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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!
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0 replies
jwelsh
Jul 1, 2025
0 replies
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Show that angle ADO = angle HAN
v_Enhance   110
N 4 hours ago by Ilikeminecraft
Source: JMO 2017 P5
Let $O$ and $H$ be the circumcenter and the orthocenter of an acute triangle $ABC$. Points $M$ and $D$ lie on side $BC$ such that $BM=CM$ and $\angle BAD = \angle CAD$. Ray $MO$ intersects the circumcircle of triangle $BHC$ in point $N$. Prove that $\angle ADO = \angle HAN$.
110 replies
v_Enhance
Apr 20, 2017
Ilikeminecraft
4 hours ago
J
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Inequalities
sqing   9
N 5 hours ago by sqing
Let $ 0\leq a \leq \frac{1}{2}$. Prove that
$$ \sqrt{\frac{2}{3}a+a^3}+\sqrt{\frac{2}{3}a+(1-2a)^3}+\sqrt{\frac{2}{3}(1-2a)+a^3} \geq \sqrt{\frac{7}{3}}$$$$ \sqrt{\frac{3}{4}a+a^3}+\sqrt{\frac{3}{4}a+(1-2a)^3}+\sqrt{\frac{3}{4}(1-2a)+a^3} \geq \frac{1}{2}\sqrt{\frac{31}{3}}$$$$\sqrt{\frac{11}{8}a+a^3}+\sqrt{\frac{11}{8}a+(1-2a)^3}+\sqrt{\frac{11}{8}(1-2a)+a^3} \geq \frac{\sqrt{2}+ \sqrt{11}+\sqrt{13}}{4}$$$$ \sqrt{\frac{31}{20}a+a^3}+\sqrt{\frac{31}{20}a+(1-2a)^3}+\sqrt{\frac{31}{20}(1-2a)+a^3} \geq \frac{5+6\sqrt{5}+\sqrt{155}}{10\sqrt{2}}$$
9 replies
sqing
Jun 3, 2025
sqing
5 hours ago
J
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introductory olympiad problems
Soupboy0   1
N Today at 4:50 AM by babyzombievillager
does anybody know any competitons with introductory olympiad questions
1 reply
Soupboy0
Today at 4:44 AM
babyzombievillager
Today at 4:50 AM
J
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Inequalities
sqing   9
N Today at 3:50 AM by massivesheep6
Suppose that $x$ and $y$ are nonzero real numbers such that $\left(x + \frac{1}{y} \right) \left(y + \frac{1}{x} \right) = 5$. Prove that
$$\left( x^2-y^2 + \frac{1}{y^2} \right) \left(y^2-x^2 + \frac{1}{x^2} \right)\leq \frac{7+3\sqrt{5}}{2}$$$$\left( x^3 -y^3+ \frac{1}{y^3} \right) \left(y^3 -x^3+ \frac{1}{x^3} \right)\leq 9+4\sqrt{5}$$
9 replies
sqing
Wednesday at 11:54 AM
massivesheep6
Today at 3:50 AM
J
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aime scores
MathBoy98   19
N Today at 3:26 AM by Andyluo
what is a good aime score if this is your first time qualifying?
19 replies
MathBoy98
Jan 10, 2023
Andyluo
Today at 3:26 AM
J
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2000 TAMU - Best Student Exam - Texas A&M University HS Mathematics Contest
parmenides51   14
N Today at 2:37 AM by imtiyas1
p1. Simplify the expression $(x-1)^4 + 4(x-1)^3 + 6(x-1)2 + 4(x-1) + 1$.


p2. Find the minimum value of $\sqrt{x^2 + y^2}$ if $6x-5y = 4$.


p3. Suppose $x, b > 0$ and $\log_{b^2} x + \log_{x^2} b = 1$.Find x.


p4. The sum of $n$ terms in an arithmetic progression is $153$, and the common difference is $2$. If the fist term is an integer, and $n > 1$, then what is the number of all possible values for $n$?


p5. Let $f$ be a function such that $f(3) = 1$ and $f(3x) = x+f (3x- 3)$ for all $x$. Find $f(300)$.


p6. Suppose $\vartriangle ABC$ is an equilateral triangle and $P$ is a point interior to $\vartriangle ABC$. If the distance from P to sides $AB$, $BC$ and $AC$ is $6$, $7$ and $8$ units respectively, what is the area of $\vartriangle ABC$?


