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k a July Highlights and 2025 AoPS Online Class Information
jwelsh   0
Jul 1, 2025
We are halfway through summer, so be sure to carve out some time to keep your skills sharp and explore challenging topics at AoPS Online and our AoPS Academies (including the Virtual Campus)!

[list][*]Over 60 summer classes are starting at the Virtual Campus on July 7th - check out the math and language arts options for middle through high school levels.
[*]At AoPS Online, we have accelerated sections where you can complete a course in half the time by meeting twice/week instead of once/week, starting on July 8th:
[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC Problem Series[/list]
[*]Plus, AoPS Online has a special seminar July 14 - 17 that is outside the standard fare: Paradoxes and Infinity
[*]We are expanding our in-person AoPS Academy locations - are you looking for a strong community of problem solvers, exemplary instruction, and math and language arts options? Look to see if we have a location near you and enroll in summer camps or academic year classes today! New locations include campuses in California, Georgia, New York, Illinois, and Oregon and more coming soon![/list]

MOP (Math Olympiad Summer Program) just ended and the IMO (International Mathematical Olympiad) is right around the corner! This year’s IMO will be held in Australia, July 10th - 20th. Congratulations to all the MOP students for reaching this incredible level and best of luck to all selected to represent their countries at this year’s IMO! Did you know that, in the last 10 years, 59 USA International Math Olympiad team members have medaled and have taken over 360 AoPS Online courses. Take advantage of our Worldwide Online Olympiad Training (WOOT) courses
and train with the best! Please note that early bird pricing ends August 19th!
Are you tired of the heat and thinking about Fall? You can plan your Fall schedule now with classes at either AoPS Online, AoPS Academy Virtual Campus, or one of our AoPS Academies around the US.

Our full course list for upcoming classes is below:
All classes start 7:30pm ET/4:30pm PT unless otherwise noted.

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0 replies
jwelsh
Jul 1, 2025
0 replies
Bogus Proof Marathon
pifinity   7755
N a minute ago by Ryanzzz
Hi!
I'd like to introduce the Bogus Proof Marathon.

In this marathon, simply post a bogus proof that is middle-school level and the next person will find the error. You don't have to post the real solution :P

Use classic Marathon format:
[hide=P#]a1b2c3[/hide]
[hide=S#]a1b2c3[/hide]


Example posts:

P(x)
-----
S(x)
P(x+1)
-----
Let's go!! Just don't make it too hard!
7755 replies
pifinity
Mar 12, 2018
Ryanzzz
a minute ago
The 24 Game, but with a twist!
PikaPika999   438
N 3 minutes ago by Ryanzzz
So many people know the 24 game, where you try to create the number 24 from using other numbers, but here's a twist:

You can only use the number 24 (up to 5 times) to try to make other numbers :)

the limit is 5 times because then people could just do $\frac{24}{24}+\frac{24}{24}+\frac{24}{24}+...$ and so on to create any number!

honestly, I feel like with only addition, subtraction, multiplication, and division, you can't get pretty far with this, so you can use any mathematical operations!

Banned functions
438 replies
PikaPika999
Jul 1, 2025
Ryanzzz
3 minutes ago
gcd of n(n+1)(n+2)(n+3)(n+4)
Marius_Avion_De_Vanatoare   13
N an hour ago by vmene
Find the greatest common divisor of the numbers: $1\cdot 2\cdot 3 \cdot 4 \cdot 5$ and $2\cdot 3 \cdot 4 \cdot 5 \cdot 6 \dots 2025 \cdot 2026 \cdot 2027 \cdot 2028 \cdot 2029$.
13 replies
Marius_Avion_De_Vanatoare
Jun 14, 2025
vmene
an hour ago
Inequality
William_Mai   2
N an hour ago by sqing
Let $a, b \geq 0$, prove it in the shortest way:
$\sqrt{\frac{a^2+b^2}{2}} + \frac{2ab}{a+b} \geq \frac{a+b}{2} + \sqrt{ab}$
2 replies
William_Mai
Today at 2:36 AM
sqing
an hour ago
Geometry — Orthocenter, Circle Intersections, and Parallel Lines
justalonelyguy   1
N an hour ago by Royal_mhyasd
Let \( ABC \) be an acute triangle with \( AB > AC \), inscribed in a circle \( (O) \). Let \( AD, BE, CF \) be the altitudes of triangle \( ABC \), and let \( H \) be their intersection (the orthocenter). Let \( X \) be the second point of intersection of line \( BH \) with circle \( (O) \) (\( X \ne B \)). Let \( Y \) be the second point of intersection of line \( XD \) with circle \( (O) \) (\( Y \ne X \)).

