Plan ahead for the next school year. Schedule your class today!

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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
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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.

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0 replies
jwelsh
Jul 1, 2025
0 replies
Troll Problem
giratina3   16
N 6 minutes ago by AbhayAttarde01
If $\frac{a}{a - 1} = \frac{b^2 + 2b - 1}{b^2 + 2b - 2}$, then what does $a$ equal in terms of $b$?

Hint 1
Hint 2
Hint 3
16 replies
1 viewing
giratina3
Jul 11, 2025
AbhayAttarde01
6 minutes ago
Square root and fraction
Silverfalcon   23
N 29 minutes ago by K1mchi_
$\sqrt{\frac{1}{9} + \frac{1}{16}} =$

$\textbf{(A)}\ \frac15 \qquad
\textbf{(B)}\ \frac14 \qquad
\textbf{(C)}\ \frac27 \qquad
\textbf{(D)}\ \frac{5}{12} \qquad
\textbf{(E)}\ \frac{7}{12}$
23 replies
Silverfalcon
Oct 22, 2005
K1mchi_
29 minutes ago
Canceling Powers of 10
AIME15   30
N 30 minutes ago by K1mchi_
$ \frac{10^7}{5 \times 10^4}=$

\[ \textbf{(A)}\ .002 \qquad
\textbf{(B)}\ .2 \qquad
\textbf{(C)}\ 20 \qquad
\textbf{(D)}\ 200 \qquad
\textbf{(E)}\ 2000
\]
30 replies
AIME15
Jan 12, 2009
K1mchi_
30 minutes ago
challenge funny
vhwwwwwwwww   2
N 30 minutes ago by K1mchi_
a,b,c>0 find min:
2 replies
vhwwwwwwwww
an hour ago
K1mchi_
30 minutes ago
Solution set of 2/x>3/3-x
EthanWYX2009   5
N 2 hours ago by booking
The solution set of the inequality \( \frac{2}{x} > \frac{3}{3-x} \) is ______.

Proposed by Baihao Lan, High School Attached to Northwest Normal University
5 replies
EthanWYX2009
Jul 22, 2025
booking
2 hours ago
420th Post Celebration
mudkip42   4
N 2 hours ago by booking
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\[ k!=(2^n-1)(2^n-2)(2^n-4)\cdots(2^n-2^{n-1}). \]
4 replies
mudkip42
Yesterday at 3:38 AM
booking
2 hours ago
Basic Inequalities Doubt
JetFire008   12
N 2 hours ago by booking
If $x+y=1$, find the maximum value of $x^2+y^2=1$.
I saw a solution to this question where Titu's lemma was applied and the answer was $\frac{1}{2}$. But my doubt is can't we apply other inequality to get the maximum result? or did they us titu's lemma because the given information can fit only in this lemma?
12 replies
JetFire008
Jul 18, 2025
booking
2 hours ago
Divisibility
Ecrin_eren   3
N 3 hours ago by P0tat0b0y


For which n is:

(1ⁿ + 2ⁿ + 3ⁿ + ... + nⁿ) divisible by n!?



3 replies
Ecrin_eren
Yesterday at 1:59 PM
P0tat0b0y
3 hours ago
Minimum value
Ecrin_eren   2
N 5 hours ago by alexheinis


Let x,y,z be positive real numbers such that
xyz=8
What is the minimum value of the expression

yz / [x²(y + z)] + xz / [y²(x + z)] + xy / [z²(x + y)] ?