p7. If $A =\begin{pmatrix} a & b\\ c & d \end{pmatrix}$ and $B =\begin{pmatrix} w & x\\ y & z \end{pmatrix}$ then the product $A \cdot B$ is defined to be $AB =\begin{pmatrix} aw + by & ax + bz\\ cw + dy & cx + dz \end{pmatrix}$.
Furthermore, $A^2 = A \cdot A$, $A^3 = A \cdot A \cdot A$, etc $...$ If $A =\begin{pmatrix} 0 & a\\ b & 0 \end{pmatrix}$, find $A^{241}$.


p8. A circle and a parabola are drawn in the $xy$-plane. The circle has its center at $(0, 5)$ with a radius of $4$, and the parabola has its vertex at $(0, 0)$ . If the circle is tangent to the parabola at two points, give the equation of the parabola.


p9. The triangle $PQR$ sits in the $xy$-plane with $P = (0, 0)$, $Q = (3, 12)$ and $R = (6, 0)$ . Suppose the x-axis represents the horizontal ground and the triangle is rotated counter clockwise around the origin (note that $P$ will stay fixed) until it reaches a position where it balances perfectly on the vertex $P$. What is the y-coordinate of the point $Q$ when the triangle is balanced?


p10. A circle is placed in the $xy$-plane and a line $L$ is drawn through the center of the circle. Suppose $P$ is a point interior to the circle which is $6$ units from the circle, $6$ units from the line $L$ and $10$ units from the closest intersection point of the line $L$ with the circle. What is the area of the circle?


p11. Five people are asked (individually) to choose a random integer in the interval $[1, 20]$. What is the probability that everyone chooses a different number?


p12. A matrix $\begin{pmatrix} a & b\\ c & d \end{pmatrix}$ is said to be singular if $ad - bc = 0$. If the matrix $\begin{pmatrix} a & b\\ c & d \end{pmatrix}$ is created at random by choosing integer values $a, b, c, d$ at random from the interval $[-3, 3]$ , what is the probability that the matrix will be singular?


p13. Consider the table of values shown below $\begin{tabular}{ | l | c | c | r| } \hline 2 & a & b & 2 \\ \hline c & 2 & 2 & d \\ \hline e & 2 & 2 & f\\ \hline 2 & g & h & 2\\ \hline \end{tabular}$.
All rows and columns of this table sum to $0$. In addition, $a + c = 5$ and $eg = 22$. Find all possible solutions $(a, b, c, d, e, f, g, h)$.


p14. Let $P > 0$ and suppose $\vartriangle ABC$ is an isosceles right triangle with area $P$ square inches. What is the radius of the circle that passes through the points $A, B$ and $C$?


p15. How must the numbers $a, b$ and $c$ be related for the following system to have at least one solution?
$$x - 2y + z = a$$$$2x + y - 2z = b$$$$x + 3y-3z = c$$

p16. Let $x$ be a real number and create a triangle having vertices $(-2, 1)$ , $(1, 3)$ and $(3x, 2x- 3)$ : Give a formula for the area of this triangle.


p17. The final race in a swimming event involves $8$ swimmers. Three of the swimmers are from one country and the other five are from different countries. Each is to be given a lane assignment between $1$ and $8$ for the race. Aside from the obvious rule that no two swimmers can be assigned to the same lane, there are two other restrictions. The first is that no two swimmers from the same country can be in adjacent lanes. The second is that the two outside lanes cannot be occupied by swimmers from the same country. With these rules, how many different lane assignments are possible for this race?


p18. Let $r > 0$. Four circles of radius $2r$ are placed in the xy-plane so that their centers are located at $(-r,-r)$ , $(-r, r)$ , $(r, r)$ and $(r,-r)$ . What is the area of the region of intersection of these circles?


