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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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jlacosta
Apr 2, 2025
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2025 Brown University Math Olympiad(BrUMO) Individual Round
fruitmonster97   8
N 4 minutes ago by rbcubed13
a la parmenides51

1. One hundred concentric circles are labelled $C_1,C_2,C_3,...,C_{100}.$ Each circle $C_n$ is inscribed within an equilateral triangle whose vertices are points on $C_{n+1}.$ Given $C_1$ has a radius of $1,$ what is the radius of $C_{100}$?

2. An infinite geometric sequence with common ratio $r$ sums to $91.$ A new sequence starting with the same term has common ratio $r^3.$ The sum of the new sequence produced is $81.$ What was the common ratio of the original sequence?

3. Let $A,B,C,D,$ and $E$ be five equally spaced points on a line in that order. Let $F,G,H,$ and $I$ all be on the same side of line $AE$ such that triangles $AFB,BGC,CHD,$ and $EID$* are equilateral with side length $1.$ Let $S$ be the region consisting of the interiors of all four triangles. Compute the length of segment $AI$ that is contained in $S.$

4. If $5f(x)-xf\left(\frac1x\right)=\frac{1}{17x^2},$ determine $f(3).$

5. How many ways are there to arrange $1,2,3,4,5,6$ such that no two consecutive numbers have the same remainder when divided by $3$?

6. Joshua is playing with his number cards. He has $9$ cards of $9$ lined up in a row. He puts a multiplication sign between two of the $9$s and calculates the product of the two strings of $9$s. For example, one possible result is $999\times999999 = 998999001.$ Let $S$ be the sum of all possible distinct results (note that $999\times999999$ yields the same result as $999999\times999$). What is the sum of digits of $S$?

7. Bruno the Bear is tasked to organize $16$ identical brown balls into $7$ bins labeled $1-7$. He must distribute the balls among the bins so that each odd-labeled bin contains an odd number of balls, and each even-labeled bin contains an even number of balls (with $0$ considered even). In how many ways can Bruno do this?

8. Let $f(n)$ be the number obtained by increasing every prime factor in $f$ by one. For instance, $f(12)=(2+1)^2(3+1)=36.$ What is the lowest $n$ such that $6^{2025}$ divides $f^{(n)}(2025),$ where $f^{(n)}$ denotes the $n$th iteration of $f$?

9. How many positive integer divisors of $63^{10}$ do not end in a $1$?

10. Bruno is throwing a party and invites $n$ guests. Each pair of party guests are either friends or enemies. Each guest has exactly $12$ enemies. All guests believe the following: the friend of an enemy is an enemy. Calculate the sum of all possible values of $n.$ (Please note: Bruno is not a guest at his own party)

11. In acute $\triangle ABC$, let $D$ be the foot of the altitude from $A$ to $BC$ and $O$ be the circumcenter. Suppose that the area of $\triangle ABD$ is equal to the area of $\triangle AOC.$ Given that $OD = 2$ and $BD = 3,$ compute $AD.$

12. Alice has $10$ gifts $g_1,g_2,...,g_{10}$ and $10$ friends $f_1,f_2,...,f_{10}.$ Gift $g_i$ can be given to friend $f_j$ if
\[i -j =-1,0, \text{ or } 1 \pmod{10}\]How many ways are there for Alice to pair the $10$ gifts with the $10$ friends such that each friend receives one gift?

13. Let $\triangle ABC$ be an equilateral triangle with side length $1.$ A real number $d$ is selected uniformly
at random from the open interval $(0,0.5).$ Points $E$ and $F$ lie on sides $AC$ and $AB,$ respectively, such that $AE=d$ and $AF=1-d.$ Let $D$ be the intersection of lines $BE$ and $CF.$ Consider line $\ell$ passing through both points of intersection of the circumcircles of triangles $\triangle DEF$ and $\triangle DBC.$ $O$ is the circumcenter of $\triangle DEF.$ Line $\ell$ intersects line $\overline{BC}$ at point $P,$ and point $Q$ lies on $AP$ such that $\angle AQB=120^\circ.$ What is the probability that the line segment $QO$ has length less than $\frac13$?

