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Rubber bands
v_Enhance 5
N
by lpieleanu
Source: OTIS Mock AIME 2024 #12
Let 
denote a triangular grid of side length 
consisting of 
pegs. Charles the Otter wishes to place some rubber bands along the pegs of such that every edge of the grid is covered by exactly one rubber band (and no rubber band traverses an edge twice). He considers two placements to be different if the sets of edges covered by the rubber bands are different or if any rubber band traverses its edges in a different order. The ordering of which bands are over and under does not matter.
For example, Charles finds there are exactly
different ways to cover 
using exactly two rubber bands; the full list is shown below, with one rubber band in orange and the other in blue.
IMAGE
Let
denote the total number of ways to cover 
with any number of rubber bands. Compute the remainder when is divided by 
.
Ethan Lee
denote a triangular grid of side length 
consisting of 
pegs. Charles the Otter wishes to place some rubber bands along the pegs of such that every edge of the grid is covered by exactly one rubber band (and no rubber band traverses an edge twice). He considers two placements to be different if the sets of edges covered by the rubber bands are different or if any rubber band traverses its edges in a different order. The ordering of which bands are over and under does not matter.For example, Charles finds there are exactly
different ways to cover 
using exactly two rubber bands; the full list is shown below, with one rubber band in orange and the other in blue.IMAGE
Let
denote the total number of ways to cover 
with any number of rubber bands. Compute the remainder when is divided by 
.Ethan Lee
5 replies
Geometry with orthocenter config
thdnder 6
N
by ohhh
Source: Own
Let 
be a triangle, and let 
be its altitudes. Let 
be its orthocenter, and let 
and 
be the circumcenters of triangles 
and 
. Let 
be the second intersection of the circumcircles of triangles 
and 
. Prove that the lines 
, 
, and 
-median of 
are concurrent.
be a triangle, and let 
be its altitudes. Let 
be its orthocenter, and let 
and 
be the circumcenters of triangles 
and 
. Let 
be the second intersection of the circumcircles of triangles 
and 
. Prove that the lines 
, 
, and 
-median of 
are concurrent.
6 replies
Strange Inequality
anantmudgal09 40
N
by starchan
Source: INMO 2020 P4
Let 
be an integer and let 
be real numbers such that 
. Prove that
Proposed by Kapil Pause
be an integer and let 
be real numbers such that 
. Prove that
Proposed by Kapil Pause
40 replies
Finding Solutions
MathStudent2002 22
N
by ihategeo_1969
Source: Shortlist 2016, Number Theory 5
Let 
be a positive integer which is not a perfect square, and consider the equation ![\[k = \frac{x^2-a}{x^2-y^2}.\]](//latex.artofproblemsolving.com/c/2/2/c223ccbc272d07c19632c5b8883571023e47a395.png)
Let be the set of positive integers 
for which the equation admits a solution in 
with 
, and let 
be the set of positive integers for which the equation admits a solution in with 
. Show that 
.
be a positive integer which is not a perfect square, and consider the equation ![\[k = \frac{x^2-a}{x^2-y^2}.\]](http://latex.artofproblemsolving.com/c/2/2/c223ccbc272d07c19632c5b8883571023e47a395.png)
Let be the set of positive integers 
for which the equation admits a solution in 
with 
, and let 
be the set of positive integers for which the equation admits a solution in with 
. Show that 
.
22 replies
USAMO 2000 Problem 3
MithsApprentice 10
N
by HamstPan38825
A game of solitaire is played with 
red cards, 
white cards, and blue cards. A player plays all the cards one at a time. With each play he accumulates a penalty. If he plays a blue card, then he is charged a penalty which is the number of white cards still in his hand. If he plays a white card, then he is charged a penalty which is twice the number of red cards still in his hand. If he plays a red card, then he is charged a penalty which is three times the number of blue cards still in his hand. Find, as a function of 
and 
the minimal total penalty a player can amass and all the ways in which this minimum can be achieved.
red cards, 
white cards, and blue cards. A player plays all the cards one at a time. With each play he accumulates a penalty. If he plays a blue card, then he is charged a penalty which is the number of white cards still in his hand. If he plays a white card, then he is charged a penalty which is twice the number of red cards still in his hand. If he plays a red card, then he is charged a penalty which is three times the number of blue cards still in his hand. Find, as a function of 
and 
the minimal total penalty a player can amass and all the ways in which this minimum can be achieved.
10 replies
Hard limits
Snoop76 7
N
by MihaiT
and 
satisfies the following recursion formulas: 
, 
and 
. Find 
and 
7 replies
Additive combinatorics (re Cauchy-Davenport)
mavropnevma 3
N
by Orzify
Source: Romania TST 3 2010, Problem 4
Let 
and 
be two finite subsets of the half-open interval 
such that 
and 
for no 
and no 
. Prove that the set 
has at least 
elements.
***
and 
be two finite subsets of the half-open interval 
such that 
and 
for no 
and no 
. Prove that the set 
has at least 
elements.***
3 replies
Ducks can play games now apparently
MortemEtInteritum 34
N
by HamstPan38825
Source: USA TST(ST) 2020 #1
Let , 
, 
be fixed positive integers. There are 
ducks sitting in a
circle, one behind the other. Each duck picks either rock, paper, or scissors, with ducks
picking rock, ducks picking paper, and ducks picking scissors.
A move consists of an operation of one of the following three forms:
[list]
[*] If a duck picking rock sits behind a duck picking scissors, they switch places.
[*] If a duck picking paper sits behind a duck picking rock, they switch places.
[*] If a duck picking scissors sits behind a duck picking paper, they switch places.
[/list]
Determine, in terms of, , and , the maximum number of moves which could take
place, over all possible initial configurations.
, 
, 
be fixed positive integers. There are 
ducks sitting in acircle, one behind the other. Each duck picks either rock, paper, or scissors, with
duckspicking rock,
ducks picking paper, and ducks picking scissors.A move consists of an operation of one of the following three forms:
[list]
[*] If a duck picking rock sits behind a duck picking scissors, they switch places.
[*] If a duck picking paper sits behind a duck picking rock, they switch places.
[*] If a duck picking scissors sits behind a duck picking paper, they switch places.
[/list]
Determine, in terms of
, , and , the maximum number of moves which could takeplace, over all possible initial configurations.
34 replies
Floor sequence
va2010 87
N
by Mathgloggers
Source: 2015 ISL N1
Determine all positive integers 
such that the sequence 
defined by ![\[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \]](//latex.artofproblemsolving.com/2/1/b/21b2a1f93b11a94b84e7b55f4b4f679aa20e36c6.png)
contains at least one integer term.
such that the sequence 
defined by ![\[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \]](http://latex.artofproblemsolving.com/2/1/b/21b2a1f93b11a94b84e7b55f4b4f679aa20e36c6.png)
contains at least one integer term.
87 replies
INMO 2019 P3
div5252 45
N
by anudeep
Let 
be distinct positive integers. Prove that
Further, determine when equality holds.
be distinct positive integers. Prove that
Further, determine when equality holds.
45 replies
