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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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x^2 + 3y^2 = 8n + 4
Ink68   2
N 2 minutes ago by Ink68
Let $n$ be a positive integer. Let $A$ be the number of pairs of integers $(x,y)$ satisfying $x^2 + 3y^2 = 8n + 4$ for odd values of $x$. Let $B$ be the number of pairs of integers $(x,y)$ satisfying $x^2 + 3y^2 = 8n + 4$. Prove that $A = \frac {2}{3} B$.
2 replies
Ink68
2 hours ago
Ink68
2 minutes ago
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G, L, H are collinear
Ink68   1
N 5 minutes ago by luci1337
Given an acute, non-isosceles triangle $ABC$. $B, C$ lie on a moving circle $(K)$. $(K)$ intersects $CA$ at $E$ and $BA$ at $F$. $BE, CF$ intersect at $G$. $KG, BC$ intersect at $D$. $L$ is the perpendicular image of $D$ with respect to $EF$. Prove that $G, L$ and the orthocenter $H$ are collinear.
1 reply
Ink68
an hour ago
luci1337
5 minutes ago
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Problem 1, BMO 2020
dangerousliri   40
N 17 minutes ago by Giant_PT
Source: Problem 1, BMO 2020
Let $ABC$ be an acute triangle with $AB=AC$, let $D$ be the midpoint of the side $AC$, and let $\gamma$ be the circumcircle of the triangle $ABD$. The tangent of $\gamma$ at $A$ crosses the line $BC$ at $E$. Let $O$ be the circumcenter of the triangle $ABE$. Prove that midpoint of the segment $AO$ lies on $\gamma$.

Proposed by Sam Bealing, United Kingdom
40 replies
dangerousliri
Nov 1, 2020
Giant_PT
17 minutes ago
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Terrifying "2018 \times 2019" board
IndoMathXdZ   20
N 39 minutes ago by HamstPan38825
Source: APMO 2019 P4
Consider a $2018 \times 2019$ board with integers in each unit square. Two unit squares are said to be neighbours if they share a common edge. In each turn, you choose some unit squares. Then for each chosen unit square the average of all its neighbours is calculated. Finally, after these calculations are done, the number in each chosen unit square is replaced by the corresponding average.
Is it always possible to make the numbers in all squares become the same after finitely many turns?
20 replies
IndoMathXdZ
Jun 11, 2019
HamstPan38825
39 minutes ago
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shadow of a cylinder, shadow of a cone
vanstraelen   3
N Yesterday at 7:35 PM by vanstraelen

a) Given is a right cylinder of height $2R$ and radius $R$.
The sun shines on this solid at an angle of $45^{\circ}$.
What is the area of the shadow that this solid casts on the plane of the botom base?

b) Given is a right cone of height $2R$ and radius $R$.
The sun shines on this solid at an angle of $45^{\circ}$.
What is the area of the shadow that this solid casts on the plane of the base?
3 replies
vanstraelen
May 9, 2025
vanstraelen
Yesterday at 7:35 PM
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Challenge Problem: triangle inequality
Bottema   7
N Yesterday at 6:02 PM by Speedysolver1
Prove that in any triangle we have:

