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Beams inside a cube
AOPS12142015 32
N
by cursed_tangent1434
Source: USOMO 2020 Problem 2, USOJMO 2020 Problem 3
An empty 
cube is given, and a 
grid of square unit cells is drawn on each of its six faces. A beam is a 
rectangular prism. Several beams are placed inside the cube subject to the following conditions:
[list=]
[*]The two
faces of each beam coincide with unit cells lying on opposite faces of the cube. (Hence, there are 
possible positions for a beam.)
[*]No two beams have intersecting interiors.
[*]The interiors of each of the four
faces of each beam touch either a face of the cube or the interior of the face of another beam.
[/list]
What is the smallest positive number of beams that can be placed to satisfy these conditions?
Proposed by Alex Zhai
cube is given, and a 
grid of square unit cells is drawn on each of its six faces. A beam is a 
rectangular prism. Several beams are placed inside the cube subject to the following conditions:[list=]
[*]The two
faces of each beam coincide with unit cells lying on opposite faces of the cube. (Hence, there are 
possible positions for a beam.)[*]No two beams have intersecting interiors.
[*]The interiors of each of the four
faces of each beam touch either a face of the cube or the interior of the face of another beam.[/list]
What is the smallest positive number of beams that can be placed to satisfy these conditions?
Proposed by Alex Zhai
32 replies
Summer internships/research opportunists in STEM
o99999 8
N
by Craftybutterfly
Hi, I am a current high school student and was looking for internships and research opportunities in STEM. Do you guys know any summer programs that do research such as RSI, but for high school freshmen that are open?
Thanks.
Thanks.
8 replies
9 Will I make JMO?
EaZ_Shadow 20
N
by Craftybutterfly
will I be able to make it... will the cutoffs will be pre-2024
20 replies
Sequence of integers
tenniskidperson3 32
N
by N3bula
Source: 2012 USAMO problem #3
Determine which integers 
have the property that there exists an infinite sequence 
of nonzero integers such that the equality ![\[a_k+2a_{2k}+\ldots+na_{nk}=0\]](//latex.artofproblemsolving.com/5/b/a/5ba1051daa520af4e7a53ece725c400b229075cf.png)
holds for every positive integer 
.
have the property that there exists an infinite sequence 
of nonzero integers such that the equality ![\[a_k+2a_{2k}+\ldots+na_{nk}=0\]](http://latex.artofproblemsolving.com/5/b/a/5ba1051daa520af4e7a53ece725c400b229075cf.png)
holds for every positive integer 
.
32 replies
Isogonal Conjugates: 2011 USAMO #5
tenniskidperson3 77
N
by N3bula
Let 
be a given point inside quadrilateral 
. Points 
and 
are located within such that
![\[\angle Q_1BC=\angle ABP,\quad\angle Q_1CB=\angle DCP,\quad\angle Q_2AD=\angle BAP,\quad\angle Q_2DA=\angle CDP.\]](//latex.artofproblemsolving.com/d/b/3/db31d11ec24322724c0e0f1485aadf8864eabf7a.png)
Prove that 
if and only if 
.
be a given point inside quadrilateral 
. Points 
and 
are located within such that![\[\angle Q_1BC=\angle ABP,\quad\angle Q_1CB=\angle DCP,\quad\angle Q_2AD=\angle BAP,\quad\angle Q_2DA=\angle CDP.\]](http://latex.artofproblemsolving.com/d/b/3/db31d11ec24322724c0e0f1485aadf8864eabf7a.png)
Prove that 
if and only if 
.
77 replies
camp/class recommendations for incoming freshman
walterboro 7
N
by Ruegerbyrd
hi guys, i'm about to be an incoming freshman, does anyone have recommendations for classes to take next year and camps this summer? i am sure that i can aime qual but not jmo qual yet. ty
7 replies
Vertices of a pentagon invariant: 2011 USAMO #2
tenniskidperson3 54
N
by N3bula
An integer is assigned to each vertex of a regular pentagon so that the sum of the five integers is 2011. A turn of a solitaire game consists of subtracting an integer 
from each of the integers at two neighboring vertices and adding 
to the opposite vertex, which is not adjacent to either of the first two vertices. (The amount and the vertices chosen can vary from turn to turn.) The game is won at a certain vertex if, after some number of turns, that vertex has the number 2011 and the other four vertices have the number 0. Prove that for any choice of the initial integers, there is exactly one vertex at which the game can be won.
from each of the integers at two neighboring vertices and adding 
to the opposite vertex, which is not adjacent to either of the first two vertices. (The amount and the vertices chosen can vary from turn to turn.) The game is won at a certain vertex if, after some number of turns, that vertex has the number 2011 and the other four vertices have the number 0. Prove that for any choice of the initial integers, there is exactly one vertex at which the game can be won.