Let \( Z \) be the intersection point of lines \( AY \) and \( HF \), and let \( T \) be the intersection point of lines \( DZ\) and \( AB \).

Prove that \( HT \parallel DF \).
1 reply
justalonelyguy
Today at 7:37 AM
Royal_mhyasd
an hour ago
Inequality
Ecrin_eren   3
N 2 hours ago by Ecrin_eren
For positive real numbers a, b, c satisfying
ab + ac + bc = 3abc, prove that

bc / (a⁴(b + c)) + ac / (b⁴(a + c)) + ab / (c⁴(a + b)) ≥ 3/2
3 replies
Ecrin_eren
Wednesday at 2:16 PM
Ecrin_eren
2 hours ago
Website to learn math
hawa   181
N 3 hours ago by TerrificHawk22
Hi, I'm kinda curious what website do yall use to learn math, like i dont find any website thats fun to learn math
181 replies
hawa
Apr 9, 2025
TerrificHawk22
3 hours ago
Protassov problem
NgoDucPhat   0
5 hours ago
**Problem 11.** Let triangle $ABC$ be circumscribed about circle $(I)$. The circle $(I)$ touches $BC$ at point $D$. A circle passing through $B$, $C$ and tangent to $(I)$ at $T$. Line $AT$ intersects the circle $(BTC)$ again at point $Q$. Prove that the quadrilateral $TIDQ$ is cyclic.
0 replies
NgoDucPhat
5 hours ago
0 replies
Prime Divisibility
radioactiverascal90210   0
Today at 5:28 AM
Let $p>3$ be a prime and consider a partition of the set $1, 2, ..., p-1$ into three disjoint subsets $A, B, C$. Prove that there exists $x, y$ and $z$ all in different subsets such that $y+z-x$ is divisible by $p$
0 replies
radioactiverascal90210
Today at 5:28 AM
0 replies
Inequality
William_Mai   1
N Today at 4:18 AM by gbatkhuu1
Let $a,b,c$ be the lengths of 3 sides of a triangle, prove that:
$(a+b+c)(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}) \geq \frac{3(a-b)(b-c)(c-a)}{abc}$
1 reply
William_Mai
Today at 3:53 AM
gbatkhuu1
Today at 4:18 AM
Inequalitis
sqing   7
N Today at 2:22 AM by sqing
Let $ a,b,c\geq  0 , a^2+b^2+c^2 =3.$ Prove that
$$a^3 +b^3 +c^3 +\frac{11}{5}abc  \leq \frac{26}{5}$$
7 replies
sqing
May 31, 2025
sqing
Today at 2:22 AM
Inequalities
sqing   2
N Today at 1:50 AM by sqing
Let $ a,b,c\geq 0, \frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c}=\frac{3}{2}.$ Prove that
$$ \left(a+b+c-\frac{17}{6}\right)^2+9abc   \geq\frac{325}{36}$$$$   \left(a+b+c-\frac{5}{2}\right)^2+12abc \geq\frac{49}{4}$$$$\left(a+b+c-\frac{14}{5}\right)^2+\frac{49}{5}abc \geq\frac{49}{5}$$
2 replies
sqing
Jun 30, 2025
sqing
Today at 1:50 AM
420th Post Celebration
mudkip42   7
N Today at 12:59 AM by maromex
Cheers to this being my 420th post :D! This is a collection of all of my favorite nice and cool problems I've solved on my journey so far. Enjoy! :)

Algebra:
2017 CMIMC A7: Let $a$, $b$, and $c$ be complex numbers satisfying the system of equations\begin{align*}\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}&=9,\\\dfrac{a^2}{b+c}+\dfrac{b^2}{c+a}+\dfrac{c^2}{a+b}&=32,\\\dfrac{a^3}{b+c}+\dfrac{b^3}{c+a}+\dfrac{c^3}{a+b}&=122.\end{align*}Find $abc$.

