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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.

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0 replies
jwelsh
Jul 1, 2025
0 replies
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Miklós Schweitzer 2022 P3
Isolemma   4
N Today at 12:11 PM by maths001Z
Source: http://www.math.u-szeged.hu/~mmaroti/schweitzer/schweitzer-2022.pdf
Original in Hungarian; translated with Google translate; polished by myself.

Let $f: [0, \infty) \to [0, \infty)$ be a function that is linear between adjacent integers, and for $n = 0, 1, \dots$ satisfies
$$f(n) = \begin{cases} 0, & \textrm{if }2\mid n,\\4^l + 1, & \textrm{if }2 \nmid n, 4^{l - 1} \leq n < 4^l(l = 1, 2, \dots).\end{cases}$$Let $f^1(x) = f(x)$, and $f^k(x) = f(f^{k - 1}(x))$ for all integers $k \geq 2$. Determine the values of $\liminf\nolimits_{k\to\infty}f^k(x)$ and $\limsup\nolimits_{k\to\infty}f^k(x)$ for almost all $x \in [0, \infty)$ under Lebesgue measure.

(Not sure whether the last sentence translates correctly; the original:
Határozzuk meg Lebesgue majdnem minden $x\in [0, \infty)$-re a $\liminf\nolimits_{k\to\infty}f^k(x)$ és $\limsup\nolimits_{k\to\infty}f^k(x)$ értékét.)
4 replies
Isolemma
Nov 22, 2022
maths001Z
Today at 12:11 PM
J
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Something Proposer Said is Easy
EthanWYX2009   0
Today at 10:57 AM
Source: 2023 September 谜之竞赛-7
For a positive integer \( n \), let \( P_n \) denote the product of all prime numbers not exceeding \( n \).

Prove that there exists a constant \( c > 0 \) such that for any sufficiently large integer \( n \), if all integers not exceeding \( P_n \) and coprime with \( P_n \) are arranged in a sequence, then there exist two adjacent numbers in this sequence whose difference is at least \( c \cdot \frac{n \cdot \ln n}{(\ln \ln n)^2} \).

Furthermore, consider whether this bound can be strengthened to \( c \cdot \frac{n \cdot \ln n \cdot \ln \ln \ln n}{(\ln \ln n)^2} \).

Created by Mucong Sun, Tianjin Experimental Binhai School
0 replies
1 viewing
EthanWYX2009
Today at 10:57 AM
0 replies
J
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Limit of the sum of the terms of a nonlinear recurrence relation
F_Adrien   2
N Today at 10:11 AM by F_Adrien
Source: I am not aware of any source related to this problem. It comes from an optimization problem with bilinear constraints.
There are two questions:
[list=1]
[*] For any positive integer $n$, let $u_1 = 1$ and for any $i \in \{1, 2, \ldots, n - 1\}$ define
\[ u_{i + 1} = \frac{u_i}{1 + n u_i^2}, \]then prove that
\[ \lim_{n \to +\infty} \sum_{i = 1}^n u_i = \sqrt{3}. \]
[*] (More challenging) For any positive integer $n$, let $u_1 = 1$ and for any $i \in \{1, 2, \ldots, n - 1\}$ define
\[ u_{i + 1} = \frac{u_i}{1 + (n - i) u_i^2}, \]then prove that
\[ \lim_{n \to +\infty} \sum_{i = 1}^n u_i = 1 + \frac{\pi}{4}. \][/list]

I would be happy if anyone can provide a reference for the above (or similar) results, if any exists. Thank you in advance.
2 replies
F_Adrien
Yesterday at 6:57 PM
F_Adrien
Today at 10:11 AM
J
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Matrix equation
Natrium   0
Today at 6:54 AM
If $A$ is a complex matrix with $AA^*A=A^3,$ prove that $A$ is self-adjoint, i.e., that $A^*=A.$
0 replies
Natrium
Today at 6:54 AM
0 replies
J
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[PMO 26 QUALS]
Shinfu   5
N Today at 1:31 AM by mathprodigy2011
An urn contains two white and two black balls. John draw two balls simultaneously from the urn. If the balls are of different colors, he stops. Otherwise, he returns both balls to the urn and then repeats the process. What is the probability that he stops after exactly three draws?
5 replies
Shinfu
Yesterday at 3:15 PM
mathprodigy2011
Today at 1:31 AM
J
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[OG Problem] Course and Cleaning
Shinfu   1
N Yesterday at 3:14 PM by Shinfu
$28$ students joined a MathDash Cohort Program that they have to attend everyday. Each day, $4$ students are scheduled to clean the classroom after each session. After the session, it was found that every pair of students had been assigned to clean the classroom exactly once. How many days does the course last for?
1 reply
Shinfu
Yesterday at 3:14 PM
Shinfu
Yesterday at 3:14 PM
J
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2002 TAMU - Best Student Exam - Texas A&M University HS Mathematics Contest
parmenides51   10
N Yesterday at 4:13 AM by imtiyas1
p1. The product of two numbers is $7$ and the sum of their reciprocals is $4$. What is the sum of these two numbers?


p2. Circles are both inscribed within and circumscribed about an equilateral triangle. The length of a side of the triangle is $4$. What is the ratio of the area of the circumscribed circle to the area of the inscribed circle?


p3. If $(x - a)(x - b)(x - c)(x - d)(x - e) = x^5 -7x^4 + 6x^3 + 5x^4 + 13x-9$ , then $a + b + c + d + e = ...$


p4. Find the area above the $x$-axis that is enclosed by the graph of the curve $f(x) = 1-2\left| x- \frac12\right|$.