54 replies
HCSSiM results
SurvivingInEnglish 69
N
by MathWizardThatCanBeatYou
Anyone already got results for HCSSiM? Are there any point in sending additional work if I applied on March 19?
69 replies
[CASH PRIZES] IndyINTEGIRLS Spring Math Competition
Indy_Integirls 1
N
by Indy_Integirls
[center]IMAGE
Greetings, AoPS! IndyINTEGIRLS will be hosting a virtual math competition on May 25,
2024 from 12 PM to 3 PM EST. Join other woman-identifying and/or non-binary "STEMinists" in solving problems, socializing, playing games, winning prizes, and more! If you are interested in competing, please register here![/center]
----------
[center]Important Information[/center]
Eligibility: This competition is open to all woman-identifying and non-binary students in and high school. Non-Indiana residents and international students are welcome as well!
Format: There will be a middle school and high school division. In each separate division, there will be an individual round and a team round, where students are grouped into teams of 3-4 and collaboratively solve a set of difficult problems. There will also be a buzzer/countdown/Kahoot-style round, where students from both divisions are grouped together to compete in a MATHCOUNTS-style countdown round! There will be prizes for the top competitors in each division.
Problem Difficulty: Our amazing team of problem writers is working hard to ensure that there will be problems for problem-solvers of all levels! The middle school problems will range from MATHCOUNTS school round to AMC 10 level, while the high school problems will be for more advanced problem-solvers. The team round problems will cover various difficulty levels and are meant to be more difficult, while the countdown/buzzer/Kahoot round questions will be similar to MATHCOUNTS state to MATHCOUNTS Nationals countdown round in difficulty.
Platform: This contest will be held virtually through Zoom. All competitors are required to have their cameras turned on at all times unless they have a reason for otherwise. Proctors and volunteers will be monitoring students at all times to prevent cheating and to create a fair environment for all students.
Prizes: At this moment, prizes are TBD, and more information will be provided and attached to this post as the competition date approaches. Rest assured, IndyINTEGIRLS has historically given out very generous cash prizes, and we intend on maintaining this generosity into our Spring Competition.
Contact & Connect With Us: Follow us on Instagram @indy.integirls, join our Discord, follow us on TikTok @indy.integirls, and email us at indy@integirls.org.
---------
[center]Help Us Out
Please help us in sharing the news of this competition! Our amazing team of officers has worked very hard to provide this educational opportunity to as many students as possible, and we would appreciate it if you could help us spread the word!
Greetings, AoPS! IndyINTEGIRLS will be hosting a virtual math competition on May 25,
2024 from 12 PM to 3 PM EST. Join other woman-identifying and/or non-binary "STEMinists" in solving problems, socializing, playing games, winning prizes, and more! If you are interested in competing, please register here![/center]
----------
[center]Important Information[/center]
Eligibility: This competition is open to all woman-identifying and non-binary students in and high school. Non-Indiana residents and international students are welcome as well!
Format: There will be a middle school and high school division. In each separate division, there will be an individual round and a team round, where students are grouped into teams of 3-4 and collaboratively solve a set of difficult problems. There will also be a buzzer/countdown/Kahoot-style round, where students from both divisions are grouped together to compete in a MATHCOUNTS-style countdown round! There will be prizes for the top competitors in each division.
Problem Difficulty: Our amazing team of problem writers is working hard to ensure that there will be problems for problem-solvers of all levels! The middle school problems will range from MATHCOUNTS school round to AMC 10 level, while the high school problems will be for more advanced problem-solvers. The team round problems will cover various difficulty levels and are meant to be more difficult, while the countdown/buzzer/Kahoot round questions will be similar to MATHCOUNTS state to MATHCOUNTS Nationals countdown round in difficulty.
Platform: This contest will be held virtually through Zoom. All competitors are required to have their cameras turned on at all times unless they have a reason for otherwise. Proctors and volunteers will be monitoring students at all times to prevent cheating and to create a fair environment for all students.
Prizes: At this moment, prizes are TBD, and more information will be provided and attached to this post as the competition date approaches. Rest assured, IndyINTEGIRLS has historically given out very generous cash prizes, and we intend on maintaining this generosity into our Spring Competition.
Contact & Connect With Us: Follow us on Instagram @indy.integirls, join our Discord, follow us on TikTok @indy.integirls, and email us at indy@integirls.org.
---------
[center]Help Us Out
Please help us in sharing the news of this competition! Our amazing team of officers has worked very hard to provide this educational opportunity to as many students as possible, and we would appreciate it if you could help us spread the word!
1 reply
Past USAMO Medals
sdpandit 4
N
by sadas123
Does anyone know where to find lists of USAMO medalists from past years? I can find the 2025 list on their website, but they don't seem to keep lists from previous years and I can't find it anywhere else. Thanks!
4 replies