2019 All-Russian Olympiad Grade 10 P1: Each point $A$ in the plane is assigned a real number $f(A).$ It is known that $f(M)=f(A)+f(B)+f(C),$ whenever $M$ is the centroid of $\triangle ABC.$ Prove that $f(A)=0$ for all points $A.$

1997 USAMO/5: Prove that, for all positive real numbers $ a$, $ b$, $ c$, the inequality
\[ \frac {1}{a^3 + b^3 + abc} + \frac {1}{b^3 + c^3 + abc} + \frac {1}{c^3 + a^3 + abc} \leq \frac {1}{abc}
\]holds.

2008 All-Russian Olympiad Grade 10 P4: The sequences $ (a_n),(b_n)$ are defined by $ a_1=1,b_1=2$ and\[a_{n + 1} = \frac {1 + a_n + a_nb_n}{b_n}, \quad b_{n + 1} = \frac {1 + b_n + a_nb_n}{a_n}.\]Show that $ a_{2008} < 5$.

2016 MP4G P12: Let $b_1$, $b_2$, $b_3$, $c_1$, $c_2$, and $c_3$ be real numbers such that for every real number $x$, we have
\[
  x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 = (x^2 + b_1 x + c_1)(x^2 + b_2 x + c_2)(x^2 + b_3 x + c_3). 
\]Compute $b_1 c_1 + b_2 c_2 + b_3 c_3$.

Combinatorics:
Chartrand-Zhang 2.36:
Find the smallest positive integer $k$ for which there exists a simple graph on $3k$ vertexes, for which exactly $k$ vertices have degree $2$, exactly $k$ vertices have degree $6$, and exactly $k$ vertices have degree $7$.

NIMO 4.3: In chess, there are two types of minor pieces, the bishop and the knight. A bishop may move along a diagonal, as long as there are no pieces obstructing its path. A knight may jump to any lattice square $\sqrt{5}$ away as long as it isn't occupied. One day, a bishop and a knight were on squares in the same row of an infinite chessboard, when a huge meteor storm occurred, placing a meteor in each square on the chessboard independently and randomly with probability $p$. Neither the bishop nor the knight were hit, but their movement may have been obstructed by the meteors. The value of $p$ that would make the expected number of valid squares that the bishop can move to and the number of squares that the knight can move to equal can be expressed as $\tfrac{a}{b}$ for relatively prime positive integers $a, b$. Compute $100a + b$.

2023 CMIMC C7: Max has a light bulb and a defective switch. The light bulb is initially off, and on the $n$th time the switch is flipped, the light bulb has a $\tfrac 1{2(n+1)^2}$ chance of changing its state (i.e. on $\to$ off or off $\to$ on). If Max flips the switch 100 times, find the probability the light is on at the end.

NIMO 5.6: Tom has a scientific calculator. Unfortunately, all keys are broken except for one row: 1, 2, 3, + and -. Tom presses a sequence of $5$ random keystrokes; at each stroke, each key is equally likely to be pressed. The calculator then evaluates the entire expression, yielding a result of $E$. Find the expected value of $E$. (Note: Negative numbers are permitted, so 13-22 gives $E = -9$. Any excess operators are parsed as signs, so -2-+3 gives $E=-5$ and -+-31 gives $E = 31$. Trailing operators are discarded, so 2++-+ gives $E=2$. A string consisting only of operators, such as -++-+, gives $E=0$.)

2015 All-Russian Olympiad Grade 11 P5: An immortal flea jumps on whole points of the number line, beginning with $0$. The length of the first jump is $3$, the second $5$, the third $9$, and so on. The length of $k^{\text{th}}$ jump is equal to $2^k + 1$. The flea decides whether to jump left or right on its own. Is it possible that sooner or later the flea will have been on every natural point, perhaps having visited some of the points more than once?

Geometry:
EGMO 1.45(Right Angles on Incircle Chord, aka Iran Lemma): The incircle of $ABC$ is tangent to $\overline{BC}, \overline{CA}, \overline{AB}$ at $D, E, F$, respectively. Let $M$ and $N$ be the midpoints of $\overline{BC}$ and $\overline{AC}$, respectively. Ray $BI$ meets line $EF$ at $K$. Show that $\overline{BK} \perp \overline{CK}$. Then show $K$ lies on line $MN$.

