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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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[*]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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China South East Mathematical Olympiad 2021 Grade11 P8
Henry_2001   2
N 8 minutes ago by parkjungmin
A sequence $\{z_n\}$ satisfies that for any positive integer $i,$ $z_i\in\{0,1,\cdots,9\}$ and $z_i\equiv i-1 \pmod {10}.$ Suppose there is $2021$ non-negative reals $x_1,x_2,\cdots,x_{2021}$ such that for $k=1,2,\cdots,2021,$ $$\sum_{i=1}^kx_i\geq\sum_{i=1}^kz_i,\sum_{i=1}^kx_i\leq\sum_{i=1}^kz_i+\sum_{j=1}^{10}\dfrac{10-j}{50}z_{k+j}.$$Determine the least possible value of $\sum_{i=1}^{2021}x_i^2.$
2 replies
Henry_2001
Aug 8, 2021
parkjungmin
8 minutes ago
J
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Killer NT that nobody solved (also my hardest NT ever created)
mshtand1   10
N 32 minutes ago by mshtand1
Source: Ukraine IMO 2025 TST P8
A positive integer number \( a \) is chosen. Prove that there exists a prime number that divides infinitely many terms of the sequence \( \{b_k\}_{k=1}^{\infty} \), where
\[ b_k = a^{k^k} \cdot 2^{2^k - k} + 1. \]
Proposed by Arsenii Nikolaev and Mykhailo Shtandenko
10 replies
mshtand1
Apr 19, 2025
mshtand1
32 minutes ago
J
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Another two parallels
jayme   0
40 minutes ago
Dear Mathlinkers,

1. ABCD a square
2. (A) the circle with center at A passing through B
3. P the points of intersection of the segment AC and (A)
4. I the midpoint of AB
5. Q the point of intersection of the segment IC and (A)
6. M the foot of the perpendicular to (AB) through Q.
7. Y the point of intersection of the segment MC and (A)
8. X the point of intersection de (AY) et (BC).

Prove : QX is parallel to AB.

Jean-Louis
0 replies
jayme
40 minutes ago
0 replies
J
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calculator operations limited, if S > x^n + 1 then S > x^n + x - 1
parmenides51   13
N an hour ago by Ducksohappi
Source: Tuymaada 2019 p4
A calculator can square a number or add $1$ to it. It cannot add $1$ two times in a row. By several operations it transformed a number $x$ into a number $S > x^n + 1$ ($x, n,S$ are positive integers). Prove that $S > x^n + x - 1$.
13 replies
parmenides51
Jul 22, 2019
Ducksohappi
an hour ago
J
No more topics!
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collinearity of intersections of diameters of 3 chords
parmenides51   1
N Jul 29, 2018 by PavelMath
Source: Mexican Mathematical Olympiad 1993 OMM P5
$OA, OB, OC$ are three chords of a circle. The circles with diameters $OA, OB$ meet again at $Z$, the circles with diameters $OB, OC$ meet again at $X$, and the circles with diameters $OC, OA$ meet again at $Y$. Show that $X, Y, Z$ are collinear.
1 reply
parmenides51
Jul 29, 2018
PavelMath
Jul 29, 2018
J
collinearity of intersections of diameters of 3 chords
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Reveal topic tags
geometry
collinear
Chords
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Source: Mexican Mathematical Olympiad 1993 OMM P5
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parmenides51
30650 posts
parmenides51
#1 h  Jul 29, 2018, 6:46 AM • 2 Y
Y by Adventure10, Mango247
$OA, OB, OC$ are three chords of a circle. The circles with diameters $OA, OB$ meet again at $Z$, the circles with diameters $OB, OC$ meet again at $X$, and the circles with diameters $OC, OA$ meet again at $Y$. Show that $X, Y, Z$ are collinear.
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PavelMath
185 posts
PavelMath
#2 h  Jul 29, 2018, 8:52 AM • 1 Y
Y by Adventure10
parmenides51 wrote:
$OA, OB, OC$ are three chords of a circle. The circles with diameters $OA, OB$ meet again at $Z$, the circles with diameters $OB, OC$ meet again at $X$, and the circles with diameters $OC, OA$ meet again at $Y$. Show that $X, Y, Z$ are collinear.

Nice!
Note that points $A$, $B$, $Z$ are colinear, points $B$, $C$, $X$ are colinear and points $A$, $C$, $Y$ are colinear. Then the line $XYZ$ is just Simson line for the triangle $ABC$.
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