PS. You should use hide for answers. Collected here.
14 replies
parmenides51
Mar 22, 2022
imtiyas1
Today at 2:37 AM
J
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Simple Angle Chasing Problem Stumping Me
WheatNeat   6
N Today at 2:31 AM by Sid-darth-vater
Isoceles triangle $ABC$ with $AC = BC$, has angle $CAB$ as $80$ degrees. Let points $E$ and $D$ be on sides $BC$ and $AC$ respectively, such that angle $CBD$ is 20 degrees and angle $CAE$ is 10 degrees. Find angle $DEA$.
6 replies
WheatNeat
Yesterday at 8:54 AM
Sid-darth-vater
Today at 2:31 AM
J
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1st Penchick Online Mathematical Olympiad 2025
aops-g5-gethsemanea2   8
N Today at 2:05 AM by happypi31415
Welcome to the first ever season of the

[center]IMAGE[br]Penchick Online Math Olympiad (POMO)![/center]

POMO is a contest ran by Filipino students in the PMO (Philippine Mathematical Olympiad) Training Community. It is a hybrid between the AMC 12 and the AIME, and is based on the format of the PMO.

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Format

The POMO 1 will run from July 4, 2025, at 9:00 am UTC+8 to July 6, 2025, at 9:00 pm UTC+8. It consists of three parts to be solved in 3 hours:
[list]
[*] Part I: 5 multiple-choice questions worth 1 point each
[*] Part II: 15 multiple-choice questions with 2 points each
[*] Part III: 8 short-answer questions (answers are integers from 1 to 999) worth 5 points each
[/list]

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Registration

To register, you need to first make an account in this website: https://penchickomo.replit.app, then go the the Contests page and select "POMO 1 (Qualifying stage)". Finally, click the register button to confirm your registration!

Please note that account creation does not automatically register you for the contest, and the registration period goes on until the end of the contest period.

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Full Details

The mechanics can be found in this link.

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Sample Problems

Here are some sample problems to help you practice for the POMO.

Sample Problem 1

Sample Problem 2

Sample Problem 3

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Wait, who's Penchick?

Penchick— short for penguin chick—is the beloved name of a distinctive penguin plush toy shown in the logo

In 2023, IMO silver and IOI bronze medalist Ephraim Wu transformed this simple toy into a symbol of good fun, good luck, and strength within the Philippine competitive programming scene. In fact, the 2024 IOI team brought their Penchicks with them to see the pyramids of Egypt! Wow!

So, our team has decided to carry the legacy and name this competition in honor of Penchick. Penchick Online Math Olympiad— HERE WE GO!

-----

[center]All further announcements will be done in this Discord server.
We hope you enjoy the contest![/center]
8 replies
aops-g5-gethsemanea2
Yesterday at 11:43 AM
happypi31415
Today at 2:05 AM
J
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min m such m2^5x 3^6x4^3x5^3x6^7 perfect square - IOQM 2020-21 p6
parmenides51   4
N Yesterday at 9:08 PM by SomeonecoolLovesMaths
What is the least positive integer by which $2^5 \cdot 3^6 \cdot 4^3 \cdot 5^3 \cdot 6^7$ should be multiplied so that, the product is a perfect square?
4 replies
parmenides51
Jan 18, 2021
SomeonecoolLovesMaths
Yesterday at 9:08 PM
J
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Expected value of OH^2 (OTIS Mock AIME 2024 #9)
v_Enhance   22
N Yesterday at 9:04 PM by Kempu33334
Let $\omega$ be a circle with center $O$ and radius $12$. Points $A$, $B$, and $C$ are chosen uniformly at random on the circumference of $\omega$. Let $H$ denote the orthocenter of $\triangle ABC$. Compute the expected value of $OH^2$.

Jiahe Liu
22 replies
v_Enhance
Jan 16, 2024
Kempu33334
Yesterday at 9:04 PM
J
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no of integer soultions of ||x| - 2020| < 5 - IOQM 2020-21 p5
parmenides51   13
N Yesterday at 9:03 PM by SomeonecoolLovesMaths
Find the number of integer solutions to $||x| - 2020| < 5$.
13 replies
parmenides51
Jan 18, 2021
SomeonecoolLovesMaths
Yesterday at 9:03 PM
J
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\sum_{k=1}^{N} (2k+1)/(k^2+k)^2}= 0.9999 - IOQM 2020-21 p3
parmenides51   4
N Yesterday at 8:58 PM by SomeonecoolLovesMaths
If $\sum_{k=1}^{N} \frac{2k+1}{(k^2+k)^2}= 0.9999$ then determine the value of $N$.
4 replies
parmenides51
Jan 18, 2021
SomeonecoolLovesMaths
Yesterday at 8:58 PM
J
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IOQM 2021 P2
RMOAspirantFaraz   10
N Yesterday at 8:55 PM by SomeonecoolLovesMaths
A number $N$ is in base 10, $503$ in base $b$ and $305$ in base $b+2$ find product of digits of $N$
10 replies
RMOAspirantFaraz
Jan 17, 2021
SomeonecoolLovesMaths
Yesterday at 8:55 PM
J
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[ABCD] = n [CDE], areas in trapezoid - IOQM 2020-21 p1
parmenides51   5
N Yesterday at 8:51 PM by SomeonecoolLovesMaths
Let $ABCD$ be a trapezium in which $AB \parallel CD$ and $AB = 3CD$. Let $E$ be then midpoint of the diagonal $BD$. If $[ABCD] = n \times [CDE]$, what is the value of $n$?

