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k a May Highlights and 2025 AoPS Online Class Information

jlacosta 0

May 1, 2025

May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following upcoming events:
[list][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
[*]May 19th, 4:30pm PT/7:30pm ET, What's Next After Beast Academy?, designed for students finishing Beast Academy and ready for Prealgebra 1.
[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies

jlacosta

May 1, 2025

0 replies

Find the expected END time for the given process

superpi 2

N 24 minutes ago by Hello_Kitty

This problem suddenly popped up in my head. But I don't know how to deal with it.

There are N bulbs. All the bulbs' available time follows same exponential distribution with parameter lambda(Or any arbitrary distribution with mean $\mu$ ). We do following operations
1. First, turn on the all $N$ bulbs
2. For each $k >= 2$ bulbs goes out, append ONE NEW BULB and turn on (This step starts and finishes immediately when kth bulb goes out)
3. Repeat 2 until all the bulbs goes out

What is the expected terminate time for the above process for given $N, k, \lambda$ ?

Or, is there any more conditions to complete the problem?

2 replies

superpi

Yesterday at 4:33 PM

Hello_Kitty

24 minutes ago

Different Paths Probability

Qebehsenuef 2

N 38 minutes ago by Etkan

Source: OBM

A mouse initially occupies cage A and is trained to change cages by going through a tunnel whenever an alarm sounds. Each time the alarm sounds, the mouse chooses any of the tunnels adjacent to its cage with equal probability and without being affected by previous choices. What is the probability that after the alarm sounds 23 times the mouse occupies cage B?

2 replies

Qebehsenuef

Apr 28, 2025

Etkan

38 minutes ago

[Sipnayan JHS] Semifinals Round B, Average, #2

LilKirb 1

N 3 hours ago by LilKirb

How many trailing zeroes are there in the base $4$ representation of $2015!$ ?

1 reply

LilKirb

4 hours ago

LilKirb

3 hours ago

36x⁴ + 12x² - 36x + 13 > 0

fxandi 4

N 4 hours ago by wh0nix

Prove that for any real $x \geq 0$ holds inequality $36x^4 + 12x^2 - 36x + 13 > 0.$

4 replies

fxandi

May 5, 2025

wh0nix

4 hours ago

2022 SMT Team Round - Stanford Math Tournament

parmenides51 5

N 4 hours ago by vanstraelen

p1. Square $ABCD$ has side length $2$ . Let the midpoint of $BC$ be $E$ . What is the area of the overlapping region between the circle centered at $E$ with radius $1$ and the circle centered at $D$ with radius $2$ ? (You may express your answer using inverse trigonometry functions of noncommon values.)

p2. Find the number of times $f(x) = 2$ occurs when $0 \le x \le 2022 \pi$ for the function $f(x) = 2^x(cos(x) + 1)$ .

p3. Stanford is building a new dorm for students, and they are looking to offer $2$ room configurations:
$\bullet$ Configuration $A$ : a one-room double, which is a square with side length of $x$ ,
$\bullet$ Configuration $B$ : a two-room double, which is two connected rooms, each of them squares with a side length of $y$ .
To make things fair for everyone, Stanford wants a one-room double (rooms of configuration $A$ ) to be exactly $1$ m $^2$ larger than the total area of a two-room double. Find the number of possible pairs of side lengths $(x, y)$ , where $x \in N$ , $y \in N$ , such that $x - y < 2022$ .

p4. The island nation of Ur is comprised of $6$ islands. One day, people decide to create island-states as follows. Each island randomly chooses one of the other five islands and builds a bridge between the two islands (it is possible for two bridges to be built between islands $A$ and $B$ if each island chooses the other). Then, all islands connected by bridges together form an island-state. What is the expected number of island-states Ur is divided into?

p5. Let $a, b,$ and $c$ be the roots of the polynomial $x^3 - 3x^2 - 4x + 5$ . Compute $\frac{a^4 + b^4}{a + b}+\frac{b^4 + c^4}{b + c}+\frac{c^4 + a^4}{c + a}$ .

p6. Carol writes a program that finds all paths on an 10 by 2 grid from cell (1, 1) to cell (10, 2) subject to the conditions that a path does not visit any cell more than once and at each step the path can go up, down, left, or right from the current cell, excluding moves that would make the path leave the grid. What is the total length of all such paths? (The length of a path is the number of cells it passes through, including the starting and ending cells.)

