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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.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following upcoming events:
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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
Classical NT FE
Kimchiks926   6
N 20 minutes ago by math-olympiad-clown
Source: Baltic Way 2022, Problem 16
Let $\mathbb{Z^+}$ denote the set of positive integers. Find all functions $f:\mathbb{Z^+} \to \mathbb{Z^+}$ satisfying the condition
$$ f(a) + f(b) \mid (a + b)^2$$for all $a,b \in \mathbb{Z^+}$
6 replies
Kimchiks926
Nov 12, 2022
math-olympiad-clown
20 minutes ago
Hagge circle, Thomson cubic, coaxal
kosmonauten3114   0
40 minutes ago
Source: My own (maybe well-known)
Let $\triangle{ABC}$ be a scalene triangle, $\triangle{M_AM_BM_C}$ its medial triangle, and $P$ a point on the Thomson cubic (= $\text{K002}$) of $\triangle{ABC}$. (Suppose that $P \notin \odot(ABC)$ ).
Let $\triangle{A'B'C'}$ be the circumcevian triangle of $P$ wrt $\triangle{ABC}$.
Let $\triangle{P_AP_BP_C}$ be the pedal triangle of $P$ wrt $\triangle{ABC}$.
Let $A_1$ be the reflection in $BC$ of $A'$. Define $B_1$, $C_1$ cyclically.
Let $A_2$ be the reflection in $M_A$ of $A'$. Define $B_2$, $C_2$ cyclically.
Let $A_3$ be the reflection in $P_A$ of $A'$. Define $B_3$, $C_3$ cyclically.

Prove that $\odot(A_1B_1C_1)$, $\odot(A_2B_2C_2)$, $\odot(A_3B_3C_3)$ and the orthocentroidal circle of $\triangle{ABC}$ are coaxal.
0 replies
+1 w
kosmonauten3114
40 minutes ago
0 replies
Combinations of lines
M11100111001Y1R   3
N 41 minutes ago by ariopro1387
Source: Iran TST 2025 Test 1 Problem 3
Suppose \( n > 10 \) lines are drawn on the plane such that no three of them are concurrent and no two are parallel. At least \( \frac{n^2}{8} + 1 \) of the bounded regions formed are colored black. A triangle formed by three lines is called a \textit{good triangle} if it lies entirely within a black region. Prove that there are at least \( \frac{n}{2} \) good triangles. (A good triangle is a bounded region with finite area.)
3 replies
M11100111001Y1R
May 27, 2025
ariopro1387
41 minutes ago
Antitours in Mahishmati
Supercali   3
N 41 minutes ago by cursed_tangent1434
Source: India TST 2023 Day 1 P1
In the fictional country of Mahishmati, there are $50$ cities, including a capital city. Some pairs of cities are connected by two-way flights. Given a city $A$, an ordered list of cities $C_1,\ldots, C_{50}$ is called an antitour from $A$ if
[list]
[*] every city (including $A$) appears in the list exactly once, and
[*] for each $k\in \{1,2,\ldots, 50\}$, it is impossible to go from $A$ to $C_k$ by a sequence of exactly $k$ (not necessarily distinct) flights.
[/list]
Baahubali notices that there is an antitour from $A$ for any city $A$. Further, he can take a sequence of flights, starting from the capital and passing through each city exactly once. Find the least possible total number of antitours from the capital city.

Proposed by Sutanay Bhattacharya
3 replies
Supercali
Jul 9, 2023
cursed_tangent1434
41 minutes ago
Trigo or Complex no.?
hzbrl   3
N Today at 8:26 AM by GreenKeeper
(a) Let $y=\cos \phi+\cos 2 \phi$, where $\phi=\frac{2 \pi}{5}$. Verify by direct substitution that $y$ satisfies the quadratic equation $2 y^2=3 y+2$ and deduce that the value of $y$ is $-\frac{1}{2}$.
(b) Let $\theta=\frac{2 \pi}{17}$. Show that $\sum_{k=0}^{16} \cos k \theta=0$
(c) If $z=\cos \theta+\cos 2 \theta+\cos 4 \theta+\cos 8 \theta$, show that the value of $z$ is $-(1-\sqrt{17}) / 4$.