(Here $[t]$ denotes the area of the geometrical figure$ t$.)
5 replies
parmenides51
Jan 18, 2021
SomeonecoolLovesMaths
Yesterday at 8:51 PM
J
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J
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Mop Qual stuff
HopefullyMcNats2025   77
N Jun 25, 2025 by ek-nalshtheguy
How good of an award/ achievement is making MOP, I adore comp math but am scared if I dedicate all my time to it I won’t get in a good college such as MIT or Harvard
77 replies
HopefullyMcNats2025
Mar 30, 2025
ek-nalshtheguy
Jun 25, 2025
J
Mop Qual stuff
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Reveal topic tags
college apps
MIT
college
Harvard
L
G H BBookmark kLocked kLocked NReply
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HopefullyMcNats2025
45 posts
HopefullyMcNats2025
#1 h  Mar 30, 2025, 11:23 PM • 3 Y
Y by RainbowJessa, Exponent11, cubres
How good of an award/ achievement is making MOP, I adore comp math but am scared if I dedicate all my time to it I won’t get in a good college such as MIT or Harvard
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Pengu14
651 posts
Pengu14
#2 h  Mar 30, 2025, 11:31 PM • 3 Y
Y by RainbowJessa, cubres, aidan0626
HopefullyMcNats2025 wrote:
How good of an award/ achievement is making MOP, I adore comp math but am scared if I dedicate all my time to it I won’t get in a good college such as MIT or Harvard

MOP basically guarantees MIT/harvard
Z K Y
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HopefullyMcNats2025
45 posts
HopefullyMcNats2025
#3 h  Mar 30, 2025, 11:35 PM • 4 Y
Y by elasticwealth, RainbowJessa, cubres, Jyuan26
Not true at all
Z K Y
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mdk2013
663 posts
mdk2013
#4 h  Mar 30, 2025, 11:38 PM • 2 Y
Y by RainbowJessa, cubres
Pengu14 wrote:
HopefullyMcNats2025 wrote:
How good of an award/ achievement is making MOP, I adore comp math but am scared if I dedicate all my time to it I won’t get in a good college such as MIT or Harvard

MOP basically guarantees MIT/harvard

def not true, a big portion of college apps comes from writing and literature
Z K Y
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hashbrown2009
197 posts
hashbrown2009
#5 h  Mar 30, 2025, 11:42 PM • 1 Y
Y by cubres
mdk2013 wrote:
Pengu14 wrote:
HopefullyMcNats2025 wrote:
How good of an award/ achievement is making MOP, I adore comp math but am scared if I dedicate all my time to it I won’t get in a good college such as MIT or Harvard

MOP basically guarantees MIT/harvard

def not true, a big portion of college apps comes from writing and literature

While I agree MOP doesn't guarantee, your statement isn't accurate either. They say it is but it actually is only like 40% of the percent. the other 60% comes from extra-curriculars, volunteer service, your grades and your achievements.
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Shan3t
467 posts
Shan3t
#6 h  Mar 30, 2025, 11:43 PM • 1 Y
Y by cubres
Pengu14 wrote:
HopefullyMcNats2025 wrote:
How good of an award/ achievement is making MOP, I adore comp math but am scared if I dedicate all my time to it I won’t get in a good college such as MIT or Harvard

MOP basically guarantees MIT/harvard

idk if this is true.

