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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

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[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
May 1, 2025
0 replies
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Divisibilty...
Sadigly   0
31 minutes ago
Source: Azerbaijan Junior NMO 2025 P2
Find all $4$ consecutive even numbers, such that the square of their product divides the sum of their squares.
0 replies
Sadigly
31 minutes ago
0 replies
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Quadratic system
juckter   35
N an hour ago by shendrew7
Source: Mexico National Olympiad 2011 Problem 3
Let $n$ be a positive integer. Find all real solutions $(a_1, a_2, \dots, a_n)$ to the system:

\[a_1^2 + a_1 - 1 = a_2\]\[ a_2^2 + a_2 - 1 = a_3\]\[\hspace*{3.3em} \vdots \]\[a_{n}^2 + a_n - 1 = a_1\]
35 replies
juckter
Jun 22, 2014
shendrew7
an hour ago
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IMO Shortlist 2012, Geometry 3
lyukhson   75
N 2 hours ago by numbertheory97
Source: IMO Shortlist 2012, Geometry 3
In an acute triangle $ABC$ the points $D,E$ and $F$ are the feet of the altitudes through $A,B$ and $C$ respectively. The incenters of the triangles $AEF$ and $BDF$ are $I_1$ and $I_2$ respectively; the circumcenters of the triangles $ACI_1$ and $BCI_2$ are $O_1$ and $O_2$ respectively. Prove that $I_1I_2$ and $O_1O_2$ are parallel.
75 replies
lyukhson
Jul 29, 2013
numbertheory97
2 hours ago
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Diophantine
TheUltimate123   31
N 2 hours ago by SomeonecoolLovesMaths
Source: CJMO 2023/1 (https://aops.com/community/c594864h3031323p27271877)
Find all triples of positive integers \((a,b,p)\) with \(p\) prime and \[a^p+b^p=p!.\]
Proposed by IndoMathXdZ
31 replies
TheUltimate123
Mar 29, 2023
SomeonecoolLovesMaths
2 hours ago
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2012 preRMO p17, roots of equation x^3 + 3x + 5 = 0
parmenides51   11
N Today at 3:29 PM by Pengu14
Let $x_1,x_2,x_3$ be the roots of the equation $x^3 + 3x + 5 = 0$. What is the value of the expression
$\left( x_1+\frac{1}{x_1} \right)\left( x_2+\frac{1}{x_2} \right)\left( x_3+\frac{1}{x_3} \right)$ ?
11 replies
parmenides51
Jun 17, 2019
Pengu14
Today at 3:29 PM
J
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Interesting question from Al-Khwarezmi olympiad 2024 P3, day1
Adventure1000   3
N Today at 2:38 PM by sqing
Find all $x, y, z \in \left (0, \frac{1}{2}\right )$ such that
$$ \begin{cases} (3 x^{2}+y^{2}) \sqrt{1-4 z^{2}} \geq z; \\ (3 y^{2}+z^{2}) \sqrt{1-4 x^{2}} \geq x; \\ (3 z^{2}+x^{2}) \sqrt{1-4 y^{2}} \geq y. \end{cases} $$Proposed by Ngo Van Trang, Vietnam
3 replies
Adventure1000
May 7, 2025
sqing
Today at 2:38 PM
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Malaysia MO IDM UiTM 2025
smartvong   1
N Today at 2:20 PM by jasperE3
MO IDM UiTM 2025 (Category C)

Contest Description

Preliminary Round
Section A
1. Given that $2^a + 2^b = 2016$ such that $a, b \in \mathbb{N}$. Find the value of $a$ and $b$.

2. Find the value of $a, b$ and $c$ such that $$\frac{ab}{a + b} = 1, \frac{bc}{b + c} = 2, \frac{ca}{c + a} = 3.$$
3. If the value of $x + \dfrac{1}{x}$ is $\sqrt{3}$, then find the value of
$$x^{1000} + \frac{1}{x^{1000}}$$.

Section B
1. Let $\mathbb{Z}$ be the set of integers. Determine all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that for all integer $a, b$:
$$f(2a) + 2f(b) = f(f(a + b))$$
2. The side lengths $a, b, c$ of a triangle $\triangle ABC$ are positive integers. Let
$$T_n = (a + b + c)^{2n} - (a - b + c)^{2n} - (a + b - c)^{2n} - (a - b - c)^{2n}$$for any positive integer $n$.
If $\dfrac{T_2}{2T_1} = 2023$ and $a > b > c$, determine all possible perimeters of the triangle $\triangle ABC$.

Final Round
Section A
1. Given that the equation $x^2 + (b - 3)x - 2b^2 + 6b - 4 = 0$ has two roots, where one is twice of the other, find all possible values of $b$.

2. Let $$f(y) = \dfrac{y^2}{y^2 + 1}.$$Find the value of $$f\left(\frac{1}{2001}\right) + f\left(\frac{2}{2001}\right) + \cdots + f\left(\frac{2000}{2001}\right) + f\left(\frac{2001}{2001}\right) + f\left(\frac{2001}{2000}\right) + \cdots + f\left(\frac{2001}{2}\right) + f\left(\frac{2001}{1}\right).$$
3. Find the smallest four-digit positive integer $L$ such that $\sqrt{3\sqrt{L}}$ is an integer.

