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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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0 replies
jlacosta
Apr 2, 2025
0 replies
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k i Adding contests to the Contest Collections
dcouchman   1
N Apr 5, 2023 by v_Enhance
Want to help AoPS remain a valuable Olympiad resource? Help us add contests to AoPS's Contest Collections.

Find instructions and a list of contests to add here: https://artofproblemsolving.com/community/c40244h1064480_contests_to_add
1 reply
dcouchman
Sep 9, 2019
v_Enhance
Apr 5, 2023
J
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k i Zero tolerance
ZetaX   49
N May 4, 2019 by NoDealsHere
Source: Use your common sense! (enough is enough)
Some users don't want to learn, some other simply ignore advises.
But please follow the following guideline:


To make it short: ALWAYS USE YOUR COMMON SENSE IF POSTING!
If you don't have common sense, don't post.


More specifically:

For new threads:


a) Good, meaningful title:
The title has to say what the problem is about in best way possible.
If that title occured already, it's definitely bad. And contest names aren't good either.
That's in fact a requirement for being able to search old problems.

Examples:
Bad titles:
- "Hard"/"Medium"/"Easy" (if you find it so cool how hard/easy it is, tell it in the post and use a title that tells us the problem)
- "Number Theory" (hey guy, guess why this forum's named that way¿ and is it the only such problem on earth¿)
- "Fibonacci" (there are millions of Fibonacci problems out there, all posted and named the same...)
- "Chinese TST 2003" (does this say anything about the problem¿)
Good titles:
- "On divisors of a³+2b³+4c³-6abc"
- "Number of solutions to x²+y²=6z²"
- "Fibonacci numbers are never squares"


b) Use search function:
Before posting a "new" problem spend at least two, better five, minutes to look if this problem was posted before. If it was, don't repost it. If you have anything important to say on topic, post it in one of the older threads.
If the thread is locked cause of this, use search function.

Update (by Amir Hossein). The best way to search for two keywords in AoPS is to input
[code]+"first keyword" +"second keyword"[/code]
so that any post containing both strings "first word" and "second form".


c) Good problem statement:
Some recent really bad post was:
[quote]$lim_{n\to 1}^{+\infty}\frac{1}{n}-lnn$[/quote]
It contains no question and no answer.
If you do this, too, you are on the best way to get your thread deleted. Write everything clearly, define where your variables come from (and define the "natural" numbers if used). Additionally read your post at least twice before submitting. After you sent it, read it again and use the Edit-Button if necessary to correct errors.


For answers to already existing threads:


d) Of any interest and with content:
Don't post things that are more trivial than completely obvious. For example, if the question is to solve $x^{3}+y^{3}=z^{3}$, do not answer with "$x=y=z=0$ is a solution" only. Either you post any kind of proof or at least something unexpected (like "$x=1337, y=481, z=42$ is the smallest solution). Someone that does not see that $x=y=z=0$ is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
Similar, posting "I have solved this problem" but not posting anything else is not welcome; it even looks that you just want to show off what a genius you are.

e) Well written and checked answers:
Like c) for new threads, check your solutions at least twice for mistakes. And after sending, read it again and use the Edit-Button if necessary to correct errors.



To repeat it: ALWAYS USE YOUR COMMON SENSE IF POSTING!


Everything definitely out of range of common sense will be locked or deleted (exept for new users having less than about 42 posts, they are newbies and need/get some time to learn).

The above rules will be applied from next monday (5. march of 2007).
Feel free to discuss on this here.
49 replies
ZetaX
Feb 27, 2007
NoDealsHere
May 4, 2019
J
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2012 RMT Team Round - Stanford Math Tournament
parmenides51   10
N 2 hours ago by soryn
p1. How many functions $f : \{1, 2, 3, 4, 5\} \to \{1, 2, 3, 4, 5\}$ take on exactly $3$ distinct values?


p2. Let $i$ be one of the numbers $0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11$. Suppose that for all positive integers $n$, the number $n^n$ never has remainder $i$ upon division by $12$. List all possible values of $i$.


p3. A card is an ordered 4-tuple $(a_1, a_2, a_3, a_4)$ where each $a_i$ is chosen from $\{0, 1, 2\}$. A line is an (unordered) set of three (distinct) cards $\{(a_1, a_2, a_3, a_4)$,$(b_1, b_2, b_3, b_4)$,$(c_1, c_2, c_3, c_4)\}$ such that for each $i$, the numbers $a_i, b_i, c_i$ are either all the same or all different. How many different lines are there?


