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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:
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0 replies
jwelsh
Jul 1, 2025
0 replies
[Modified] 12th PMO National Stage (Written Phase)
monsterbob   1
N an hour ago by monsterbob
Find the sum of all $x$ such that $f(x) = 0$ if $f : \mathbb{R} \setminus 1 \rightarrow \mathbb{R}$ satisfies the following functional equation:

\[
x + f(x) + 2f \left( \frac{x + 2009}{x - 1} \right) = 2010
\].
1 reply
monsterbob
an hour ago
monsterbob
an hour ago
12th PMO National Stage (Oral Phase)
monsterbob   1
N an hour ago by monsterbob
Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a function that satisfies the functional equation
\[
f(x - y) = 2009f(x)f(y)
\]
for all $x, y \in \mathbb{R}$. If $f(x)$ is never zero, what is $f(\sqrt{2009})$?
1 reply
monsterbob
an hour ago
monsterbob
an hour ago
11th PMO National Stage (Written Phase)
monsterbob   1
N an hour ago by monsterbob
The sequence $\{ a_0, a_1, a_2, \dots \}$ of real numbers satisfies the recursive relation
\[
n(n + 1)a_{n + 1} + (n - 2) a_{n - 1} = n(n - 1)a_n
\]for every positive integer $n$, where $a_0 = a_1 = 1$. Calculate the sum:
\[
\frac{a_0}{a_1} + \frac{a_1}{a_2} + \cdots + \frac{a_{2008}}{a_{2009}}.
\]
1 reply
monsterbob
an hour ago
monsterbob
an hour ago
Geometry Problem from Albanian Second Round Math Olympiad
Deomad123   0
an hour ago
Let $O$ be the circumcenter of isosceles triangle $ABC (AB=BC)$. Point $M$ lies on segment $BO$ and point $M'$ is the symmetric of point $M$ with respect to the midpoint of the side $AB.$ The intersection of the lines $M'O$ and $AB$ is denoted by $K$. The point $L$ on the side $BC$ is such that $\angle CLO = \angle BLM$
Prove that the points $O, K, B, L$ lie on a circle.
0 replies
Deomad123
an hour ago
0 replies
Number Theory
Mr_GermanCano   4
N an hour ago by DensSv
Determine all pairs of natural numbers (x, n) that satisfy the equation x^3 + 2x + 1 = 2^n.
4 replies
Mr_GermanCano
Oct 5, 2024
DensSv
an hour ago
number theory
menseggerofgod   4
N 2 hours ago by HAL9000sk
find the maximun integer value of k , when X and Y are natural numbers
X /Y + Y/3 +3/X = k
4 replies
menseggerofgod
Today at 3:42 AM
HAL9000sk
2 hours ago
Interesting Summation
fjm30   19
N 3 hours ago by cortex_classes
What is the value of $ { \sum_{1 \le i< j \le 10}(i+j)}_{i+j=odd} $ $ - { \sum_{1 \le i< j \le 10}(i+j)}_{i+j=even} $
19 replies
fjm30
Jun 8, 2019
cortex_classes
3 hours ago
2018 preRMO p15, 2a-b, a-2b and a+b all distinct squares
