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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
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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
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Inequality with Max
nsato   7
N 13 minutes ago by Nari_Tom
Source: 2012 Baltic Way, Problem 2
Let $a$, $b$, $c$ be real numbers. Prove that
\[ab + bc + ca + \max\{|a - b|, |b - c|, |c - a|\} \le 1 + \frac{1}{3} (a + b + c)^2.\]
7 replies
nsato
Nov 22, 2012
Nari_Tom
13 minutes ago
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The six faces of a cube are painted in a manner that no two adjacent faces have
Vulch   0
an hour ago
The six faces of a cube are painted in a manner that no two adjacent faces have the same colour.The three colours used in painting are red, blue and green.The cube is then cut into 36 smaller cubes in a manner that 32 cubes are of one size and the rest of a bigger size and each of the bigger cube has no red side.How many cubes only have one side coloured?
0 replies
Vulch
an hour ago
0 replies
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Quadratic division
giangtruong13   0
an hour ago
Let $x,y,z$ be integer numbers satisfy that: $x^2-3y^2-z^2=xy+3xz-8yz$.Prove that: $$44|5x+19y+15z$$
0 replies
giangtruong13
an hour ago
0 replies
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2012 RMT Team Round - Stanford Math Tournament
parmenides51   7
N an hour 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.
7 replies
parmenides51
Jan 24, 2022
soryn
an hour ago
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Number Theory Chain!
JetFire008   5
N an hour ago by whwlqkd
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
5 replies
JetFire008
Today at 7:14 AM
whwlqkd
an hour ago
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<KCL wanted, K,L on hypotenuse AB of right isosceles ,AK: KL: LB = 1: 2: \sqrt3
parmenides51   1
N an hour ago by Mathzeus1024
Source: 2015 SPbU finals, grades 10-11 p3 v8 - Saint Petersburg State University School Olympiad
On the hypotenuse $AB$ of an isosceles right-angled triangle $ABC$ such $K$ and $L$ are marked, such that $AK: KL: LB = 1: 2: \sqrt3$. Find $\angle KCL$.
1 reply
parmenides51
Jan 24, 2021
Mathzeus1024
an hour ago
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Problems with Progression and Series
SomeonecoolLovesMaths   2
N an hour ago by fruitmonster97
These are a few questions I wasn't able to solve, any help will be appreciated!

$1.$ If $a_{n+1} = \frac{1}{1- a_n}$ for $n \geq 1$ and $a_3 = a_1$, then find the value of $(a_{2001})^{2001}$.

$2.$ If the $p$th term of an A.P. is $q$ and the $q$th term is $p$, then find its $r$th term.

$3.$ If $x$ is a positive real number different from $1$, then prove that the numbers $\frac{1}{1 + \sqrt{x}}, \frac{1}{1-x} , \frac{1}{1- \sqrt{x}}, \cdots$ are in A.P. Also find their common difference.

My Progress
2 replies
SomeonecoolLovesMaths
2 hours ago
fruitmonster97
an hour ago
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Problem3
samithayohan   113
N an hour ago by VideoCake
Source: IMO 2015 problem 3
Let $ABC$ be an acute triangle with $AB > AC$. Let $\Gamma $ be its circumcircle, $H$ its orthocenter, and $F$ the foot of the altitude from $A$. Let $M$ be the midpoint of $BC$. Let $Q$ be the point on $\Gamma$ such that $\angle HQA = 90^{\circ}$ and let $K$ be the point on $\Gamma$ such that $\angle HKQ = 90^{\circ}$. Assume that the points $A$, $B$, $C$, $K$ and $Q$ are all different and lie on $\Gamma$ in this order.

Prove that the circumcircles of triangles $KQH$ and $FKM$ are tangent to each other.

Proposed by Ukraine
113 replies
samithayohan
Jul 10, 2015
VideoCake
an hour ago
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Hard problem
Tendo_Jakarta   0
2 hours ago
Let \(x,y,z,t\) be positive real numbers. Find the minimum value of
\[ T = (x+y+z+t)^2.\left[\dfrac{1}{x(y+z+t)}+\dfrac{1}{y(z+t+x)}+\dfrac{1}{z(t+x+y)}+\dfrac{1}{t(x+y+z)}\right] \]
0 replies
Tendo_Jakarta
2 hours ago
0 replies
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Incenter and concurrency
jenishmalla   4
N 2 hours ago by Double07
Source: 2025 Nepal ptst p3 of 4
Let the incircle of $\triangle ABC$ touch sides $BC$, $CA$, and $AB$ at points $D$, $E$, and $F$, respectively. Let $D'$ be the diametrically opposite point of $D$ with respect to the incircle. Let lines $AD'$ and $AD$ intersect the incircle again at $X$ and $Y$, respectively. Prove that the lines $DX$, $D'Y$, and $EF$ are concurrent, i.e., the lines intersect at the same point.

