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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.

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[*]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
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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
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2022 SMT Team Round - Stanford Math Tournament
parmenides51   4
N 26 minutes ago by P162008
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.
4 replies
parmenides51
Jun 30, 2022
P162008
26 minutes ago
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a tst 2013 test
Math2030   0
an hour ago
Given the sequence $(a_n): a_1=1, a_2=11$ and $a_{n+2}=a_{n+1}+5a_{n}, n \geq 1$
. Prove that $a_n $not is a perfect square for all $n > 3$.
0 replies
Math2030
an hour ago
0 replies
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Serbian selection contest for the IMO 2025 - P2
OgnjenTesic   10
N an hour ago by Mahdi_Mashayekhi
Source: Serbian selection contest for the IMO 2025
Let $ABC$ be an acute triangle. Let $A'$ be the reflection of point $A$ over the line $BC$. Let $O$ and $H$ be the circumcenter and the orthocenter of triangle $ABC$, respectively, and let $E$ be the midpoint of segment $OH$. Let $D$ and $L$ be the points where the reflection of line $AA'$ with respect to line $OA'$ intersects the circumcircle of triangle $ABC$, where point $D$ lies on the arc $BC$ not containing $A$. If \( M \) is a point on the line \( BC \) such that \( OM \perp AD \), prove that \( \angle MAD = \angle EAL \).

Proposed by Strahinja Gvozdić
10 replies
OgnjenTesic
Thursday at 4:02 PM
Mahdi_Mashayekhi
an hour ago
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find all Polynomials
andria   9
N an hour ago by A.H.H
Source: Iranian third round 2015 algebra problem 5
Find all polynomials $p(x)\in\mathbb{R}[x]$ such that for all $x\in \mathbb{R}$:
$p(5x)^2-3=p(5x^2+1)$ such that:
$a) p(0)\neq 0$
$b) p(0)=0$
9 replies
andria
Sep 8, 2015
A.H.H
an hour ago
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AT // BC wanted
parmenides51   105
N an hour ago by Adywastaken
Source: IMO 2019 SL G1
Let $ABC$ be a triangle. Circle $\Gamma$ passes through $A$, meets segments $AB$ and $AC$ again at points $D$ and $E$ respectively, and intersects segment $BC$ at $F$ and $G$ such that $F$ lies between $B$ and $G$. The tangent to circle $BDF$ at $F$ and the tangent to circle $CEG$ at $G$ meet at point $T$. Suppose that points $A$ and $T$ are distinct. Prove that line $AT$ is parallel to $BC$.

(Nigeria)
105 replies
parmenides51
Sep 22, 2020
Adywastaken
an hour ago
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Two lines concur on (ABC)
amar_04   19
N 2 hours ago by Giant_PT
Source: XVII Sharygin Corespondnce Round P13
In triangle $ABC$ with circumcircle $\Omega$ and incenter $I$, point $M$ bisects arc $BAC$ and line $\overline{AI}$ meets $\Omega$ at $N\ne A$. The excircle opposite to $A$ touches $\overline{BC}$ at point $E$. Point $Q\ne I$ on the circumcircle of $\triangle MIN$ is such that $\overline{QI}\parallel\overline{BC}$. Prove that the lines $\overline{AE}$ and $\overline{QN}$ meet on $\Omega$.
19 replies
amar_04
Mar 2, 2021
Giant_PT
2 hours ago
J
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Nice problem of concurrency
deraxenrovalo   0
2 hours ago
Let $(I)$ be an inscribed circle of $\triangle$$ABC$ and touching $BC$, $CA$, $AB$ at $D$, $E$, $F$ respectively. Let $EE'$ and $FF'$ be diameters of $(I)$. Let $X$ and $Y$ be the pole of $DE'$ and $DF'$ with respect to $(I)$, respectively. $BE$ cuts $(I)$ again at $K$. $CF$ cuts $(I)$ again at $L$. The tangent at $K$ of $(I)$ cuts $AX$ at $M$. The tangent at $L$ of $(I)$ cuts $AY$ at $N$. Let $U$ and $V$ be midpoint of $IM$ and $IN$, respectively.

Show that : $UV$, $E'F'$ and perpendicular bisector of $ID$ are concurrent.
0 replies
deraxenrovalo
2 hours ago
0 replies
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A hunter and an invisible rabbit are playing again...
Phorphyrion   1
N 2 hours ago by JARP091
Source: 2021 Discord CCCC P4
A hunter and an invisible rabbit play a game in a $2021\times 2021$ grid. The rabbit's starting square is $P_0$ (unknown to the hunter), and after $n-1$ rounds, the rabbit is at square $P_{n-1}$. In the $n$-th round of the game, two things occur in order:

(i) The rabbit moves invisibly to a square $P_n$ which shares a point with $P_{n-1}$ (There are up to eight of these).

(ii) A tracking device searches $k$ squares of the hunter's choosing. If the rabbit is in one of these squares, the rabbit is captured and the game ends.