Mit/Harvard looks 4 science, coding, literature, etc. But if ur like a GOD at maths, aka Luke Robitaille level, you can definitely make those colleges.
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Pengu14
651 posts
Pengu14
#7 h  Mar 30, 2025, 11:43 PM • 1 Y
Y by cubres
mdk2013 wrote:
Pengu14 wrote:
HopefullyMcNats2025 wrote:
How good of an award/ achievement is making MOP, I adore comp math but am scared if I dedicate all my time to it I won’t get in a good college such as MIT or Harvard

MOP basically guarantees MIT/harvard

def not true, a big portion of college apps comes from writing and literature

I meant it in the sense that around 95% of MOPpers get into MIT. Also, why would literature matter at all for college apps?
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mdk2013
663 posts
mdk2013
#8 h  Mar 30, 2025, 11:43 PM • 2 Y
Y by Exponent11, cubres
bruv english is important
This post has been edited 1 time. Last edited by mdk2013, Jun 25, 2025, 5:11 PM
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LearnMath_105
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LearnMath_105
#9 h  Mar 30, 2025, 11:45 PM • 1 Y
Y by cubres
Not sure about harvard but from what I've heard its a near lock for MIT as long as your GPA isn't bad
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Pengu14
651 posts
Pengu14
#10 h  Mar 30, 2025, 11:46 PM • 1 Y
Y by cubres
Shan3t wrote:
Pengu14 wrote:
HopefullyMcNats2025 wrote:
How good of an award/ achievement is making MOP, I adore comp math but am scared if I dedicate all my time to it I won’t get in a good college such as MIT or Harvard

MOP basically guarantees MIT/harvard

idk if this is true.

Mit/Harvard looks 4 science, coding, literature, etc. But if ur like a GOD at maths, aka Luke Robitaille level, you can definitely make those colleges.

They look for achievements and dedication in things you’re passionate about.
This post has been edited 1 time. Last edited by Pengu14, Mar 30, 2025, 11:46 PM
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scannose
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scannose
#11 h  Mar 30, 2025, 11:46 PM • 1 Y
Y by cubres
it should at the very least increase your chances by a lot
mit does value qualification for mop, and they should admit you as long as your gpa is good and nothing else is problematic (although what counts as a "good" gpa is debatable)
i've also heard that mit doesn't care that much about essays, but from how my source turned out last time take this will possibly less than a grain of salt
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hashbrown2009
197 posts
hashbrown2009
#12 h  Mar 30, 2025, 11:47 PM • 1 Y
Y by cubres
Shan3t wrote:
Pengu14 wrote:
HopefullyMcNats2025 wrote:
How good of an award/ achievement is making MOP, I adore comp math but am scared if I dedicate all my time to it I won’t get in a good college such as MIT or Harvard

MOP basically guarantees MIT/harvard

idk if this is true.

Mit/Harvard looks 4 science, coding, literature, etc. But if ur like a GOD at maths, aka Luke Robitaille level, you can definitely make those colleges.

Note that nearly all people who apply for MIT, Princeton, Harvard etc. who have made MOP usually apply for math or computer science and nearly all make it in, according to statistics of the MOPpers.

Edit: Also, please note that very few people are as good as Luke Robitaille, so don't use him as a comparison.
This post has been edited 1 time. Last edited by hashbrown2009, Mar 30, 2025, 11:48 PM
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HopefullyMcNats2025
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HopefullyMcNats2025
#13 h  Mar 30, 2025, 11:54 PM • 1 Y
Y by cubres
I would say no one aside from Evan Chen, because luke made imo gold medal in 9th and won mc 2x in a row, his achievements are unique to him mostly
This post has been edited 1 time. Last edited by HopefullyMcNats2025, Mar 30, 2025, 11:55 PM
Reason: Typo
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Pengu14
651 posts
Pengu14
#14 h  Mar 30, 2025, 11:58 PM • 1 Y
Y by cubres
HopefullyMcNats2025 wrote:
I would say no one aside from Evan Chen, because luke made imo gold medal in 9th and won mc 2x in a row, his achievements are unique to him mostly

dotted is also most likely going to be a 4x gold medalist
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awesomeming327.
1780 posts
awesomeming327.
#15 h  Mar 31, 2025, 12:03 AM • 9 Y
Y by scannose, Pengu14, Marcus_Zhang, aidan0626, Radio2, cubres, aidensharp, Sedro, KevinYang2.71
If you really love math competitions, you should do it regardless of whether it looks good on a college application. This is because if you end up not making MOP, you'll feel sad but rewarded by the experience overall, as opposed to only feeling sad.

But for what it's worth, making MOP is nearly a direct ticket.
This post has been edited 1 time. Last edited by awesomeming327., Mar 31, 2025, 12:04 AM
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