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k a June Highlights and 2025 AoPS Online Class Information
jlacosta   0
Jun 2, 2025
Congratulations to all the mathletes who competed at National MATHCOUNTS! If you missed the exciting Countdown Round, you can watch the video at this link. Are you interested in training for MATHCOUNTS or AMC 10 contests? How would you like to train for these math competitions in half the time? We have accelerated sections which meet twice per week instead of once starting on July 8th (7:30pm ET). These sections fill quickly so enroll today!

[list][*]MATHCOUNTS/AMC 8 Basics
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[*]AMC 10 Problem Series[/list]
For those interested in Olympiad level training in math, computer science, physics, and chemistry, be sure to enroll in our WOOT courses before August 19th to take advantage of early bird pricing!

Summer camps are starting this 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 a transformative 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)!

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0 replies
jlacosta
Jun 2, 2025
0 replies
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tangent circles
m4thbl3nd3r   1
N 7 minutes ago by Lil_flip38
Let $O,H,T$ be circumcenter, orthocenter and A-HM point of triangle $ABC$. Let $AH,AT$ intersect $(O)$ at $K,N$, respectively. Let $XYZ$ be the triangle formed by $TH,BC,KN$. Prove that $(XYZ)$ is tangent to $(O).$
1 reply
m4thbl3nd3r
an hour ago
Lil_flip38
7 minutes ago
J
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inequality
SunnyEvan   2
N 14 minutes ago by sqing
Let $ a,b > 0 ,$ such that : $ a+b \geq \frac{3(a^4+b^4)}{a^2+b^2+1}\sqrt{\frac{\frac{1}{a}+\frac{1}{b}}{a+b}}.$
Prove that: $$ \frac{a^2+b^2+2}{a^6b^2+a^2b^6} \geq 2 $$
2 replies
SunnyEvan
Today at 6:53 AM
sqing
14 minutes ago
J
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inequality
SunnyEvan   8
N 18 minutes ago by sqing
Let $ x,y \geq 0 ,$ such that : $ \frac{x^2}{x^3+y}+\frac{y^2}{x+y^3} \geq 1 .$
Prove that : $$ x^2+y^2-xy \leq x+y $$$$ (x+\frac{1}{2})^2+(x+\frac{1}{2})^2 \leq \frac{5}{2} $$$$ (x+1)^2+(y+1)^2 \leq 5 $$$$ (x+2)^2+(y+2)^2 \leq 13 $$
8 replies
+2 w
SunnyEvan
Yesterday at 1:51 PM
sqing
18 minutes ago
J
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Why did all the old Japanese math Olympiad questions disappear?
parkjungmin   0
26 minutes ago
Why did all the old Japanese math Olympiad questions disappear?

Did the administrator delete it?
0 replies
parkjungmin
26 minutes ago
0 replies
J
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Counting problem
lgx57   1
N 2 hours ago by HAL9000sk
Calculate the number of $n$ that meet the following conditions:
1. $585 \mid n$
2.$0 \sim 7$ appears exactly once in each octal digit of $n$
1 reply
lgx57
5 hours ago
HAL9000sk
2 hours ago
J
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Geometry
Arytva   0
3 hours ago
Let \(ABC\) be an acute triangle, and let its circumcircle be \(\Gamma\). On side \(BC\), pick a point \(D\) (distinct from \(B\) and \(C\)). The lines through \(D\) tangent to \(\Gamma\) (other than \(DA\), if \(A\) lies inside the angle at \(D\)) touch \(\Gamma\) again at points \(E\) and \(F\). Let \(BE\) meet \(AC\) at \(P\), and let \(CF\) meet \(AB\) at \(Q\). Prove that the three lines \(AP\), \(AQ\), and \(EF\) are concurrent.
0 replies
Arytva
3 hours ago
0 replies
J
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Easy Function
Darealzolt   1
N 4 hours ago by alexheinis
Let \( f(x+y) = f(x^2y)\) for all real numbers \(x,y\), hence find the value of \(f(3)\) if \(f(2023)=26\).
1 reply
Darealzolt
4 hours ago
alexheinis
4 hours ago
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Spot-It inspired question
alexroberts   1
N Today at 6:25 AM by alexroberts
Oscar bought a set of blank playing cards. He puts stamps on each card such that
1. Each card has $k\geq 4$ different stamps each.
2. Every two cards have exactly one stamp in common.
3. Every stamp is used at least twice.