p4. We say that the pair of positive integers $(x, y)$, where $x < y$, is a $k$-tangent pair if we have
$\arctan \frac{1}{k} = arctan\frac{1}{x}+ arctan\frac{1}{y}$ . Compute the second largest integer that appears in a $2012$-tangent pair.


p5. Regular hexagon $A_1A_2A_3A_4A_5A_6$ has side length $1$. For $i = 1, ..., 6$, choose $B_i$ to be a point on the segment $A_iA_{i+1}$ uniformly at random, assuming the convention that $A_{j+6} = A_j$ for all integers $j$. What is the expected value of the area of hexagon $B_1B_2B_3B_4B_5B_6$?


p6. Evaluate $\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\frac{1}{nm(n + m + 1)}$.


p7. A plane in $3$-dimensional space passes through the point $(a_1, a_2, a_3)$, with $a_1$, $a_2$, and $a_3$ all positive. The plane also intersects all three coordinate axes with intercepts greater than zero (i.e. there exist positive numbers $b_1$, $b_2$, $b_3$ such that $(b_1, 0, 0)$, $(0, b_2, 0)$, and $(0, 0, b_3)$ all lie on this plane). Find, in terms of $a_1$, $a_2$, $a_3$, the minimum possible volume of the tetrahedron formed by the origin and these three intercepts.


p8. The left end of a rubber band e meters long is attached to a wall and a slightly sadistic child holds on to the right end. A point-sized ant is located at the left end of the rubber band at time $t = 0$, when it begins walking to the right along the rubber band as the child begins stretching it. The increasingly tired ant walks at a rate of $1/(ln(t + e))$ centimeters per second, while the child uniformly stretches the rubber band at a rate of one meter per second. The rubber band is infinitely stretchable and the ant and child are immortal. Compute the time in seconds, if it exists, at which the ant reaches the right end of the rubber band. If the ant never reaches the right end, answer $+\infty$.


p9. We say that two lattice points are neighboring if the distance between them is $1$. We say that a point lies at distance d from a line segment if $d$ is the minimum distance between the point and any point on the line segment. Finally, we say that a lattice point $A$ is nearby a line segment if the distance between $A$ and the line segment is no greater than the distance between the line segment and any neighbor of $A$. Find the number of lattice points that are nearby the line segment connecting the origin and the point $(1984, 2012)$.


p10. A permutation of the first n positive integers is valid if, for all $i > 1$, $i$ comes after $\left\lfloor \frac{i}{2} \right\rfloor $ in the permutation. What is the probability that a random permutation of the first $14$ integers is valid?


p11. Given that $x, y, z > 0$ and $xyz = 1$, find the range of all possible values of
$\frac{x^3 + y^3 + z^3 - x^{-3} - y^{-3} - z^{-3}}{x + y + z - x^{-1} - y^{-1} - z^{-1}}$.


p12. A triangle has sides of length $\sqrt2$, $3 + \sqrt3$, and $2\sqrt2 + \sqrt6$. Compute the area of the smallest regular polygon that has three vertices coinciding with the vertices of the given triangle.


p13. How many positive integers $n$ are there such that for any natural numbers $a, b$, we have $n | (a^2b + 1)$ implies $n | (a^2 + b)$?


p14. Find constants $a$ and $c$ such that the following limit is finite and nonzero: $c = \lim_{n \to \infty} \frac{e\left( 1- \frac{1}{n}\right)^n - 1}{n^a}$.
Give your answer in the form $(a, c)$.


p15. Sean thinks packing is hard, so he decides to do math instead. He has a rectangular sheet that he wants to fold so that it fits in a given rectangular box. He is curious to know what the optimal size of a rectangular sheet is so that it’s expected to fit well in any given box. Let a and b be positive reals with $a \le b$, and let $m$ and $n$ be independently and uniformly distributed random variables in the interval $(0, a)$. For the ordered $4$-tuple $(a, b, m, n)$, let $f(a, b, m, n)$ denote the ratio between the area of a sheet with dimension a×b and the area of the horizontal cross-section of the box with dimension $m \times n$ after the sheet has been folded in halves along each dimension until it occupies the largest possible area that will still fit in the box (because Sean is picky, the sheet must be placed with sides parallel to the box’s sides). Compute the smallest value of b/a that maximizes the expectation $f$.