I could solve (a) and (b). Can anyone help me with the 3rd part please?
3 replies
hzbrl
May 27, 2025
GreenKeeper
Today at 8:26 AM
3rd AKhIMO for university students, P3
UzbekMathematician   1
N Today at 3:07 AM by pineconee
Source: AKhIMO 2025, P3
Two points are chosen randomly - independently with uniform probability - from a semicircular arc with radius 1. A third point is chosen randomly - independently with uniform probability - from the diameter that connects the endpoints of the arc. What is expected value of the area of the triangle with the three chosen points as its vertices?
1 reply
UzbekMathematician
Yesterday at 1:57 PM
pineconee
Today at 3:07 AM
D1038 : A generalization of Jensen
Dattier   4
N Yesterday at 11:21 PM by Dattier
Source: les dattes à Dattier
Let $f \in C^1([0,1]), g \in C^2(f([0;1]))$.

Is it true that

$$\min(|g''|)\times \min(|f'|^2) \leq 24 \times\left|\int_0^1g(f(x)) \text{d}x- g(\int_0^1 f(x) \text{d}x) \right| \leq \max(|g''|)\times \max(|f'|^2)$$?
4 replies
Dattier
Yesterday at 12:15 PM
Dattier
Yesterday at 11:21 PM
3rd AKhIMO for University Students, P2
UzbekMathematician   1
N Yesterday at 9:40 PM by grupyorum
Source: AKhIMO 2025, P2
Find all possible values of $gcd(a^{2m}+1, a^n+1)$, where $a, m, n$ are positive integers and $n$ is odd.
1 reply
UzbekMathematician
Yesterday at 1:48 PM
grupyorum
Yesterday at 9:40 PM
Problem 2, Grade 12th RMO Shortlist - Year 2002
sticknycu   5
N Yesterday at 4:07 PM by P_Fazioli
Let $A \in M_2(C), A \neq O_2, A \neq I_2, n \in \mathbb{N}^*$ and $S_n = \{ X \in M_2(C) | X^n = A \}$.
Show:
a) $S_n$ with multiplication of matrixes operation is making an isomorphic-group structure with $U_n$.
b) $A^2 = A$.

Marian Andronache
5 replies
sticknycu
Jan 3, 2020
P_Fazioli
Yesterday at 4:07 PM
3rd AKhIMO for University Students, P1
UzbekMathematician   1
N Yesterday at 3:42 PM by KAME06
Source: AKhIMO 2025, P1
There are two circles in the $xy-$plane centered on the $y-$axis which are tangent to both the parabola $y=x^2$ and the line $y=2025$. Determine the lengths of the circles' diameters.
1 reply
UzbekMathematician
Yesterday at 1:44 PM
KAME06
Yesterday at 3:42 PM
3rd AKhIMO for university students, p4
UzbekMathematician   1
N Yesterday at 2:29 PM by grupyorum
Source: AKhIMO 2025, P4
Define a sequence $a_1, a_2, a_3, ... $ by $a_1=2, a_2=5$ and $a_{n+2}=f(a_n, a_{n+1})$ for all $n \ge 1$, where $$f(x,y)=5(x+y)+2\sqrt{6x^2+15xy+6y^2}.$$Show that $a_n$ is an integer for all $n\ge 1$.
1 reply
UzbekMathematician
Yesterday at 2:03 PM
grupyorum
Yesterday at 2:29 PM
Double Integral
namesis   3
N Yesterday at 2:27 PM by Mathzeus1024
The area of integration, $D$, is defined in plane polar coordinates $(r, \phi)$ by the inequality
$r-2 \leq r \leq r_1$, where $r_1 = 1 + \cos(\phi)$ and $r_2 = 3/2$.

Evaluate:

$\iint_D \frac{x+y+xy}{x^2 + y^2} dx dy $

I tried evaluating the integral in polar, with $r$ from $\frac{3}{2}$ to $1+ \cos \phi$ and $\phi$ from $- \frac{\pi}{3}$ to $ \frac{\pi}{3}$ but in vain.
3 replies
namesis
Dec 16, 2015
Mathzeus1024
Yesterday at 2:27 PM
Japanese Olympiad
parkjungmin   8
N Yesterday at 2:22 PM by parkjungmin
It's about the Japanese Olympiad

I can't solve it no matter how much I think about it.