2 replies
Ecrin_eren
Yesterday at 2:08 PM
alexheinis
5 hours ago
Fe in Q* with nice condition
Tofa7a._.36   5
N 5 hours ago by Ikbal_gk
Find all surjective functions $f : \mathbb{Q^*} \to \mathbb{Q^*}$ such that: $$(f (x) + f (y))f (x + y) = f (xy)$$for all $x,y \in \mathbb{Q^*}$ with $x+y\ne 0$.
5 replies
Tofa7a._.36
Jul 15, 2025
Ikbal_gk
5 hours ago
Inversive Geo
Tofa7a._.36   2
N 5 hours ago by Ikbal_gk
Let $ABC$ be an acute and scalene triangle with circumcircle $\omega$. The perpendicular bisector of the segment $AB$ intersects the lines $BC$ and $AC$ in points $D$ and $E$, respectively, such that $E$ lies outside segment $AC$. The perpendicular to line $BC$ from $D$ intersects $(BCE)$ at a point $X$ outside $\triangle ABC$. Line $DX$ intersects line $AC$ at $Y$ and $\omega$ at points $Z$ and $T$ such that $Z$ lies on the arc $AC$ that does not contain $B$. The circumcircle of triangle $\triangle ZET$ intersects the side $BC$ and the circumcircle of triangle $\triangle YDE$ in $P$ and $Q$, respectively.
Prove that the tangent to $(YZQ)$ from $Z$, the tangent to $(YTQ)$ from $T$, and the line $PX$ meet at one point.
2 replies
Tofa7a._.36
Jul 15, 2025
Ikbal_gk
5 hours ago
Nothing but a game
AlexCenteno2007   2
N 6 hours ago by vanstraelen
Let ABCD be a trapeze with AD ∥ BC. M and N are the midpoints of CD and BC
respectively, and P is the common point of the lines AM and DN. If PM/AP = 4, show that
ABCD is a parallelogram.
2 replies
AlexCenteno2007
Yesterday at 5:11 PM
vanstraelen
6 hours ago
Function
Ecrin_eren   1
N Today at 8:06 AM by alexheinis

Is there a function from non-negative real numbers to non-negative real numbers satisfying

f(f(x)) = |x - 1| for all x ≥ 0?











1 reply
Ecrin_eren
Yesterday at 2:26 PM
alexheinis
Today at 8:06 AM
Polynomials
Roots_Of_Moksha   5
N Today at 7:04 AM by vanstraelen
The polynomial $x^3 - 3(1+\sqrt{2})x^2 + (6\sqrt{2}-55)x -(7+5\sqrt{2})$ has three distinct real roots $\alpha$, $\beta$ and $\gamma$. The polynomial $p(x)=x^3+ax^2+bx+c$ has roots $\sqrt[3]{\alpha}$, $\sqrt[3]{\beta}$, $\sqrt[3]{\gamma}$. Find the integer closest to $a^2 + b^2 + c^2$.
Answer
5 replies
Roots_Of_Moksha
Jul 20, 2025
vanstraelen
Today at 7:04 AM
math olympiad question - please help
senboy   2
N May 7, 2025 by senboy
I got stumped on this math olympiad question



Some perfect $5^\text{th}$ powers of positive integers have all distinct digits, and some do not. For example, $5^5 = 3125$ has all distinct digits, while $6^5 = 7776$ does not. Show that the maximum number of perfect $5^\text{th}$ powers of positive integers with distinct digits is $89$.

2 replies
senboy
May 7, 2025
senboy
May 7, 2025
math olympiad question - please help
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senboy
32 posts
#1 • 1 Y
Y by PikaPika999
I got stumped on this math olympiad question



Some perfect $5^\text{th}$ powers of positive integers have all distinct digits, and some do not. For example, $5^5 = 3125$ has all distinct digits, while $6^5 = 7776$ does not. Show that the maximum number of perfect $5^\text{th}$ powers of positive integers with distinct digits is $89$.
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ostriches88
1541 posts
#2 • 1 Y
Y by PikaPika999
senboy wrote:
I got stumped on this math olympiad question



Some perfect $5^\text{th}$ powers of positive integers have all distinct digits, and some do not. For example, $5^5 = 3125$ has all distinct digits, while $6^5 = 7776$ does not. Show that the maximum number of perfect $5^\text{th}$ powers of positive integers with distinct digits is $89$.

hint

solution
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senboy
32 posts
#3
Y by
thank you!
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