PS. You had better use hide for answers.
10 replies
1 viewing
parmenides51
Jan 24, 2022
soryn
2 hours ago
J
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IMO ShortList 2002, geometry problem 1
orl   47
N 2 hours ago by Avron
Source: IMO ShortList 2002, geometry problem 1
Let $B$ be a point on a circle $S_1$, and let $A$ be a point distinct from $B$ on the tangent at $B$ to $S_1$. Let $C$ be a point not on $S_1$ such that the line segment $AC$ meets $S_1$ at two distinct points. Let $S_2$ be the circle touching $AC$ at $C$ and touching $S_1$ at a point $D$ on the opposite side of $AC$ from $B$. Prove that the circumcentre of triangle $BCD$ lies on the circumcircle of triangle $ABC$.
47 replies
orl
Sep 28, 2004
Avron
2 hours ago
J
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2^2^n+2^2^{n-1}+1-Iran 3rd round-Number Theory 2007
Amir Hossein   5
N 2 hours ago by SomeonecoolLovesMaths
Prove that $2^{2^{n}}+2^{2^{{n-1}}}+1$ has at least $n$ distinct prime divisors.
5 replies
Amir Hossein
Jul 28, 2010
SomeonecoolLovesMaths
2 hours ago
J
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This problem has unintended solution, found by almost all who solved it :(
mshtand1   5
N 3 hours ago by iliya8788
Source: Ukrainian Mathematical Olympiad 2025. Day 2, Problem 11.7
Given a triangle \(ABC\), an arbitrary point \(D\) is chosen on the side \(AC\). In triangles \(ABD\) and \(CBD\), the angle bisectors \(BK\) and \(BL\) are drawn, respectively. The point \(O\) is the circumcenter of \(\triangle KBL\). Prove that the second intersection point of the circumcircles of triangles \(ABL\) and \(CBK\) lies on the line \(OD\).

Proposed by Anton Trygub
5 replies
mshtand1
Mar 14, 2025
iliya8788
3 hours ago
J
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3 var inquality
sqing   1
N 3 hours ago by hashtagmath
Source: Own
Let $ a,b,c>0 $ and $ \dfrac{a}{bc}+\dfrac{2b}{ca}+\dfrac{5c}{ab}\leq 12.$ Prove that$$ a^2+b^2+c^2\geq 1$$
1 reply
sqing
Apr 6, 2025
hashtagmath
3 hours ago
J
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a_n >= 1/n if a_{n+1}^2 + a_{n+1} = a_n, a_1=1 , a_i>=0
parmenides51   11
N 3 hours ago by richrow12
Source: Canadian Junior Mathematical Olympiad - CJMO 2020 p1
Let $a_1, a_2, a_3, . . .$ be a sequence of positive real numbers that satisfies $a_1 = 1$ and $a^2_{n+1} + a_{n+1} = a_n$ for every natural number $n$. Prove that $a_n \ge \frac{1}{n}$ for every natural number $n$.
11 replies
parmenides51
Jul 15, 2020
richrow12
3 hours ago
J
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Don't bite me for this straightforward sequence
Assassino9931   4
N 4 hours ago by RagvaloD
Source: Bulgaria National Olympiad 2025, Day 1, Problem 1
Determine all infinite sequences $a_1, a_2, \ldots$ of real numbers such that
\[ a_{m^2 + m + n} = a_{m}^2 + a_m + a_n\]for all positive integers $m$ and $n$.
4 replies
Assassino9931
Today at 1:47 PM
RagvaloD
4 hours ago
J
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Orthocenter config once again
Assassino9931   4
N 4 hours ago by cj13609517288
Source: Bulgaria National Olympiad 2025, Day 2, Problem 4
Let \( ABC \) be an acute triangle with \( AB < AC \), midpoint $M$ of side $BC$, altitude \( AD \) (\( D \in BC \)), and orthocenter \( H \). A circle passes through points \( B \) and \( D \), is tangent to line \( AB \), and intersects the circumcircle of triangle \( ABC \) at a second point \( Q \). The circumcircle of triangle \( QDH \) intersects line \( BC \) at a second point \( P \). Prove that the lines \( MH \) and \( AP \) are perpendicular.
4 replies
Assassino9931
Today at 1:53 PM
cj13609517288
4 hours ago
J
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a functional equation on positive reals
littletush   10
N 4 hours ago by Frd_19_Hsnzde
Source: Czech and Slovak third round,2004,p6
Find all functions $f:\mathbb R^+ \rightarrow \mathbb R^+$ such that for all positive real numbers $x,y$,
\[x^2[f(x)+f(y)]=(x+y)f(yf(x)).\]
10 replies
littletush
Mar 3, 2012
Frd_19_Hsnzde
4 hours ago
J
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Set with a property
socrates   4
N 4 hours ago by sadat465
Let $n\in \Bbb{N}, n \geq 4.$ Determine all sets $ A = \{a_1, a_2, . . . , a_n\} \subset \Bbb{N}$ containing $2015$ and having the property that $ |a_i - a_j|$ is prime, for all distinct $i, j\in \{1, 2, . . . , n\}.$
4 replies
socrates
May 29, 2015
sadat465
4 hours ago
J
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Number Theory Chain!
JetFire008   21
N 4 hours ago by Primeniyazidayi
I will post a question and someone has to answer it. Then they have to post a question and someone else will answer it and so on. We can only post questions related to Number Theory and each problem should be more difficult than the previous. Let's start!