For what $k$ can the rabbit avoid capture indefinitely?
1 reply
Phorphyrion
Dec 24, 2023
JARP091
2 hours ago
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SL 2015 G1: Prove that IJ=AH
Problem_Penetrator   137
N 2 hours ago by heheman
Source: IMO 2015 Shortlist, G1
Let $ABC$ be an acute triangle with orthocenter $H$. Let $G$ be the point such that the quadrilateral $ABGH$ is a parallelogram. Let $I$ be the point on the line $GH$ such that $AC$ bisects $HI$. Suppose that the line $AC$ intersects the circumcircle of the triangle $GCI$ at $C$ and $J$. Prove that $IJ = AH$.
137 replies
Problem_Penetrator
Jul 7, 2016
heheman
2 hours ago
J
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Inequalities
sqing   1
N 2 hours ago 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
3 hours ago
sqing
2 hours ago
J
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IMO Shortlist 2011, G2
WakeUp   30
N 3 hours ago by ezpotd
Source: IMO Shortlist 2011, G2
Let $A_1A_2A_3A_4$ be a non-cyclic quadrilateral. Let $O_1$ and $r_1$ be the circumcentre and the circumradius of the triangle $A_2A_3A_4$. Define $O_2,O_3,O_4$ and $r_2,r_3,r_4$ in a similar way. Prove that
\[\frac{1}{O_1A_1^2-r_1^2}+\frac{1}{O_2A_2^2-r_2^2}+\frac{1}{O_3A_3^2-r_3^2}+\frac{1}{O_4A_4^2-r_4^2}=0.\]

Proposed by Alexey Gladkich, Israel
30 replies
WakeUp
Jul 13, 2012
ezpotd
3 hours ago
J
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2-var inequality
sqing   1
N 3 hours ago by sqing
Source: Own
Let $ a,b\geq 0 ,\frac{a}{b+2}+\frac{b}{a+2}+ \frac{ab}{3}\leq 1.$ Prove that
$$ a^2+b^2 +\frac{5}{3}ab \leq 4$$
1 reply
sqing
3 hours ago
sqing
3 hours ago
J
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Inequalities
sqing   19
N 3 hours ago 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
3 hours ago
J
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Rational Points in n-Dimensional Space
steven_zhang123   0
4 hours ago
Let \( T = (x_1, x_2, \ldots, x_n) \), where \( x_i \) is rational for \( i = 1, 2, \ldots, n \). A vector \( T \) is called a rational point in \( n \)-dimensional space. Denote the set of all such vectors \( T \) as \( S \). For \( A = (x_1, x_2, \ldots, x_n) \) and \( B = (y_1, y_2, \ldots, y_n) \) in \( S \), define the distance between points \( A \) and \( B \) as \( d(A, B) = \sqrt{(x_1 - y_1)^2 + (x_2 - y_2)^2 + \cdots + (x_n - y_n)^2} \). We say that point \( A \) can move to point \( B \) if and only if there is a unit distance between two points in \( S \).

Prove:
(1) If \( n \leq 4 \), there exists a point that cannot be reached from the origin via a finite number of moves.
(2) If \( n \geq 5 \), any point in \( S \) can be reached from any other point via moves.
0 replies
steven_zhang123
4 hours ago
0 replies
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Right Triangle
phiReKaLk6781   4
N Apr 13, 2025 by lpieleanu
A right triangle has sides of integer length. One side has length 11. What is the area of the triangle?
4 replies
phiReKaLk6781
Mar 16, 2010
lpieleanu
Apr 13, 2025
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Right Triangle
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Reveal topic tags
geometry
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phiReKaLk6781
2074 posts
phiReKaLk6781
#1 h  Mar 16, 2010, 10:30 PM • 4 Y
Y by Adventure10, Kizaruno, and 2 other users
A right triangle has sides of integer length. One side has length 11. What is the area of the triangle?
This post has been edited 1 time. Last edited by phiReKaLk6781, Mar 16, 2010, 10:48 PM
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isabella2296
4829 posts
isabella2296
#2 h  Mar 16, 2010, 10:41 PM • 5 Y
Y by Adventure10, Mango247, Kizaruno, and 2 other users
Suppose 11 is a leg. We have $ 121 + b^2 = c^2 \implies 121 = (c+b)(c-b)$. The factors of 121 are (1,121) and (11,11). For (11,11), there's no way for $ c$ and $ b$ to be integers, so we could use (1,121) instead. This gives $ c = 61$ and $ b = 60$.

Thus, the area is $ 30 \times 11 = 330$.
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cmtappu96
274 posts
cmtappu96
#3 h  Apr 3, 2010, 12:44 PM • 2 Y
Y by Adventure10, Kizaruno
isabella2296 wrote:
For (11,11), there's no way for $ c$ and $ b$ to be integers

Correct solution, but just want to point out a mistake-
You said there is no way for $ c$ and $ b$ to be INTEGERS.
But if $ c = 11$ and $ b = 0$, $ c$ and $ b$ are integers, and $ (c + b, c - b) = (11,11)$
Only that degenerates the triangle.
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Abhishek009
1 post
Abhishek009
#4 h  Apr 6, 2010, 11:42 AM • 2 Y
Y by Adventure10, Kizaruno
You can solve this directly if you are aware of Pythagorean triple rule.

For your reference I am providing you with the link. http://en.wikipedia.org/wiki/Pythagorean_triple
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lpieleanu
3002 posts
lpieleanu
#5 h  Apr 13, 2025, 12:13 AM • 1 Y
Y by Kizaruno
Solution
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