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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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0 replies
jwelsh
Jul 1, 2025
0 replies
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Peru IMO TST 2023
diegoca1   1
N an hour ago by grupyorum
Source: Peru IMO TST 2023 pre-selection P4
Prove that, for every integer $n \geq 3$, there exist $n$ positive composite integers that form an arithmetic progression and are pairwise coprime.

Note: A positive integer is called composite if it can be expressed as the product of two integers greater than 1.
1 reply
diegoca1
Yesterday at 7:38 PM
grupyorum
an hour ago
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Quad formed by orthocenters has same area (all 7's!)
v_Enhance   38
N an hour ago by Kempu33334
Source: USA January TST for the 55th IMO 2014
Let $ABCD$ be a cyclic quadrilateral, and let $E$, $F$, $G$, and $H$ be the midpoints of $AB$, $BC$, $CD$, and $DA$ respectively. Let $W$, $X$, $Y$ and $Z$ be the orthocenters of triangles $AHE$, $BEF$, $CFG$ and $DGH$, respectively. Prove that the quadrilaterals $ABCD$ and $WXYZ$ have the same area.
38 replies
1 viewing
v_Enhance
Apr 28, 2014
Kempu33334
an hour ago
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Number of distinct residues of x^n+ax mod p
gghx   4
N an hour ago by dolphinday
Source: SMO Open 2022
Let $n\ge 2$ be a positive integer. For any integer $a$, let $P_a(x)$ denote the polynomial $x^n+ax$. Let $p$ be a prime number and define the set $S_a$ as the set of residues mod $p$ that $P_a(x)$ attains. That is, $$S_a=\{b\mid 0\le b\le p-1,\text{ and there is }c\text{ such that }P_a(c)\equiv b \pmod{p}\}.$$Show that the expression $\frac{1}{p-1}\sum\limits_{a=1}^{p-1}|S_a|$ is an integer.

Proposed by fattypiggy123
4 replies
gghx
Jul 2, 2022
dolphinday
an hour ago
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Balkan MO 2022/1 is reborn
Assassino9931   11
N an hour ago by lendsarctix280
Source: Bulgaria EGMO TST 2023 Day 1, Problem 1
Let $ABC$ be a triangle with circumcircle $k$. The tangents at $A$ and $C$ intersect at $T$. The circumcircle of triangle $ABT$ intersects the line $CT$ at $X$ and $Y$ is the midpoint of $CX$. Prove that the lines $AX$ and $BY$ intersect on $k$.
11 replies
+1 w
Assassino9931
Feb 7, 2023
lendsarctix280
an hour ago
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Find the value of angle C
markosa   12
N 4 hours ago by sunken rock
Given a triangle ABC with base BC

angle B = 3x
angle C = x
AP is the bisector of base BC (i.e.) BP = PC
angle APB = 45 degrees

Find x

I know there are multiple methods to solve this problem using cosine law, coord geo
But is there any pure geometrical solution?
12 replies
markosa
Yesterday at 12:45 PM
sunken rock
4 hours ago
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10 Problems
Sedro   57
N Today at 3:25 PM by Sedro
Title says most of it. I've been meaning to post a problem set on HSM since at least a few months ago, but since I proposed the most recent problems I made to the 2025 SSMO, I had to wait for that happen. (Hence, most of these problems will probably be familiar if you participated in that contest, though numbers and wording may be changed.) The problems are very roughly arranged by difficulty. Enjoy!

Problem 1: An sequence of positive integers $u_1, u_2, \dots, u_8$ has the property for every positive integer $n\le 8$, its $n^\text{th}$ term is greater than the mean of the first $n-1$ terms, and the sum of its first $n$ terms is a multiple of $n$. Let $S$ be the number of such sequences satisfying $u_1+u_2+\cdots + u_8 = 144$. Compute the remainder when $S$ is divided by $1000$.

Problem 2 (solved by fruitmonster97): Rhombus $PQRS$ has side length $3$. Point $X$ lies on segment $PR$ such that line $QX$ is perpendicular to line $PS$. Given that $QX=2$, the area of $PQRS$ can be expressed as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.