Section B
1. Given that $\tan A : \tan B : \tan C$ is $1 : 2 : 3$ in triangle $\triangle ABC$, find the ratio of the side length $AC$ to the side length $AB$.

2. Prove that $\cos{\frac{2\pi}{5}} + \cos{\frac{4\pi}{5}} = -\dfrac{1}{2}.$
1 reply
smartvong
Today at 1:01 PM
jasperE3
Today at 2:20 PM
J
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Nice problem
gasgous   2
N Today at 1:47 PM by vincentwant
Given that the product of three integers is $60$.What is the least possible positive sum of the three integers?
2 replies
gasgous
Today at 1:30 PM
vincentwant
Today at 1:47 PM
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Angle Formed by Points on the Sides of a Triangle
xeroxia   1
N Today at 1:28 PM by vanstraelen

In triangle $ABC$, points $D$, $E$, and $F$ lie on sides $BC$, $CA$, and $AB$, respectively, such that
$BD = 20$, $DC = 15$, $CE = 13$, $EA = 8$, $AF = 6$, $FB = 22$.

What is the measure of $\angle EDF$?


1 reply
xeroxia
Today at 10:28 AM
vanstraelen
Today at 1:28 PM
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Inequalities
sqing   1
N Today at 1:08 PM by sqing
Let $ a,b,c\geq 0 , (a+8)(b+c)=9.$ Prove that
$$\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}\geq \frac{38}{23}$$Let $ a,b,c\geq 0 , (a+2)(b+c)=3.$ Prove that
$$\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}\geq \frac{2(2\sqrt{3}+1)}{5}$$
1 reply
sqing
Today at 12:50 PM
sqing
Today at 1:08 PM
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Is this true?
Entrepreneur   1
N Today at 12:56 PM by revol_ufiaw
Define the $\text{\textcolor{red}{Pell Sequence}}$ as $$P_0=0,P_1=1,\;P_{n+2}=2P_{n+1}+P_n.$$Prove that $4P_{2k}^2+1$ is prime for all $k\in\mathbb N.$
1 reply
Entrepreneur
Today at 9:32 AM
revol_ufiaw
Today at 12:56 PM
J
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Geometry
AlexCenteno2007   4
N Today at 12:05 PM by Raul_S_Baz
Let ABC be an acute triangle and let D, E and F be the feet of the altitudes from A, B and C respectively. The straight line EF and the circumcircle of ABC intersect at P such that F is between E and P, the straight lines BP and DF intersect at Q. Show that if ED = EP then CQ and DP are parallel.
4 replies
AlexCenteno2007
Apr 28, 2025
Raul_S_Baz
Today at 12:05 PM
J
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Inequalities
sqing   1
N Today at 11:45 AM by sqing
Let $ 0\leq x,y,z\leq 2. $ Prove that
$$-48\leq (x-yz)( 3y-zx)(z-xy)\leq 9$$$$-144\leq (3x-yz)(y-zx)(3z-xy)\leq\frac{81}{64}$$$$-144\leq (3x-yz)(2y-zx)(3z-xy)\leq\frac{81}{16}$$
1 reply
sqing
Yesterday at 8:50 AM
sqing
Today at 11:45 AM
J
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Concurrent in a pyramid
vanstraelen   0
Today at 7:13 AM

Given a pyramid $(T,ABCD)$ where $ABCD$ is a parallelogram.
The intersection of the diagonals of the base is point $S$.
Point $A$ is connected to the midpoint of $[CT]$, point $B$ to the midpoint of $[DT]$,
point $C$ to the midpoint of $[AT]$ and point $D$ to the midpoint of $[BT]$.
a) Prove: the four lines are concurrent in a point $P$.
b) Calulate $\frac{TS}{TP}$.
0 replies
vanstraelen
Today at 7:13 AM
0 replies
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J
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Find maximum area of right triangle
jl_   1
N Apr 23, 2025 by navier3072
Source: Malaysia IMONST 2 2023 (Primary) P4
Given a right-angled triangle with hypothenuse $2024$, find the maximal area of the triangle.
1 reply
jl_
Apr 23, 2025
navier3072
Apr 23, 2025
J
Find maximum area of right triangle
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Reveal topic tags
algebra
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G H BBookmark kLocked kLocked NReply
Source: Malaysia IMONST 2 2023 (Primary) P4
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jl_
9 posts
jl_
#1 h  Apr 23, 2025, 10:33 AM
Y by
Given a right-angled triangle with hypothenuse $2024$, find the maximal area of the triangle.
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navier3072
115 posts
navier3072
#2 h  Apr 23, 2025, 11:18 AM
Y by
$a^2+b^2=2024^2$, so minimum area is $\frac{1}{2} ab \leq \frac{a^2+b^2}{4}=1012^2=(1000+12)^2=1000000+24000+144=1024144$
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