parmenides51   7
N 3 hours ago by cortex_classes
Let $a$ and $b$ be natural numbers such that $2a-b$, $a-2b$ and $a+b$ are all distinct squares. What is the smallest possible value of $b$ ?
7 replies
parmenides51
Aug 7, 2019
cortex_classes
3 hours ago
Inequalities
sqing   14
N 3 hours ago by sqing
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
$$\left( x^2-y^2 + \frac{1}{y^2} \right) \left(y^2-x^2 + \frac{1}{x^2} \right)\leq  \frac{7+3\sqrt{5}}{2}$$$$\left( x^3 -y^3+ \frac{1}{y^3} \right) \left(y^3 -x^3+ \frac{1}{x^3} \right)\leq  9+4\sqrt{5}$$
14 replies
sqing
Jul 2, 2025
sqing
3 hours ago
Inequalities
sqing   3
N 4 hours ago by sqing
Let $ a, b, c $ be real numbers such that $ a + b + c = 0 $ and $ abc = -16 $. Prove that$$ \frac{a^2 + b^2}{c} + \frac{b^2 + c^2}{a} + \frac{c^2 + a^2}{b} -a^2\geq 2$$$$ \frac{a^2 + b^2}{c} + \frac{b^2 + c^2}{a} + \frac{c^2 + a^2}{b}-a^2-bc\geq -2$$Equality holds when $a=-4,b=c=2.$
3 replies
sqing
Today at 1:55 AM
sqing
4 hours ago
Bounding With Powers
Shreyasharma   9
N Today at 3:54 AM by MathRook7817
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
9 replies
Shreyasharma
Jul 11, 2023
MathRook7817
Today at 3:54 AM
Inequalities
sqing   18
N Today at 2:18 AM by sqing
Let $ 0\leq a \leq \frac{1}{2}$. Prove that
$$  \sqrt{\frac{2}{3}a+a^3}+\sqrt{\frac{2}{3}a+(1-2a)^3}+\sqrt{\frac{2}{3}(1-2a)+a^3}  \geq \sqrt{\frac{7}{3}}$$$$ \sqrt{\frac{3}{4}a+a^3}+\sqrt{\frac{3}{4}a+(1-2a)^3}+\sqrt{\frac{3}{4}(1-2a)+a^3} \geq \frac{1}{2}\sqrt{\frac{31}{3}}$$$$\sqrt{\frac{11}{8}a+a^3}+\sqrt{\frac{11}{8}a+(1-2a)^3}+\sqrt{\frac{11}{8}(1-2a)+a^3} \geq \frac{\sqrt{2}+ \sqrt{11}+\sqrt{13}}{4}$$$$ \sqrt{\frac{31}{20}a+a^3}+\sqrt{\frac{31}{20}a+(1-2a)^3}+\sqrt{\frac{31}{20}(1-2a)+a^3} \geq \frac{5+6\sqrt{5}+\sqrt{155}}{10\sqrt{2}}$$
18 replies
sqing
Jun 3, 2025
sqing
Today at 2:18 AM
Inequalities
sqing   15
N Today at 2:13 AM by sqing
Let $ x,y \geq 0 ,  \frac{x}{x^2+y}+\frac{y}{x+y^2} \geq 2 .$ Prove that
$$ (x-\frac{1}{2})^2+(y+\frac{1}{2})^2 \leq \frac{5}{4} $$$$ (x-1)^2+(y+1)^2 \leq  \frac{13}{4} $$$$ (x-2)^2+(y+2)^2 \leq \frac{41}{4} $$$$ (x-\frac{1}{2})^2+(y+1)^2 \leq \frac{5}{2}  $$$$ (x-1)^2+(y+2)^2 \leq  \frac{29}{4} $$$$ (x-\frac{1}{2})^2+(y+2)^2 \leq \frac{13}{2}  $$
15 replies
sqing
Jun 6, 2025
sqing
Today at 2:13 AM
2001 TAMU - Best Student Exam - Texas A&M University HS Mathematics Contest
parmenides51   10
N Today at 2:10 AM by aaravdodhia
p1. Find the area of an equilateral triangle which is inscribed in a circle of area $\pi$ .