p7. Consider the sequence of integers an defined by $a_1 = 1$ , $a_p = p$ for prime $p$ and $a_{mn} = ma_n + na_m$ for $m, n > 1$ . Find the smallest $n$ such that $\frac{a_n^2}{2022}$ is a perfect power of $3$ .

p8. Let $\vartriangle ABC$ be a triangle whose $A$ -excircle, $B$ -excircle, and $C$ -excircle have radii $R_A$ , $R_B$ , and $R_C$ , respectively. If $R_AR_BR_C = 384$ and the perimeter of $\vartriangle ABC$ is $32$ , what is the area of $\vartriangle ABC$ ?

p9. Consider the set $S$ of functions $f : \{1, 2, . . . , 16\} \to \{1, 2, . . . , 243\}$ satisfying:
(a) $f(1) = 1$
(b) $f(n^2) = n^2f(n)$ ,
(c) $n |f(n)$ ,
(d) $f(lcm(m, n))f(gcd(m, n)) = f(m)f(n)$ .
If $|S|$ can be written as $p^{\ell_1}_1 \cdot p^{\ell_2}_2 \cdot ... \cdot p^{\ell_k}_k$ where $p_i$ are distinct primes, compute $p_1\ell_1+p_2\ell_2+. . .+p_k\ell_k$ .

p10. You are given that $\log_{10}2 \approx 0.3010$ and that the first (leftmost) two digits of $2^{1000}$ are 10. Compute the number of integers $n$ with $1000 \le n \le 2000$ such that $2^n$ starts with either the digit $8$ or $9$ (in base $10$ ).

p11. Let $O$ be the circumcenter of $\vartriangle ABC$ . Let $M$ be the midpoint of $BC$ , and let $E$ and $F$ be the feet of the altitudes from $B$ and $C$ , respectively, onto the opposite sides. $EF$ intersects $BC$ at $P$ . The line passing through $O$ and perpendicular to $BC$ intersects the circumcircle of $\vartriangle ABC$ at $L$ (on the major arc $BC$ ) and $N$ , and intersects $BC$ at $M$ . Point $Q$ lies on the line $LA$ such that $OQ$ is perpendicular to $AP$ . Given that $\angle BAC = 60^o$ and $\angle AMC = 60^o$ , compute $OQ/AP$ .

p12. Let $T$ be the isosceles triangle with side lengths $5, 5, 6$ . Arpit and Katherine simultaneously choose points $A$ and $K$ within this triangle, and compute $d(A, K)$ , the squared distance between the two points. Suppose that Arpit chooses a random point $A$ within $T$ . Katherine plays the (possibly randomized) strategy which given Arpit’s strategy minimizes the expected value of $d(A, K)$ . Compute this value.

p13. For a regular polygon $S$ with $n$ sides, let $f(S)$ denote the regular polygon with $2n$ sides such that the vertices of $S$ are the midpoints of every other side of $f(S)$ . Let $f^{(k)}(S)$ denote the polygon that results after applying f a total of k times. The area of $\lim_{k \to \infty} f^{(k)}(P)$ where $P$ is a pentagon of side length $1$ , can be expressed as $\frac{a+b\sqrt{c}}{d}\pi^m$ for some positive integers $a, b, c, d, m$ where $d$ is not divisible by the square of any prime and $d$ does not share any positive divisors with $a$ and $b$ . Find $a + b + c + d + m$ .

p14. Consider the function $f(m) = \sum_{n=0}^{\infty}\frac{(n - m)^2}{(2n)!}$ . This function can be expressed in the form $f(m) = \frac{a_m}{e} +\frac{b_m}{4}e$ for sequences of integers $\{a_m\}_{m\ge 1}$ , $\{b_m\}_{m\ge 1}$ . Determine $\lim_{n \to \infty}\frac{2022b_m}{a_m}$ .

p15. In $\vartriangle ABC$ , let $G$ be the centroid and let the circumcenters of $\vartriangle BCG$ , $\vartriangle CAG$ , and $\vartriangle ABG$ be $I, J$ , and $K$ , respectively. The line passing through $I$ and the midpoint of $BC$ intersects $KJ$ at $Y$ . If the radius of circle $K$ is $5$ , the radius of circle $J$ is $8$ , and $AG = 6$ , what is the length of $KY$ ?

