another problem
by kjhgyuio, May 1, 2025, 12:46 AM
Attachments:
problem interesting
by Cobedangiu, Apr 30, 2025, 5:06 AM
Let 
Prove that: 
is the sum of 
square numbers
Let 
and 
is the sum of square numbers. Prove that: 
is the sum of square numbers
Prove that: 
is the sum of 
square numbers
Let 
and 
is the sum of square numbers. Prove that: 
is the sum of square numbersThis post has been edited 1 time. Last edited by Cobedangiu, Yesterday at 5:06 AM
BMO 2024 SL A3
by MuradSafarli, Apr 27, 2025, 12:42 PM
A3.
Find all triples
of positive real numbers that satisfy the system:
![\[
\begin{aligned}
11bc - 36b - 15c &= abc \\
12ca - 10c - 28a &= abc \\
13ab - 21a - 6b &= abc.
\end{aligned}
\]](//latex.artofproblemsolving.com/1/7/0/170e1cfaee90f5c37428fa75192ac889261a8f83.png)
Find all triples
of positive real numbers that satisfy the system:![\[
\begin{aligned}
11bc - 36b - 15c &= abc \\
12ca - 10c - 28a &= abc \\
13ab - 21a - 6b &= abc.
\end{aligned}
\]](http://latex.artofproblemsolving.com/1/7/0/170e1cfaee90f5c37428fa75192ac889261a8f83.png)
BMO 2024 SL A1
by MuradSafarli, Apr 27, 2025, 12:24 PM
A1.
Let
be positive reals. Prove that there is a cyclic permutation 
of 
such that the inequality:
![\[
\frac{a}{xa + yb + zc} + \frac{b}{xb + yc + za} + \frac{c}{xc + ya + zb} \geq \frac{3}{x + y + z}
\]](//latex.artofproblemsolving.com/5/5/3/553527fc40a1b7d29dbdfb4b5e63635308f46c85.png)
holds for all positive real numbers
and 
.
Let
be positive reals. Prove that there is a cyclic permutation 
of 
such that the inequality:![\[
\frac{a}{xa + yb + zc} + \frac{b}{xb + yc + za} + \frac{c}{xc + ya + zb} \geq \frac{3}{x + y + z}
\]](http://latex.artofproblemsolving.com/5/5/3/553527fc40a1b7d29dbdfb4b5e63635308f46c85.png)
holds for all positive real numbers
and 
.This post has been edited 1 time. Last edited by MuradSafarli, Apr 27, 2025, 12:26 PM
2^x+3^x = yx^2
by truongphatt2668, Apr 22, 2025, 3:38 PM
Prove that the following equation has infinite integer solutions:
Yet another domino problem
by juckter, Apr 9, 2019, 11:12 AM
Let 
be a positive integer. Dominoes are placed on a 
board in such a way that every cell of the board is adjacent to exactly one cell covered by a domino. For each , determine the largest number of dominoes that can be placed in this way.
(A domino is a tile of size
or 
. Dominoes are placed on the board in such a way that each domino covers exactly two cells of the board, and dominoes do not overlap. Two cells are said to be adjacent if they are different and share a common side.)
be a positive integer. Dominoes are placed on a 
board in such a way that every cell of the board is adjacent to exactly one cell covered by a domino. For each , determine the largest number of dominoes that can be placed in this way.(A domino is a tile of size
or 
. Dominoes are placed on the board in such a way that each domino covers exactly two cells of the board, and dominoes do not overlap. Two cells are said to be adjacent if they are different and share a common side.)Bounding is hard
by whatshisbucket, Jun 28, 2018, 7:13 AM
Let 
be a finite sequence of positive integers. Prove that there exist nonnegative integers 
and 
such that 
holds for all integers 
Proposed by Carl Schildkraut
be a finite sequence of positive integers. Prove that there exist nonnegative integers 
and 
such that 
holds for all integers 
Proposed by Carl Schildkraut
This post has been edited 1 time. Last edited by whatshisbucket, Jun 29, 2018, 1:06 AM
Show that XD and AM meet on Gamma
by MathStudent2002, Jul 19, 2017, 4:32 PM
Let 
be a triangle with circumcircle 
and incenter 
and let 
be the midpoint of 
. The points 
, 
, 
are selected on sides , 
, 
such that 
, 
, and 
. Suppose that the circumcircle of 
intersects at a point 
other than 
. Prove that lines 
and 
meet on .
Proposed by Evan Chen, Taiwan
be a triangle with circumcircle 
and incenter 
and let 
be the midpoint of 
. The points 
, 
, 
are selected on sides , 
, 
such that 
, 
, and 
. Suppose that the circumcircle of 
intersects at a point 
other than 
. Prove that lines 
and 
meet on .Proposed by Evan Chen, Taiwan
This post has been edited 2 times. Last edited by v_Enhance, May 6, 2019, 1:36 PM
Medium geometry with AH diameter circle
by v_Enhance, Jun 28, 2016, 2:25 PM
Let be a scalene triangle with orthocenter 
and circumcenter 
. Denote by , the midpoints of 
, . Suppose the circle 
with diameter meets the circumcircle of at 
, and meets line 
at a point 
. The tangent to at 
meets line 
at 
. Show that the circumcircles of 
and 
intersect at a point 
on 
.
Proposed by Evan Chen
be a scalene triangle with orthocenter 
and circumcenter 
. Denote by , the midpoints of 
, . Suppose the circle 
with diameter meets the circumcircle of at 
, and meets line 
at a point 
. The tangent to at 
meets line 
at 
. Show that the circumcircles of 
and 
intersect at a point 
on 
.Proposed by Evan Chen
are positive real numbers such that 
,![\[ a^{b+c} b^{c+a} c^{a+b} \leq 1 . \]](http://latex.artofproblemsolving.com/0/c/e/0ce097b0b507403c959c2992cae1217ecdd7e581.png)
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