Problem 3 (solved by Math-lover1): Positive integers $a$ and $b$ satisfy $a\mid b^2$, $b\mid a^3$, and $a^3b^2 \mid 2025^{36}$. If the number of possible ordered pairs $(a,b)$ is equal to $N$, compute the remainder when $N$ is divided by $1000$.

Problem 4 (solved by CubeAlgo15): Let $ABC$ be a triangle. Point $P$ lies on side $BC$, point $Q$ lies on side $AB$, and point $R$ lies on side $AC$ such that $PQ=BQ$, $CR=PR$, and $\angle APB<90^\circ$. Let $H$ be the foot of the altitude from $A$ to $BC$. Given that $BP=3$, $CP=5$, and $[AQPR] = \tfrac{3}{7} \cdot [ABC]$, the value of $BH\cdot CH$ can be expressed as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.

Problem 5 (solved by maromex): Anna has a three-term arithmetic sequence of integers. She divides each term of her sequence by a positive integer $n>1$ and tells Bob that the three resulting remainders are $20$, $52$, and $R$, in some order. For how many values of $R$ is it possible for Bob to uniquely determine $n$?

Problem 6 (solved by Mathsll-enjoy): There is a unique ordered triple of positive reals $(x,y,z)$ satisfying the system of equations \begin{align*} x^2 + 9 &= (y-\sqrt{192})^2 + 4 \\ y^2 + 4 &= (z-\sqrt{192})^2 + 49 \\ z^2 + 49 &= (x-\sqrt{192})^2 + 9. \end{align*}The value of $100x+10y+z$ can be expressed as $p\sqrt{q}$, where $p$ and $q$ are positive integers such that $q$ is square-free. Compute $p+q$.

Problem 7 (solved by sami1618): Let $S$ be the set of all monotonically increasing six-term sequences whose terms are all integers between $0$ and $6$ inclusive. We say a sequence $s=n_1, n_2, \dots, n_6$ in $S$ is symmetric if for every integer $1\le i \le 6$, the number of terms of $s$ that are at least $i$ is $n_{7-i}$. The probability that a randomly chosen element of $S$ is symmetric is $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Compute $p+q$.

Problem 8: For a positive integer $n$, let $r(n)$ denote the value of the binary number obtained by reading the binary representation of $n$ from right to left. Find the smallest positive integer $k$ such that the equation $n+r(n)=2k$ has at least ten positive integer solutions $n$.

Problem 9 (solved by Math-lover1, sami1618): Let $p$ be a quadratic polynomial with a positive leading coefficient. There exists a positive real number $r$ such that $r < 1 < \tfrac{5}{2r} < 5$ and $p(p(x)) = x$ for $x \in \{ r,1, \tfrac{5}{2r} , 5\}$. Compute $p(20)$.