(Kritesh Dhakal, Nepal)
4 replies
jenishmalla
Mar 15, 2025
Double07
2 hours ago
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Ratio of lengths in right-angled triangle
DylanN   1
N 2 hours ago by Mathzeus1024
Source: South African Mathematics Olympiad 2021, Problem 2
Let $PAB$ and $PBC$ be two similar right-angled triangles (in the same plane) with $\angle PAB = \angle PBC = 90^\circ$ such that $A$ and $C$ lie on opposite sides of the line $PB$. If $PC = AC$, calculate the ratio $\frac{PA}{AB}$.
1 reply
+1 w
DylanN
Aug 11, 2021
Mathzeus1024
2 hours ago
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Pythagorean new journey
XAN4   4
N 2 hours ago by XAN4
Source: Inspired by sarjinius
The number $4$ is written on the blackboard. Every time, Carmela can erase the number $n$ on the black board and replace it with a new number $m$, if and only if $|n^2-m^2|$ is a perfect square. Prove or disprove that all positive integers $n\geq4$ can be written exactly once on the blackboard.
4 replies
XAN4
Yesterday at 3:41 AM
XAN4
2 hours ago
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wu2481632 Mock Geometry Olympiad problems
wu2481632   14
N 2 hours ago by bin_sherlo
To avoid clogging the fora with a horde of geometry problems, I'll post them all here.

Day I

Day II

Enjoy the problems!
14 replies
wu2481632
Mar 13, 2017
bin_sherlo
2 hours ago
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inequalities 070425
pennypc123456789   5
N Today at 6:21 AM by Sadigly
Let $a,b,c$ be positive real numbers . Prove that :
$$\dfrac{2ab}{a^2+b^2} + \dfrac{2bc}{b^2+c^2} + \dfrac{2ac}{a^2+c^2} \ge \dfrac{24abc}{(a+b)(b+c)(a+c)} $$
5 replies
pennypc123456789
Today at 4:24 AM
Sadigly
Today at 6:21 AM
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Inequality
JK1603JK   1
N Apr 2, 2025 by lbh_qys
Let $a,b,c\ge 0: a+b+c=3$ then prove \frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}+\frac{1}{2}\cdot\frac{abc}{ab+bc+ca}\ge \frac{5}{3}.$$
1 reply
JK1603JK
Apr 2, 2025
lbh_qys
Apr 2, 2025
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Inequality
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inequalities
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JK1603JK
44 posts
JK1603JK
#1 h  Apr 2, 2025, 9:12 AM
Y by
Let $a,b,c\ge 0: a+b+c=3$ then prove \frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}+\frac{1}{2}\cdot\frac{abc}{ab+bc+ca}\ge \frac{5}{3}.$$
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lbh_qys
482 posts
lbh_qys
#2 h  Apr 2, 2025, 9:52 AM
Y by
JK1603JK wrote:
Let $a,b,c\ge 0: a+b+c=3$ then prove $$\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}+\frac{1}{2}\cdot\frac{abc}{ab+bc+ca}\ge \frac{5}{3}.$$

Since
\[ \sum \frac{1}{a+b} = \frac{1}{3}\sum \frac{a+b+c}{a+b} = 1 + \frac{1}{3}\sum \frac{c}{a+b}, \]and by the Cauchy–Schwarz inequality,
\[ \sum \frac{c}{a+b} \geq \frac{(\sum c)^2}{\sum c(a+b)} = \frac{9}{2\sum ab}, \]it suffices to prove that
\[ 1 + \frac{3}{2\sum ab} + \frac{abc}{2\sum ab} \geq \frac{5}{3}, \]that is,
\[ abc \geq \frac{4\sum ab - 9}{3}. \]Which is Schur’s inequality.
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