Show that the maximum number of different stamps $v$ he can use is in the range $$k^2-2k+5 \leq v \leq k^2-k+1$$
1 reply
alexroberts
Wednesday at 8:46 PM
alexroberts
Today at 6:25 AM
J
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Floor of Cube Root
Magdalo   2
N Today at 6:18 AM by RedFireTruck
Find the amount of natural numbers $n<1000$ such that $\lfloor \sqrt[3]{n}\rfloor\mid n$.
2 replies
Magdalo
Jun 2, 2025
RedFireTruck
Today at 6:18 AM
J
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Find the value of m
Darealzolt   2
N Today at 6:13 AM by RedFireTruck
Let \(m\) be a positive integer, such that \(m\) fulfills
\[ \frac{1}{m^2+3m+2}+\frac{1}{m^2+5m+6}+\frac{1}{m^2+7m+12}+\dots +\frac{1}{m^2+15m+56}+\frac{1}{m^2+17m+72} = \frac{8}{33} \]Hence find the value of \(m\).
2 replies
Darealzolt
Yesterday at 11:38 AM
RedFireTruck
Today at 6:13 AM
J
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Polynomials
P162008   2
N Today at 6:02 AM by RedFireTruck
P1. Find $p(0) + p(5)$ where $p$ is a monic polynomial of degree $4$ satisfying $p(r) = 2^r ; r = 1,2,3,4.$

P2. Find $p(1), p(-1)$ where $p$ is a polynomial of smallest degree possible satisfying $p(r) = \frac{1}{r^2 - 1}; r = 2,3,4,\cdots, 10.$

P3. Find $k$ and $p(0)$, if polynomial $p$ satisfies $x.p(x) + 1 = k\left(\prod_{i = 1}^{5} (x - i)\right).$

P4. Find $p(0)$ where $p$ is a polynomial of smallest degree satisfying $p(r) = \frac{1}{r}; r = 1,2,3,\cdots,10.$

P5. Find $p(0),p(6),k$ and $\alpha$ if polynomial $p$ satisfies $(x^2 - 6x).p(x) + 1 = k\left(\prod_{i=1}^{5} (x - i)\right)(x - \alpha).$

P6. If $f(x)$ is a polynomial of degree $50$ such that $f(x) = \frac{x}{x + 1}; x = 0,1,2,\cdots,50.$ Evaluate $f(-1).$
2 replies
P162008
Today at 12:17 AM
RedFireTruck
Today at 6:02 AM
J
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common tangents
gasgous   2
N Today at 5:26 AM by gasgous
Find the equations of the common tangents to the circles:$\left(x-1\right)^2+{(y+2)}^2=16$ and $\left(x+2\right)^2+{(y-3)}^2=36$.
2 replies
gasgous
Jun 4, 2025
gasgous
Today at 5:26 AM
J
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the Basics
wpdnjs   6
N Today at 4:56 AM by kyEEcccccc
given that log base 3 of 2 is approximately 0.631, fin the smallest positivie integer a such that 3^a > 2^102.



somebody anyone pls help :wacko:
6 replies
wpdnjs
Today at 3:00 AM
kyEEcccccc
Today at 4:56 AM
J
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Find sets of positive integers (a,b) such that there exists a positive integer c
kyotaro   1
N Today at 3:19 AM by alexheinis
Find sets of positive integers (a,b) such that there exists a positive integer c such that $$a^n+b^{n+9} =c (mod 13),\forall n\in \mathbb N^*$$and $0<a,b<13$.
1 reply
kyotaro
Today at 1:29 AM
alexheinis
Today at 3:19 AM
J
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J
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Gen func 8s somewhere in the balkans
Assassino9931   1
N Apr 27, 2025 by MR.1
Source: Balkan MO 2024 Shortlist A2
Prove that there is a positive integer such that the decimal representation of
$$ \sum_{k=1}^{\lfloor n/3 \rfloor} \binom{n}{3k}8^k $$has all of its last $2023$ digits equal to $8$.
1 reply
Assassino9931
Apr 27, 2025
MR.1
Apr 27, 2025
J
Gen func 8s somewhere in the balkans
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Reveal topic tags
algebra
generating functions
complex numbers
Summation
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G H BBookmark kLocked kLocked NReply
Source: Balkan MO 2024 Shortlist A2
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Assassino9931
1390 posts
Assassino9931
#1 h  Apr 27, 2025, 12:38 PM
Y by
Prove that there is a positive integer such that the decimal representation of
$$ \sum_{k=1}^{\lfloor n/3 \rfloor} \binom{n}{3k}8^k $$has all of its last $2023$ digits equal to $8$.
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MR.1
138 posts
MR.1
#2 h  Apr 27, 2025, 12:41 PM
Y by
also georgia bmo tst p2
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