Problem 10 (solved by aaravdodhia, sami1618): Find the number of ordered triples of positive integers $(a,b,c)$ such that $a+b+c=995$ and $ab+bc+ca$ is a multiple of $995$.
57 replies
Sedro
Jul 10, 2025
Sedro
Today at 3:25 PM
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Trigonometry prove
Studying_geometry   2
N Today at 2:39 PM by CuriousMathBoy72
Prove that $ sin36^\circ - cos18^\circ = \frac{1}{2} $
2 replies
Studying_geometry
Today at 2:18 PM
CuriousMathBoy72
Today at 2:39 PM
J
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PROVINCIAL MATHEMATICS 9 MATH QUESTIONS FOR REFERENCE
tuananh_vvvbb   0
Today at 1:33 PM
Hello friends, I would like to share with you a reference to an HSG question for me to get a score of 70% or more, which is quite difficult. Readers, please refer to me for an explanation. Thank you all very much. Good health.
0 replies
tuananh_vvvbb
Today at 1:33 PM
0 replies
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Inequalitis
sqing   11
N Today at 10:07 AM by sqing
Let $ a,b,c\geq 0 , a^2+b^2+c^2 =3.$ Prove that
$$a^3 +b^3 +c^3 +\frac{11}{5}abc \leq \frac{26}{5}$$
11 replies
sqing
May 31, 2025
sqing
Today at 10:07 AM
J
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Inequalities of integers
nhathhuyyp5c   3
N Today at 9:07 AM by Pal702004
Let $m,n$ be positive integers, $m$ is even such that $\sqrt{2}<\dfrac{m}{n}<\sqrt{2}+\dfrac{1}{2}$. Prove that there exist positive integers $k,l$ satisfying $$\left|\frac{k}{l}-\sqrt{2}\right|<\frac{m}{n}-\sqrt{2}.$$
3 replies
nhathhuyyp5c
Jun 14, 2025
Pal702004
Today at 9:07 AM
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Inequalities
sqing   5
N Today at 8:05 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
$$ \left(a+b+c-\frac{17}{6}\right)^2+9abc \geq\frac{325}{36}$$$$ \left(a+b+c-\frac{5}{2}\right)^2+12abc \geq\frac{49}{4}$$$$\left(a+b+c-\frac{14}{5}\right)^2+\frac{49}{5}abc \geq\frac{49}{5}$$
5 replies
sqing
Jun 30, 2025
sqing
Today at 8:05 AM
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x+y+z+1/x+1/y+1/z=0
nhathhuyyp5c   1
N Today at 7:48 AM by sqing
Let $x,y,z$ be reals such that $|x|,|y|,|z|\geq1$ and $x+y+z+\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=0$. Find $\max P=x+y+z$.
1 reply
nhathhuyyp5c
Today at 6:38 AM
sqing
Today at 7:48 AM
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On a generalization of a^3+b^3+c^3-3abc
Sivin   1
N Today at 4:05 AM by Sivin
We note that $x_1^2+x_2^2-2x_1x_2=(x_1-x_2)^2$ and
$${x_1^3+x_2^3+x_3^3-3x_1x_2x_3=(x_1+x_2+x_3)(x_1^2+x_2^2+x_3^2-x_1x_2-x_2x_3-x_3x_1)}$$are reducible polynomials in $\mathbb{C}.$ However, $x_1^4+x_2^4+x_3^4+x_4^4-4x_1x_2x_3x_4$ is an irreducible polynomial in $\mathbb{C}.$ So what are all the $n$ such that the polynomial:
$$f_n(x_1,x_2,\dots,x_n)=x_1^n+x_2^n+\dots+x_n^n-nx_1x_2\dots x_n$$is reducible in $\mathbb{C}$?
1 reply
Sivin
Dec 3, 2024
Sivin
Today at 4:05 AM
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Inequalities
sqing   11
N Today at 1:50 AM by sqing
Let $ a,b> 0 $ and $2a+2b+ab=5. $ Prove that
$$ \frac{a^4}{b^4}+\frac{1}{a^4}+42ab-a^4\geq 43$$$$ \frac{a^5}{b^5}+\frac{1}{a^5}+64ab-a^5\geq 65$$$$ \frac{a^6}{b^6}+\frac{1}{a^6}+90ab-a^6\geq 91$$$$ \frac{a^7}{b^7}+\frac{1}{a^7}+121ab-a^7\geq 122$$
11 replies
sqing
May 28, 2025
sqing
Today at 1:50 AM
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D1036 : Composition of polynomials
Dattier   3
N May 26, 2025 by mohabstudent1
Source: les dattes à Dattier
Find all $A \in \mathbb Q[x]$ with $\exists Q \in \mathbb Q[x], Q(A(x))= x^{2025!+2}+x^2+x+1$ and $\deg(A)>1$.
3 replies
Dattier
May 24, 2025
mohabstudent1
May 26, 2025
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D1036 : Composition of polynomials
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Reveal topic tags
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Source: les dattes à Dattier
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Dattier
1560 posts
Dattier
#1 h  May 24, 2025, 1:52 PM
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Find all $A \in \mathbb Q[x]$ with $\exists Q \in \mathbb Q[x], Q(A(x))= x^{2025!+2}+x^2+x+1$ and $\deg(A)>1$.
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Dattier
1560 posts
Dattier
#3 h  May 26, 2025, 7:15 AM
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No one ?
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mohabstudent1
7 posts
mohabstudent1
#4 h  May 26, 2025, 8:55 PM
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Coming to solve it!
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mohabstudent1
7 posts
mohabstudent1
#5 h  May 26, 2025, 8:55 PM
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I have biology exam tomorrow, so I will solve it when I get up.
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