p5. Let $g$ be a function that satisfies the following properties:
(i) $g(0) = 2$ .
(ii) $g(1) = 3$.
(iii) $g(x + y) + g(x-y) = g(x)g(y)$ for all integers $x$ and $y$ .
Find $g(5)$ .


p6. The center of a circle of radius $4$ is located at the center of a square table with side $16$. A coin with radius $1/8$ is randomly thrown onto the table. What is the probability that the coin comes to rest on the boundary of the circle?


p7. Find the value of
$$\frac{3}{(1 \cdot 2)^2} + \frac{5}{(2 \cdot 3)^2} +\frac{7}{(3 \cdot 4)^2} +\frac{9}{(4 \cdot 5)^2} + ...+\frac{2001}{ (1000 \cdot 1001)^2}$$

p8. Two candles of equal length start burning at the same time. One of the candles will burn in $4$ hours, and the other in $5$ hours. How long in hours will they have to burn before one candle is $3$ times the length of the other?


p9. Three integers form a geometric progression. Their sum is $21$ and the sum of their reciprocals is $\frac{7}{12}$. Find the largest integer.


p10. There are eight men in a room. Each one shakes hands with each of the others once. How many handshakes are there?


p11. Two squares (shown below), each with side $12$, are placed so that a corner of one lies at the center of the other. Find the area of quadrilateral $EJCK$ if $BJ= 4$.
IMAGE


p12. In a $10$-team baseball league, each team plays each of the others $18$ times. The season ends, not in a tie, with each team the same number of games ahead of the following team. What is the greatest number of games that the last team could have won?


p13. Find the unique pair of real numbers $(x, y)$ such that $(4x^2 + 6x + 4)(4y^2 - 12y + 25) = 28$ .


p14. A man and his grandson have the same birthday. For six consecutive birthdays the man is an integral number of times as old as his grandson. How old is the man at the sixth of these birthdays?


p15. In the product $9 \cdot HATBOX = 4 \cdot BOXHAT$ , find the six-digit number $BOXHAT$ .


p16. If $n$ is an even integer, express in terms of $n$ the number of solutions in positive integers of $2x+y+z = n$ .


p17. A sequence is defined by $x_1 = 2$ and $x_{n+1} =\frac{x_n}{1+ x_n}$ for all $n \ge 1$ . Find $x_{10,000 }$.


p18. If $a =\frac{x}{x^2 + y^2}$ and $b =\frac{y}{x^2 + y^2}$ , find $x + y$ in terms of $a$ and $b$ . Express your answer as a common fraction.




PS. You should use hide for answers. Collected here.
10 replies
parmenides51
Mar 23, 2022
imtiyas1
Yesterday at 4:13 AM
J
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rectangle 15 x n from specific pieces - Chile 2002 L2 P2
parmenides51   3
N Yesterday at 12:15 AM by Squirrel7O
Determine all natural numbers $n$ for which it is possible to construct a rectangle of sides $15$ and $n$, with pieces congruent to:

IMAGE

The squares of the pieces have side $1$ and the pieces cannot overlap or leave free spaces
3 replies
parmenides51
Sep 1, 2022
Squirrel7O
Yesterday at 12:15 AM
J
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2001 TAMU - Best Student Exam - Texas A&M University HS Mathematics Contest
parmenides51   10
N Jul 9, 2025 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
Jul 9, 2025
J
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[PMO27 Areas] I.6 Factorial series
aops-g5-gethsemanea2   3
N Jul 6, 2025 by Siopao_Enjoyer
Determine the value of $\log_4 x$ if $$x=\frac{60!}{59!}+\frac{60!}{3!\cdot57!}+\frac{60!}{5!\cdot55!}+\dots+\frac{60!}{27!\cdot33!}+\frac{60!}{29!\cdot31!}.$$
Answer confirmation
3 replies
aops-g5-gethsemanea2
Jan 25, 2025
Siopao_Enjoyer
Jul 6, 2025
J
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2000 TAMU - Best Student Exam - Texas A&M University HS Mathematics Contest
parmenides51   14
N Jul 4, 2025 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
Jul 4, 2025
J
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2018 preRMO p11, teacups in the kitchen, some wiht handles
parmenides51   3
N Jul 3, 2025 by cortex_classes
There are several teacups in the kitchen, some with handles and the others without handles. The number of ways of selecting two cups without a handle and three with a handle is exactly $1200$. What is the maximum possible number of cups in the kitchen?
3 replies
parmenides51
Aug 8, 2019
cortex_classes
Jul 3, 2025
J
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Average divisors
Magdalo   1
N Jul 2, 2025 by Magdalo
I have a number $n$, where the the greatest common divisor of $10!$ and $n$ is equal to the least common multiple of $5!$ and $n$. What is the average amount of divisors of $n$?
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Magdalo
Jul 2, 2025
Magdalo
Jul 2, 2025
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Penchick Porridge
Magdalo   1
N Jul 2, 2025 by Magdalo
Penrick has $9$ cans of porridge, with $3$ each of red, blue, and green cans. He places them randomly in a $3\times3$ grid. What is the expected sum of distinct colors per column? For example, an arrangement with $3$ of the same color per column has a sum of $3$.
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Magdalo
Jul 2, 2025
Magdalo
Jul 2, 2025
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analysis
ay19bme   0
Jul 2, 2025
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ay19bme
Jul 2, 2025
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ay19bme
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ay19bme
#1 h  Jul 2, 2025, 12:06 PM
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