If there are people who are good at math

Please help me.
8 replies
parkjungmin
May 10, 2025
parkjungmin
Yesterday at 2:22 PM
3rd AKhIMO for university students, P5
UzbekMathematician   0
Yesterday at 2:10 PM
Source: AKhIMO 2025, P5
Show that for every positive integer $n$ there exist nonnegative integers $p, q$ and integers $a_1, a_2, ... , a_p, b_1, b_2, ... , b_q \ge 2$ such that $$ n=\frac{(a_1^3-1)(a_2^3-1)...(a_p^3-1)}{(b_1^3-1)(b_2^3-1)...(b_q^3-1)} $$
0 replies
UzbekMathematician
Yesterday at 2:10 PM
0 replies
k Geometry Marathon-Olympiad level
vaibhav2903   1435
N Aug 10, 2013 by ThirdTimeLucky
Source: 0
Hi mathlinkers!
I want to organise a Geometry marathon. This is to improve our skills in geometry. Whoever answers a pending problem can post a new problem. Let us start with easy ones and then go to complicated ones.

Different approaches are most welcome :).

Here is an easy one

1. Inradius of a triangle is equal to $ 1$.find the sides of the triangle and P.T one of its angle is $ 90$

[moderator edit: topic capitalized. :)]
1435 replies
vaibhav2903
Feb 14, 2010
ThirdTimeLucky
Aug 10, 2013
Geometry Marathon-Olympiad level
G H J
Source: 0
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vaibhav2903
306 posts
#1 • 20 Y
Y by gold46, math_lover81, Arshia.esl, Adventure10, Mango247, and 15 other users
Hi mathlinkers!
I want to organise a Geometry marathon. This is to improve our skills in geometry. Whoever answers a pending problem can post a new problem. Let us start with easy ones and then go to complicated ones.

Different approaches are most welcome :).

Here is an easy one

1. Inradius of a triangle is equal to $ 1$.find the sides of the triangle and P.T one of its angle is $ 90$

[moderator edit: topic capitalized. :)]
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Stephen
402 posts
#2 • 3 Y
Y by Adventure10 and 2 other users
Well, I think you mean that the sides of the triangle are intergers.

Let the sides of the triangle are $ x, y, z$.

Then by heron, $ \frac{s(s-x)(s-y)(s-z)}{s^2}=1^2=1$ where $ s=\frac{x+y+z}{2}$.

So if we let $ s-x=a, s-y=b, s-z=c$, then $ a+b+c=abc$.

This is a famous problem. Since $ a, b, c$ are positive, $ {a, b, c}={1, 2, 3}$.

So $ {x, y, z}={3, 4, 5}$.

And it is obvious that one of that triangle's angle is 90 :)

Problem 2

In a convex hexagon $ ABCDEF$, triangles $ ABC, CDE, EFA$ are similar.

Find conditions on these triangles under which triangle $ ACE$ is equilateral if and only if so is $ BDF$.
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vaibhav2903
306 posts
#3 • 5 Y
Y by Adventure10 and 4 other users
can you just tell the problem clearly
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Stephen
402 posts
#4 • 3 Y
Y by Adventure10, Mango247, and 1 other user
Sorry. :(

I'll change the problem.

Problem 2

Let $ O$ be the circumcenter of an acute triangle $ ABC$ and let $ k$ be the circle with center $ S$ that is tangent to $ O$ at $ A$ and tangent to side $ BC$ at $ D$.

Circle $ k$ meets $ AB$ and $ AC$ again at $ E$ and $ F$ respectively. The lines $ OS$ and $ ES$ meet $ k$ again at $ I$ and $ G$.

Lines $ BO$ and $ IG$ intersect at $ H$.