a^{2}+b^{2}+c^{2} \geq 4sqrt{3}S
7 replies
Bottema
May 12, 2004
Speedysolver1
Yesterday at 6:02 PM
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2022 MARBLE - Mock ARML I -8 \frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}=32
parmenides51   2
N Yesterday at 4:33 PM by Kempu33334
Let $a,b,c$ complex numbers with $ab+ +bc+ca = 61$ such that
$$\frac{1}{b+c}+\frac{1}{c+a}+\frac{1}{a+b}= 5$$$$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}=32.$$Find the value of $abc$.
2 replies
parmenides51
Jan 14, 2024
Kempu33334
Yesterday at 4:33 PM
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Inequalities
sqing   13
N Yesterday at 3:03 PM by sqing
Let $ a,b,c>0 , a+b+c +abc=4$. Prove that
$$ \frac {a}{a^2+2}+\frac {b}{b^2+2}+\frac {c}{c^2+2} \leq 1$$Let $ a,b,c>0 , ab+bc+ca+abc=4$. Prove that
$$ \frac {a}{a^2+2}+\frac {b}{b^2+2}+\frac {c}{c^2+2} \leq 1$$
13 replies
sqing
May 15, 2025
sqing
Yesterday at 3:03 PM
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Geomettry ez
AnhIsGod   1
N Yesterday at 1:17 PM by Soupboy0
Let two circles (O) and (O') intersect at two points (one of which is called A). The common tangent CD (with C belonging to (O) and D belonging to (O')) lies on the same side as A with respect to the line OO', intersecting OO' at S. The line segment SA intersects circle (O) at E (different from A). Prove that EC is parallel to AD.
1 reply
AnhIsGod
Yesterday at 12:43 PM
Soupboy0
Yesterday at 1:17 PM
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Minimum and Maximum of Complex Numbers
pythagorazz   1
N Yesterday at 8:39 AM by alexheinis
Let $a,b,$ and $c$ be complex numbers. For a complex number $z=p+qi$ where $i=\sqrt(-1)$, define the norm $|z|$ to be the distance of $z$ from the origin, or $|z|=\sqrt(p^2+q^2 )$. Let $m$ be the minimum value and $M$ be the maximum value of $\frac{(|a+b|+|b+c|+|c+a|)}{(|a|+|b|+|c| )}$ for all complex numbers $a,b,c$ where $|a|+|b|+|c|\ne 0$. Find $M+m$.
1 reply
pythagorazz
Apr 14, 2025
alexheinis
Yesterday at 8:39 AM
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Folklore
Osim_09   2
N Yesterday at 8:36 AM by pigeon123
Let ABCD be a circumscribed quadrilateral, which is also cyclic. Let I be the incenter, O the circumcenter, and E the intersection point of the diagonals of the quadrilateral. Prove that the points O, I, and E are collinear.
2 replies
Osim_09
Jan 21, 2025
pigeon123
Yesterday at 8:36 AM
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Bounding With Powers
Shreyasharma   5
N Yesterday at 2:20 AM by jacosheebay
Is this a valid solution for the following problem (St. Petersburg 1996):

Find all positive integers $n$ such that,

$$ 3^{n-1} + 5^{n-1} | 3^n + 5^n$$
Solution
5 replies
Shreyasharma
Jul 11, 2023
jacosheebay
Yesterday at 2:20 AM
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2021 SMT Guts Round 5 p17-20 - Stanford Math Tournament
parmenides51   7
N Friday at 8:05 PM by Rombo
p17. Let the roots of the polynomial $f(x) = 3x^3 + 2x^2 + x + 8 = 0$ be $p, q$, and $r$. What is the sum $\frac{1}{p} +\frac{1}{q} +\frac{1}{r}$ ?


p18. Two students are playing a game. They take a deck of five cards numbered $1$ through $5$, shuffle them, and then place them in a stack facedown, turning over the top card next to the stack. They then take turns either drawing the card at the top of the stack into their hand, showing the drawn card to the other player, or drawing the card that is faceup, replacing it with the card on the top of the pile. This is repeated until all cards are drawn, and the player with the largest sum for their cards wins. What is the probability that the player who goes second wins, assuming optimal play?


p19. Compute the sum of all primes $p$ such that $2^p + p^2$ is also prime.


p20. In how many ways can one color the $8$ vertices of an octagon each red, black, and white, such that no two adjacent sides are the same color?


PS. You should use hide for answers. Collected here.
7 replies
parmenides51
Feb 11, 2022
Rombo
Friday at 8:05 PM
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2024 Mock AIME 1 ** p15 (cheaters' trap) - 128 | n^{\sigma (n)} - \sigma(n^n)
parmenides51   6
N Friday at 7:32 PM by NamelyOrange
Let $N$ be the number of positive integers $n$ such that $n$ divides $2024^{2024}$ and $128$ divides
$$n^{\sigma (n)} - \sigma(n^n)$$where $\sigma (n)$ denotes the number of positive integers that divide $n$, including $1$ and $n$. Find the remainder when $N$ is divided by $1000$.
6 replies
parmenides51
Jan 29, 2025
NamelyOrange
Friday at 7:32 PM
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J
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a^{2m}+a^{n}+1 is perfect square
kmh1   2
N Mar 30, 2025 by cuden
Source: own
Find all positive integer triplets $(a,m,n)$ such that $2m>n$ and $a^{2m}+a^{n}+1$ is a perfect square.
2 replies
kmh1
Mar 20, 2025
cuden
Mar 30, 2025
J
a^{2m}+a^{n}+1 is perfect square
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kmh1
105 posts
kmh1
#1 h  Mar 20, 2025, 1:34 AM
Y by
Find all positive integer triplets $(a,m,n)$ such that $2m>n$ and $a^{2m}+a^{n}+1$ is a perfect square.
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kmh1
105 posts
kmh1
#2 h  Mar 23, 2025, 5:07 PM
Y by
Bump this! It's a fun(and challenging) problem.
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cuden
122 posts
cuden
#3 h  Mar 30, 2025, 10:58 AM
Y by
bump thisss :blush:
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