14. Define sequence $\{a_n\}^{\infty}_{n=1}$ such that $a_1=\frac{\pi}{3}$ and $a_{n+1}=\cot^{-1}(\csc(a_n))$ for all positive integers $n.$ Find the value of
\[\frac{1}{\cos(a_1)\cos(a_2)\cos(a_3)\cdots\cos(a_{16})}.\]
15. Define $\{x\}$ to be the fractional part of $x.$ For example, $\{20.25\}=0.25$ and $\{\pi\}=\pi-3.$ Let
$A=\sum_{a=1}^{96}\sum_{n=1}^{96}\left\{\frac{a^n}{97}\right\},$ where $\{x\}$ denotes the fractional part of $x.$ Compute $A$ rounded to the nearest integer.
8 replies
fruitmonster97
5 hours ago
rbcubed13
4 minutes ago
J
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n^2 + n + p = 1996 diophantine (1996 Greece Junior 4b)
parmenides51   3
N 26 minutes ago by Congruence
Determine whether exist a prime number $p$ and natural number $n$ such that $n^2 + n + p = 1996$.
3 replies
parmenides51
Mar 17, 2020
Congruence
26 minutes ago
J
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Algebra Problem
ReticulatedPython   1
N an hour ago by mathprodigy2011
The graphs of $ax^2=y$ and $x^2+y^2-2ry=0$ intersect at exactly one point. Prove that for all real numbers $a$, the inequality $$|r| \le \frac{1}{2a}$$is always valid.
1 reply
ReticulatedPython
3 hours ago
mathprodigy2011
an hour ago
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Inequality minimum
Deomad123   1
N 2 hours ago by alexheinis
Let $a,b,c$ be positive real numbers with $\sqrt{ab}+\sqrt{bc}+\sqrt{ac}=1$ Find the minimum value of the expression.$$E=\frac{a^2}{a+b}+\frac{b^2}{b+c}+\frac{c^2}{c+a}$$
1 reply
Deomad123
2 hours ago
alexheinis
2 hours ago
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Inequalities (Please help me!!!)
yt12   6
N Apr 1, 2025 by lamhihi1234
Let $a,b,c$ be reals with $a+b+c=1$and $ a,b,c \ge \frac{-3}{ 4}$. Prove that
$$\frac{a}{a^2+1}+\frac{b}{b^2+1}+\frac{c}{ c^2+1} \le \frac{9}{ 10}$$
6 replies
yt12
Mar 4, 2023
lamhihi1234
Apr 1, 2025
J
Inequalities (Please help me!!!)
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yt12
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yt12
#1 h  Mar 4, 2023, 2:45 PM
Y by
Let $a,b,c$ be reals with $a+b+c=1$and $ a,b,c \ge \frac{-3}{ 4}$. Prove that
$$\frac{a}{a^2+1}+\frac{b}{b^2+1}+\frac{c}{ c^2+1} \le \frac{9}{ 10}$$
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the.math.king
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the.math.king
#2 h  Mar 4, 2023, 3:48 PM
Y by
If $a,b,c>=0$, the Jensen work fine for the function $f(x)=\frac{x}{{{x}^{2}}+1},f:\left[ -\frac{3}{4};1 \right]\to \mathbb{R}$ this is concave. For negative $a,b,c$ the expression it is less than for the positive values.
This post has been edited 2 times. Last edited by the.math.king, Mar 4, 2023, 3:50 PM
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sqing
41457 posts
sqing
#3 h  Mar 5, 2023, 1:11 AM
Y by
yt12 wrote:
Let $a,b,c$ be reals with $a+b+c=1$and $ a,b,c \ge \frac{-3}{ 4}$. Prove that
$$\frac{a}{a^2+1}+\frac{b}{b^2+1}+\frac{c}{ c^2+1} \le \frac{9}{ 10}$$
https://artofproblemsolving.com/community/c6h120128p23425009
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youthdoo
1312 posts
youthdoo
#4 h  Mar 5, 2023, 5:26 AM • 1 Y
Y by teomihai
We can prove $\frac x{x^2+1}\le\frac{3+36x}{50}$, where $x\ge-\frac34$.
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sqing
41457 posts
sqing
#5 h  Mar 5, 2023, 5:39 AM
Y by
youthdoo wrote:
We can prove $\frac x{x^2+1}\le\frac{3+36x}{50}$, where $x\ge-\frac34$.
https://artofproblemsolving.com/community/c6h120128p24022319
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sqing
41457 posts
sqing
#6 h  Mar 5, 2023, 5:50 AM
Y by
Let $a,b,c\ge -\frac{3}{ 4}$ and $a+bc=2$. Prove that
$$- \frac{1928}{3925}\le \frac{a}{a^2+1}+\frac{b}{b^2+1}+\frac{c}{ c^2+1} \le \frac{3}{2}$$
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lamhihi1234
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lamhihi1234
#7 h  Apr 1, 2025, 4:16 PM
Y by
iiuuffff
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