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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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[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
Apr 2, 2025
0 replies
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IMO Shortlist 2014 N5
hajimbrak   60
N 6 minutes ago by sansgankrsngupta
Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$.

Proposed by Belgium
60 replies
hajimbrak
Jul 11, 2015
sansgankrsngupta
6 minutes ago
J
H
n variables with n-gon sides
mihaig   0
7 minutes ago
Source: Own
Let $n\geq3$ and let $a_1,a_2,\ldots, a_n\geq0$ be reals such that $\sum_{i=1}^{n}{\frac{1}{2a_i+n-2}}=1.$
Prove
$$\frac{24}{(n-1)(n-2)}\cdot\sum_{1\leq i<j<k\leq n}{a_ia_ja_k}\geq3\sum_{i=1}^{n}{a_i}+n.$$
0 replies
mihaig
7 minutes ago
0 replies
J
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4 variables with quadrilateral sides
mihaig   3
N 28 minutes ago by mihaig
Source: VL
Let $a,b,c,d\geq0$ satisfying
$$\frac1{a+1}+\frac1{b+1}+\frac1{c+1}+\frac1{d+1}=2.$$Prove
$$4\left(abc+abd+acd+bcd\right)\geq3\left(a+b+c+d\right)+4.$$
3 replies
mihaig
Today at 5:11 AM
mihaig
28 minutes ago
J
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Calculate the distance of chess king!!
egxa   5
N an hour ago by Tesla12
Source: All Russian 2025 9.4
A chess king was placed on a square of an \(8 \times 8\) board and made $64$ moves so that it visited all squares and returned to the starting square. At every moment, the distance from the center of the square the king was on to the center of the board was calculated. A move is called $\emph{pleasant}$ if this distance becomes smaller after the move. Find the maximum possible number of pleasant moves. (The chess king moves to a square adjacent either by side or by corner.)
5 replies
egxa
Apr 18, 2025
Tesla12
an hour ago
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No more topics!
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J
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Prove that the segments do not intersect for each k, n
Amir Hossein   0
Sep 21, 2010
Let $A_0,A_1, \ldots , A_n$ be points in a plane such that
(i) $A_0A_1 \leq \frac{1}{ 2} A_1A_2 \leq \cdots \leq \frac{1}{2^{n-1} } A_{n-1}A_n$ and
(ii) $0 < \measuredangle A_0A_1A_2 < \measuredangle A_1A_2A_3 < \cdots < \measuredangle A_{n-2}A_{n-1}A_n < 180^\circ,$
where all these angles have the same orientation. Prove that the segments $A_kA_{k+1},A_mA_{m+1}$ do not intersect for each $k$ and $n$ such that $0 \leq k \leq m - 2 < n- 2.$
0 replies
Amir Hossein
Sep 21, 2010
0 replies
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Prove that the segments do not intersect for each k, n
G H J
Reveal topic tags
geometry
point set
euclidean distance
Intersection
angles
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Amir Hossein
5452 posts
Amir Hossein
#1 h  Sep 21, 2010, 11:21 AM • 1 Y
Y by Adventure10
Let $A_0,A_1, \ldots , A_n$ be points in a plane such that
(i) $A_0A_1 \leq \frac{1}{ 2} A_1A_2 \leq \cdots \leq \frac{1}{2^{n-1} } A_{n-1}A_n$ and
(ii) $0 < \measuredangle A_0A_1A_2 < \measuredangle A_1A_2A_3 < \cdots < \measuredangle A_{n-2}A_{n-1}A_n < 180^\circ,$
where all these angles have the same orientation. Prove that the segments $A_kA_{k+1},A_mA_{m+1}$ do not intersect for each $k$ and $n$ such that $0 \leq k \leq m - 2 < n- 2.$
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