#2 h  Apr 27, 2014, 8:00 AM • 2 Y
Y by Adventure10, Mango247
$AM=BN$ means $2(AB^2+AC^2)-BC^2=2(AB^2+BD^2)-AD^2$ and $CN=DM$ means $2(CD^2+CA^2)-AD^2=2(DB^2+DC^2)-BC^2$ and summing these relations we obtain $AC^2=BD^2$, done.
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