p2. The probability that Patty passes a driving test is $p$ and the probability that she fails is $6p^2$ . Find the value $p$ .


p3. A cylindrical can without a top holds $100$ cm$^3$ of a liquid when completely full. The radius of the base of the can is $r$ cm. Express the surface area of the can in square centimeters as a function of $r$ ,


p4. If $x $, $2x+2$ , $3x+3$,$...$ are nonzero terms in a geometric progression (geometric sequence), what is the fourth term?


p5. The numbers in the figure shown below are called triangular numbers:
IMAGE
Let $a_n$ be the $n$ th triangular number, with $a_1 = 1$, $a_2 = 3$, $a_3 = 6$ etc. What is $a^2_n - a^2_{n-1}$ ?


p6. Let $P$ be the point $(3, 2)$ . Let $Q$ be the reflection of $P$ about the $x$-axis, $R$ the reflection of $Q$ about the line $y = -x$ and $S$ the reflection of $R$ through the origin. $PQRS$ is a convex quadrilateral. Find its area.


p7. Find the minimum value of $f(x) = 2|2x-1| - |3x-1|+|4x-3|$ on the interval $[0, 1]$ .


p8. A regular dodecagon ($12$ sides) is inscribed in a circle of radius $r$ inches. What is the area of the dodecagon in square inches?


p9. Let $S = \{a, b, c, d, e\}$ . Find $\sum_{A \subseteq S} n(A)$ , where $n(A)$ is the number of elements in the set $A$.


p10. The sum of two numbers is $4$ and their product is $1$. Find the sum of their cubes.


p11. Triangle $ABC$ is a right triangle, $D$ is a point on the leg $BC$ and $E$ is the foot of the perpendicular from $D$ to the hypotenuse $AB$ . If the segments $AC$ , $AE$ and $EB$ are $10$, $14$ and $12$ respectively, find the length of segment $BD$.
IMAGE


p12. The batting average for a baseball player is determined by dividing his total number of hits for the season by his total number of official at bats for the season. A baseball player had an average of $0.250$ prior to his game yesterday. The player had $0$ hits in $4$ official at bats in his game yesterday and his average dropped to $0.2475$. How many hits does this player have for the season?


p13. If $f$ is twice continuously differentiable, find $\lim_{h\to 0^+}\frac{f(a +\sqrt{h}) - 2f(a) + f(a-\sqrt{h})}{h}$.


p14. Let $w$ be a solution to $x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 = 0$ . Find $w^{14}$ .


p15. Let $s$ be the limiting sum of the series $4 - \frac83 + \frac{16}{9}- \frac{32}{27}+ ...$ .Then $s$ equals ?


p16. Any five points are taken inside or on a square of side length $1$. Let $a$ be the smallest possible number with the property that it is always possible to select one pair of points from these five such that the distance between them is less than or equal to $a$ . What is the value of $a$ ?


p17. Let $n$ be the number of pairs of numbers $b$ and $c$ such that $3x + by + c = 0$ and $cx - 2y + 12 = 0$ have the same graph. Find $n$ .


p18. Find three integers which form a geometric progression if their sum is $21$ and the sum of their reciprocals is $\frac{7}{12}$.


p19. $$\lim_{x \to \infty} \frac{\sqrt{x^2 + 3x}0\sqrt{x^2 - 3x}}{\sqrt{x^2 + 9x}-\sqrt{x^2 -9x}}=$$

p20. Suppose that $x$ satisfies the equation $\sin 2x - \cos 3x = 0$ . Find the smallest possible value of $\cos x$.


p21. Among all ordered pairs of real numbers $(x, y)$ which satisfy $x^4 + y^4 = x^2 + y^2$ ,what is the largest value of $x$ ?



PS. You should use hide for answers. Collected here.
10 replies
parmenides51
Mar 23, 2022
aaravdodhia
Today at 2:10 AM
Minimum and Maximum of Complex Numbers
pythagorazz   1
N May 17, 2025 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
May 17, 2025
Minimum and Maximum of Complex Numbers
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pythagorazz
382 posts
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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 $M+m$.
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alexheinis
10731 posts
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We can fix $|a|+|b|+|c|=1$ and we have to min/max the quantity $f:=|a+b|+|b+c|+|c+a|$.
We have $f\le |a|+|b|+|b|+|c|+|c|+|a|=2$ with equality when $a=b=c=1/3$. We also have
$f=|-a-b|+|b+c|+|c+a|\ge |(-a-b)+(b+c)+(c+a)|=2|c|$ and with the two other cases we find $f\ge 2\max(|a|,|b|,|c|)\ge 2/3$. Equality when $a=b=1/3,c=-1/3$.
This post has been edited 2 times. Last edited by alexheinis, May 17, 2025, 9:03 AM
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