Prove that $ GH = \frac{DF^2}{AF}$.
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vaibhav2903
306 posts
#5 • 3 Y
Y by Adventure10, Mango247, and 1 other user
is the fig correct?[geogebra]29cd5e332189655049720ed1a0c527f8d822c43d[/geogebra]
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Stephen
402 posts
#6 • 3 Y
Y by Adventure10, Mango247, and 1 other user
vaibhav2903 wrote:
is the fig correct?[geogebra]29cd5e332189655049720ed1a0c527f8d822c43d[/geogebra]

Sorry, I cannot see the picture. :( Can't you post it again by a file or something?

P.S. Since you sent me to post a solution, I'll give you some hints(if you can't understand it, sorry for my poor English).

In circle $ O$, $ < BAO = < ABO$. And in circle $ k$, $ < EAS = < AES$.

Since $ < BAO = < EAS$, $ < ABO = < AES$. So $ ES$ and $ BO$ are parallel.

And $ < EAS = < AES = < AEG = <AIG$. So $ AE$ and $ GI$ are parallel too.

So we can know that $ EBHG$ is a parallelogram. We can know that $ GH = BE$.

Hint: $ GH = BE$
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vaibhav2903
306 posts
#7 • 3 Y
Y by Adventure10, Mango247, and 1 other user
since no 1 is able to post a solution fr this problem
lets do the next

problem 3:$ {ABCD}$ is parellelogram and a st. line cuts $ AB$ at $ \frac {AB} {3}$ and $ AD$ at $ \frac {AD} {4}$ and AC at $ xAC$.find $ x$.
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SOURBH
212 posts
#8 • 4 Y
Y by Adventure10, Mango247, and 2 other users
Solution to Problem number 3

Let the line $ l$ intersect $ AB$ at $ K$,$ AD$ at $ L$ and $ AC$ at $ M$.

As given $ AK = \frac {AB}{3}$ and $ AL = \frac {AD}{4}$

Extend line $ l$ to meet $ DC$ at $ Z$ (Obviously, line $ l$ is not parallel to $ DC$)

$ \triangle{ZDL}\sim{\triangle{KAL}}$

Hence, $ \frac {AL}{LD} = \frac {AK}{ZD}$

$ \frac {1}{3} = \frac {\frac {x}{3}}{ZD}$

$ ZD = x$

By Menelaus Theorem , we get

$ \frac {DZ}{ZC}.\frac {CM}{MA}.\frac {AL}{LD} = 1$

(Note that here I am not taking into consideration directed line segments)

$ \frac {1}{2}.\frac {CM}{MA}.\frac {1}{3} = 1$

$ \frac{AM}{MC}=\frac{1}{6}$

Problem 4

In $ \triangle{ABC}$, $ \angle{BAC} = 120$ , Let $ AD$ be the angle bisector of $ \angle{BAC}$

Express $ AD$ in terms of $ AB$ and $ BC$
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vaibhav2903
306 posts
#9 • 5 Y
Y by vprasad_nalluri, Adventure10, Mango247, and 2 other users
we know that

Click to reveal hidden text

problem 5

In a triangle $ ABC$,$ AD$ is the feet of perpendicular to $ BC$.the inradii of $ ADC$,$ ADB$ and $ ABC$ are $ x,y,z$ .find the relation between $ x,y,z$?
This post has been edited 3 times. Last edited by vaibhav2903, Feb 19, 2010, 2:37 AM
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SOURBH
212 posts
#10 • 3 Y
Y by Adventure10 and 2 other users
Click to reveal hidden text
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Agr_94_Math
881 posts
#11 • 4 Y
Y by Adventure10, Mango247, and 2 other users
$ (x+y) c \sin B  + x AC + y AB + x b \cos C + y c \cos B = 2zs$.
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gokussj3
223 posts
#12 • 4 Y
Y by Adventure10, Mango247, and 2 other users
Is there anything simpler? :maybe:
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SOURBH
212 posts
#13 • 4 Y
Y by Adventure10, Mango247, and 2 other users
Click to reveal hidden text

Problem 6

Prove that the third pedal triangle is similar to the original triangle.[/hide]
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vaibhav2903
306 posts
#14 • 4 Y
Y by Adventure10, Mango247, and 2 other users
hey,it is wrong the relation is $ x^{2} + y^{2}=z^{2}$
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vaibhav2903
306 posts
#15 • 4 Y
Y by Adventure10, Mango247, and 2 other users
still no one can answer it?
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