2008 IMO P1: Let $ H$ be the orthocenter of an acute-angled triangle $ ABC$. The circle $ \Gamma_{A}$ centered at the midpoint of $ BC$ and passing through $ H$ intersects the sideline $ BC$ at points $ A_{1}$ and $ A_{2}$. Similarly, define the points $ B_{1}$, $ B_{2}$, $ C_{1}$ and $ C_{2}$. Prove that the six points $ A_{1}$, $ A_{2}$, $ B_{1}$, $ B_{2}$, $ C_{1}$ and $ C_{2}$ are concyclic.

1993 USAMO P2: Let $\, ABCD \,$ be a convex quadrilateral such that diagonals $\, AC \,$ and $\, BD \,$ intersect at right angles, and let $\, E \,$ be their intersection. Prove that the reflections of $\, E \,$ across $\, AB, \, BC, \, CD, \, DA \,$ are concyclic.

Unknown: Consider two circles $\Gamma_1$ and $\Gamma_2$ which are internally tangent at $P.$ A line intersects $\Gamma_1$ and $\Gamma_2$ at four distinct points $A, B, C, D$ in that order. Prove that $\angle APB = \angle CPD.$

2009 IMO P4: Let $ ABC$ be a triangle with $ AB = AC$ . The angle bisectors of $ \angle C AB$ and $ \angle AB C$ meet the sides $ B C$ and $ C A$ at $ D$ and $ E$ , respectively. Let $ K$ be the incentre of triangle $ ADC$. Suppose that $ \angle B E K = 45^\circ$ . Find all possible values of $ \angle C AB$ .


Number Theory:
2011 USAJMO P1: Find, with proof, all positive integers $n$ for which $2^n + 12^n + 2011^n$ is a perfect square.

2005 China National Olympiad P6:
Find all nonnegative integer solutions $(x,y,z,w)$ of the equation\[2^x\cdot3^y-5^z\cdot7^w=1.\]
USAMTS 5/3/36: Find all ordered triples of nonnegative integers $(a,b,c)$ satisfying $2^a \cdot 5^b - 3^c = 1.$

1990 IMO P3: Determine all integers $ n > 1$ such that
\[ \frac {2^n + 1}{n^2}
\]is an integer.

2019 IMO P4: Find all pairs $(k,n)$ of positive integers such that\[ k!=(2^n-1)(2^n-2)(2^n-4)\cdots(2^n-2^{n-1}). \]
7 replies
mudkip42
Jul 23, 2025
maromex
Today at 12:59 AM
Find the secret document
dodo7   3
N Yesterday at 11:50 PM by greenturtle3141
Dudu has 729 drawers numbered from 1 to 729. Exactly one of them contains a secret document.

Dudu can open drawers to try to find it, but with the following restrictions:

Dudu can open up to 6 drawers in total.

After opening an empty drawer, Dudu receives a hint; however, this hint can be false in up to 1 of his 6 attempts (that is, when opening an empty drawer, the safe may lie and say “the document is in a drawer with a higher number” when it is actually in a lower-numbered drawer, or vice versa).

If Dudu opens the drawer containing the document, he finds it and the search ends.

(a) Show that, despite the possibility of 1 lie, it is still possible to guarantee locating the document within 6 openings.

(b) Explain the strategy that allows overcoming this lie and finding the document safely.
3 replies
dodo7
Yesterday at 11:25 PM
greenturtle3141
Yesterday at 11:50 PM
Help to make it clear on basic Concept
Miranda2829   2
N May 22, 2025 by UberPiggy
Where I use extraneous solution in equation?
Why square of both side we will have plus and minus answer?
2 replies
Miranda2829
May 22, 2025
UberPiggy
May 22, 2025
Help to make it clear on basic Concept
G H J
G H BBookmark kLocked kLocked NReply
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Miranda2829
210 posts
#1
Y by
Where I use extraneous solution in equation?
Why square of both side we will have plus and minus answer?
Z K Y
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iwastedmyusername
224 posts
#2
Y by
Extraneous solutions are basically created when you perform an operation to both sides of an equation.

For example:

$\sqrt{x+6}=x$
$x+6=x^2$
$x^2-x-6=0$
$x=3,-2$

However, notice that plugging $-2$ into the original equation doesn't satisfy the equation
Z K Y
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UberPiggy
56 posts
#3
Y by
Extraneous solutions are created when certain operations you apply are not reversible. For example, in @above's example, both sides of the equation are squared. This means that if you reverse the steps, the $x$ values are actually the solutions to $\sqrt{x+6} = \pm x$.
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