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NT game with products
Kimchiks926 4
N
by math-olympiad-clown
Source: Baltic Way 2022, Problem 20
Ingrid and Erik are playing a game. For a given odd prime 
, the numbers 
are written on a blackboard. The players take turns making moves with Ingrid starting. A move consists of one of the players crossing out a number on the board that has not yet been crossed out. If the product of all currently crossed out numbers is 
after the move, the player whose move it was receives one point, otherwise, zero points are awarded. The game ends after all numbers have been crossed out.
The player who has received the most points by the end of the game wins. If both players have the same score, the game ends in a draw. For each, determine which player (if any) has a winning strategy
, the numbers 
are written on a blackboard. The players take turns making moves with Ingrid starting. A move consists of one of the players crossing out a number on the board that has not yet been crossed out. If the product of all currently crossed out numbers is 
after the move, the player whose move it was receives one point, otherwise, zero points are awarded. The game ends after all numbers have been crossed out.The player who has received the most points by the end of the game wins. If both players have the same score, the game ends in a draw. For each
, determine which player (if any) has a winning strategy
4 replies
set with c+2a>3b
VicKmath7 49
N
by wangyanliluke
Source: ISL 2021 A1
Let 
be a positive integer. Given is a subset 
of 
with 
elements. Prove that there exist three elements 
from such that 
.
Proposed by Dominik Burek and Tomasz Ciesla, Poland
be a positive integer. Given is a subset 
of 
with 
elements. Prove that there exist three elements 
from such that 
.Proposed by Dominik Burek and Tomasz Ciesla, Poland
49 replies
interesting geo config (2/3)
Royal_mhyasd 8
N
by Royal_mhyasd
Source: own
Let 
be an acute triangle and 
its orthocenter. Let 
be a point on the parallel through to 
such that 
. Define 
and 
as points on the parallels through 
to 
and through 
to 
similarly. If 
are positioned around the sides of as in the given configuration, prove that are collinear.
be an acute triangle and 
its orthocenter. Let 
be a point on the parallel through to 
such that 
. Define 
and 
as points on the parallels through 
to 
and through 
to 
similarly. If 
are positioned around the sides of as in the given configuration, prove that are collinear.
8 replies
Problem 10
SlovEcience 4
N
by SlovEcience
Let 
be positive real numbers satisfying
![\[ xy + yz + zx = 3xyz. \]](//latex.artofproblemsolving.com/9/2/0/9200874fd6bf9a32ff0c4e82382a450be68cc444.png)
Prove that
![\[
\sqrt{\frac{x}{3y^2z^2 + xyz}} + \sqrt{\frac{y}{3x^2z^2 + xyz}} + \sqrt{\frac{z}{3x^2y^2 + xyz}} \le \frac{3}{2}.
\]](//latex.artofproblemsolving.com/f/5/f/f5fb85d81e93935601ceb5f30c25aea7aa3f23b2.png)
be positive real numbers satisfying![\[ xy + yz + zx = 3xyz. \]](http://latex.artofproblemsolving.com/9/2/0/9200874fd6bf9a32ff0c4e82382a450be68cc444.png)
Prove that![\[
\sqrt{\frac{x}{3y^2z^2 + xyz}} + \sqrt{\frac{y}{3x^2z^2 + xyz}} + \sqrt{\frac{z}{3x^2y^2 + xyz}} \le \frac{3}{2}.
\]](http://latex.artofproblemsolving.com/f/5/f/f5fb85d81e93935601ceb5f30c25aea7aa3f23b2.png)
4 replies
IMO ShortList 2003, combinatorics problem 4
darij grinberg 39
N
by ThatApollo777
Source: Problem 5 of the German pre-TST 2004, written in December 03
Let 
and 
be real numbers. Let 
be the matrix with entries ![\[a_{ij} = \begin{cases}1,&\text{if }x_i + y_j\geq 0;\\0,&\text{if }x_i + y_j < 0.\end{cases}\]](//latex.artofproblemsolving.com/c/0/8/c0856fe7b2b113180659952c4d216a01d8849f64.png)
Suppose that is an 
matrix with entries 
, 
such that the sum of the elements in each row and each column of is equal to the corresponding sum for the matrix . Prove that 
.
and 
be real numbers. Let 
be the matrix with entries ![\[a_{ij} = \begin{cases}1,&\text{if }x_i + y_j\geq 0;\\0,&\text{if }x_i + y_j < 0.\end{cases}\]](http://latex.artofproblemsolving.com/c/0/8/c0856fe7b2b113180659952c4d216a01d8849f64.png)
Suppose that is an 
matrix with entries 
, 
such that the sum of the elements in each row and each column of is equal to the corresponding sum for the matrix . Prove that 
.
39 replies
greatest volume
hzbrl 4
N
by hzbrl
Source: purple comet
A large sphere with radius 7 contains three smaller balls each with radius 3 . The three balls are each externally tangent to the other two balls and internally tangent to the large sphere. There are four right circular cones that can be inscribed in the large sphere in such a way that the bases of the cones are tangent to all three balls. Of these four cones, the one with the greatest volume has volume 
. Find .
. Find .
4 replies
Projective geo
drmzjoseph 1
N
by Luis González
Any pure projective solution? I mean no metrics, Menelaus, Ceva, bary, etc
Only pappus, desargues, dit, etc
Btw prove that
are collinear, and 
are arbitrary points
Only pappus, desargues, dit, etc
Btw prove that
are collinear, and 
are arbitrary points
1 reply
2019 Iberoamerican Mathematical Olympiad, P1
jbaca 9
N
by jordiejoh
For each positive integer , let 
be the sum of the squares of the digits of . For example, 
. Determine all integers 
such that 
.
, let 
be the sum of the squares of the digits of . For example, 
. Determine all integers 
such that 
.
9 replies
Conditional geo with centroid
a_507_bc 7
N
by Tkn
Source: Singapore Open MO Round 2 2023 P1
In a scalene triangle 
with centroid 
and circumcircle 
centred at 
, the extension of 
meets at 
; lines and 
intersect at ; and lines and 
intersect at . Suppose the circumcentre 
of the triangle 
lies on and 
are collinear. Prove that 
.
with centroid 
and circumcircle 
centred at 
, the extension of 
meets at 
; lines and 
intersect at ; and lines and 
intersect at . Suppose the circumcentre 
of the triangle 
lies on and 
are collinear. Prove that 
.
7 replies
People live in Kansas?
jj_ca888 13
N
by Ilikeminecraft
Source: SMO 2020/5
In triangle , let 
and 
be points on sides and , respectively, such that 
is cyclic. Let lines 
and 
intersect at point , and and 
be the midpoints of 
and 
, respectively. If 
is the foot of the perpendicular from to , and the circumcircles of triangles 
and 
intersect at second point 
different from , prove that 
and are collinear.
Proposed by Andrew Wen and William Yue
, let 
and 
be points on sides and , respectively, such that 
is cyclic. Let lines 
and 
intersect at point , and and 
be the midpoints of 
and 
, respectively. If 
is the foot of the perpendicular from to , and the circumcircles of triangles 
and 
intersect at second point 
different from , prove that 
and are collinear.Proposed by Andrew Wen and William Yue
13 replies