Question 1
21 replies
JetFire008
Yesterday at 7:14 AM
Primeniyazidayi
4 hours ago
J
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No bash for this inequality
giangtruong13   1
N 5 hours ago by no_room_for_error
Let $x,y,z$ be positive real number satisfy that: $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=1$.Find the minimum: $$ \sum_{cyc} \frac{(xy)^2}{z(x^2+y^2)} $$
1 reply
1 viewing
giangtruong13
Today at 3:08 PM
no_room_for_error
5 hours ago
J
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Problem practice
Deomad123   1
N 5 hours ago by no_room_for_error
Let $a,b,c$ be real numbers so that $a+2b+3c=2$ and $2ab+3ac+6bc=1$ Show that $a\in[0,\frac{4}{3}]$,$b\in[0,\frac{2}{3}]$,$c\in[0,\frac{4}{9}]$
1 reply
Deomad123
Today at 3:58 PM
no_room_for_error
5 hours ago
J
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2025 Brown University Math Olympiad(BrUMO) Individual Round
fruitmonster97   10
N 5 hours ago by lpieleanu
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 $5f(x)-xf\left(\frac1x\right)=\frac{1}{17}x^2,$ 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.
10 replies
fruitmonster97
Yesterday at 6:32 PM
lpieleanu
5 hours ago
J
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J
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A tight inequality
jokehim   1
N Mar 28, 2025 by liyufish
Let $a,b,c\ge 0: ab+bc+ca>0.$ Prove that $$\frac{a+b}{2a^2+3ab+2b^2}+\frac{b+c}{2b^2+3bc+2c^2}+\frac{c+a}{2c^2+3ca+2a^2}\ge \frac{18}{7(a+b+c)}.$$
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jokehim
Mar 27, 2025
liyufish
Mar 28, 2025
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A tight inequality
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jokehim
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jokehim
#1 h  Mar 27, 2025, 3:03 PM
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Let $a,b,c\ge 0: ab+bc+ca>0.$ Prove that $$\frac{a+b}{2a^2+3ab+2b^2}+\frac{b+c}{2b^2+3bc+2c^2}+\frac{c+a}{2c^2+3ca+2a^2}\ge \frac{18}{7(a+b+c)}.$$
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liyufish
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liyufish
#2 h  Mar 28, 2025, 2:08 PM
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$$ \begin{aligned} L.H.S & =\sum_{cyc}\frac{1}{2(a+b)-\frac{ab}{a+b}} \\ & \ge \sum_{cyc}\frac{1}{2(2\sqrt{ab})-\frac{ab}{2\sqrt{ab}}} \\ & =\sum_{cyc}\frac{2}{7\sqrt{ab}} \end{aligned} $$Now we consider proving$\displaystyle \sum_{cyc} \frac{1}{\sqrt{ab}} \cdot \sum_{cyc} a \ge 9$
From AM-GM inequality it's trivial.
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