PS. You should use hide for answers. Collected here.

5 replies

parmenides51

Jun 30, 2022

vanstraelen

4 hours ago

[Sipnayan SHS] Finals Round, Difficult

LilKirb 1

N Today at 6:59 AM by LilKirb

Let $f$ be a polynomial with nonnegative integer coefficients. If $f(1) = 11$ and $f(11) = 2311$ , what is the remainder when $f(10)$ is divided by $1000?$

1 reply

LilKirb

Today at 6:49 AM

LilKirb

Today at 6:59 AM

Inequalities

sqing 1

N Today at 3:50 AM by sqing

Let $a,b> 0 ,\frac{a}{2b+1}+\frac{b}{3}+\frac{1}{2a+1} \leq 1.$ Prove that
$a^2+b^2 -ab\leq 1$ $a^2+b^2 +ab \leq3$ Let $a,b,c> 0 , \frac{a}{2b+1}+\frac{b}{2c+1}+\frac{c}{2a+1} \leq 1.$ Prove that
$a +b +c +abc \leq 4$

1 reply

sqing

Today at 3:11 AM

sqing

Today at 3:50 AM

Inequalities

sqing 19

N Today at 2:50 AM by sqing

Let $a,b>0$ . Prove that
$\frac{a}{a^2+a +2b+1}+ \frac{b}{b^2+2a +b+1} \leq \frac{2}{5}$ $\frac{a}{a^2+2a +b+1}+ \frac{b}{b^2+a +2b+1} \leq \frac{2}{5}$

19 replies

sqing

May 13, 2025

sqing

Today at 2:50 AM

inequality

luckvoltia.112 9

N Today at 2:35 AM by Shan3t

Given that $a, b, c, d$ are nonzero real numbers, find the minimum value of the expression
$P = \left| \frac{b + c + d}{a} \right| + \left| \frac{c + d + a}{b} \right| + \left| \frac{d + a + b}{c} \right| + \left| \frac{a + b + c}{d} \right|.$

9 replies

luckvoltia.112

Today at 12:45 AM

Shan3t

Today at 2:35 AM

2024 Miklós-Schweitzer problem 3

Martin.s 3

N Today at 1:30 AM by naenaendr

Do there exist continuous functions $f, g: \mathbb{R} \to \mathbb{R}$ , both nowhere differentiable, such that $f \circ g$ is differentiable?

3 replies

Martin.s

Dec 5, 2024

naenaendr

Today at 1:30 AM

prove that

luckvoltia.112 0

Today at 12:52 AM

Let $a, b, c$ be non-negative real numbers such that $a + b + c > 0$ and
$\frac{25a + 36b + 49c}{5a + 6b + 7c} + \frac{25b + 36c + 49a}{5b + 6c + 7a} + \frac{25c + 36a + 49b}{5c + 6a + 7b} = 18.$ Prove that exactly two of the numbers $a, b, c$ are equal to 0.

0 replies

luckvoltia.112

Today at 12:52 AM

0 replies

Geometry Trigonometry Olympiads

Foxellar 0

Yesterday at 11:07 PM

Let $\triangle ABC$ be a triangle such that $\angle ABC = 120^\circ$ . Points $X, Y, Z$ lie on segments $BC, CA, AB$ , respectively, such that lines $AX, BY,$ and $CZ$ are the angle bisectors of triangle $ABC$ . Find the measure of angle $\angle XYZ$ .

0 replies

Foxellar

Yesterday at 11:07 PM

0 replies

Proof of ramsey number

smadadi1000 1

N Yesterday at 10:35 PM by smadadi1000

How do you prove that r(n,2)=n using the pigeonhole principle?

1 reply

smadadi1000

Yesterday at 10:31 PM

smadadi1000

Yesterday at 10:35 PM

Minimize

lgx57 1

N Yesterday at 5:25 PM by Math-lover1

Minimize $\sqrt{\cos^2 x+(2-\sin x)^2}+\dfrac{1}{2}\sqrt{(\sqrt 3-\cos x)^2+(\sin x+1)^2}$

1 reply

lgx57

Yesterday at 1:29 PM

Math-lover1

Yesterday at 5:25 PM

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