and 
where 
denotes greatest integer function.
and 
and 
be distinct real numbers such that 
and 
Y by
In triangle 
, prove that
![\[ a = b\cos C + c\cos B. \]](//latex.artofproblemsolving.com/7/e/d/7ed699f18ede191e5b89bc79d7f0592294347818.png)
, prove that![\[ a = b\cos C + c\cos B. \]](http://latex.artofproblemsolving.com/7/e/d/7ed699f18ede191e5b89bc79d7f0592294347818.png)
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Orthocentroidal circle, orthotransversal, concurrent lines
kosmonauten3114 0
Source: My own
Let 
be a scalene oblique triangle, and 
a point on the orthocentroidal circle of (
).
Prove that the orthotransversal of, trilinear polar of the polar conjugate (
-isoconjugate) of , Droz-Farny axis of are concurrent.
The definition of the Droz-Farny axis of with respect to is as follows:
For a point
, there exists a pair of orthogonal lines 
, 
through such that the midpoints of the 3 segments cut off by , from the sidelines of are collinear. The line through these 3 midpoints is the Droz-Farny axis of wrt .
be a scalene oblique triangle, and 
a point on the orthocentroidal circle of (
).Prove that the orthotransversal of
, trilinear polar of the polar conjugate (
-isoconjugate) of , Droz-Farny axis of are concurrent.The definition of the Droz-Farny axis of
with respect to is as follows:For a point
, there exists a pair of orthogonal lines 
, 
through such that the midpoints of the 3 segments cut off by , from the sidelines of are collinear. The line through these 3 midpoints is the Droz-Farny axis of wrt .
0 replies
[PMO27 Areas] I.11 Polynomial interpolation
aops-g5-gethsemanea2 3
N
by BinariouslyRandom
A polynomial 
with nonnegative integer coefficients satisfies 
and 
. Find 
.
with nonnegative integer coefficients satisfies 
and 
. Find 
.3 replies
Is problem true?!?!?!?
giangtruong13 0
Let be a triangle, 
is incenter of triangle . Draw 
perpendicular to 
at 
and 
perpendicular to 
at 
, 
. Prove that: Area of triangle 
be a triangle, 
is incenter of triangle . Draw 
perpendicular to 
at 
and 
perpendicular to 
at 
, 
. Prove that: Area of triangle 
0 replies
Easy one
irregular22104 1
N
by IceyCold
Given two positive integers 
written on the board. We apply the following rule: At each step, we will add all the numbers that are the sum of the two numbers on the board so that the sum does not appear on the board. For example, if the two initial numbers are 
; then the numbers on the board after step 1 are 
; after step 2 are 
1) With
; 
, prove that the number 2024 cannot appear on the board.
2) With
; 
, prove that the number 2024 can appear on the board.
written on the board. We apply the following rule: At each step, we will add all the numbers that are the sum of the two numbers on the board so that the sum does not appear on the board. For example, if the two initial numbers are 
; then the numbers on the board after step 1 are 
; after step 2 are 
1) With
; 
, prove that the number 2024 cannot appear on the board.2) With
; 
, prove that the number 2024 can appear on the board.
1 reply
[PMO27 Qualis] III.4 Grid path (sort of)
aops-g5-gethsemanea2 3
N
by tapilyoca
How many ways are there to write each integer from 
to 
on a different unit square of a 
square grid, such that consecutive integers are on adjacent squares, and is not adjacent to ? (Note that adjacent squares are squares that share a common side.)
to 
on a different unit square of a 
square grid, such that consecutive integers are on adjacent squares, and is not adjacent to ? (Note that adjacent squares are squares that share a common side.)3 replies
NT problem
toanrathay 0
Let 
be a prime and 
be positive integers such that 
and 
is prime. Prove that 
be a prime and 
be positive integers such that 
and 
is prime. Prove that 
0 replies
[PMO19 Areas I.18] easy combi
tapilyoca 2
N
by tapilyoca
A railway passes through four towns 
, 
, 
, and 
. The railway forms a complete loop, as shown below, and trains go in both directions. Suppose that a trip between two adjacent towns costs one ticket. Using exactly eight tickets, how many distinct ways are there of traveling from town and ending at town ? (You may pass through town in the middle).
IMAGE
, 
, 
, and 
. The railway forms a complete loop, as shown below, and trains go in both directions. Suppose that a trip between two adjacent towns costs one ticket. Using exactly eight tickets, how many distinct ways are there of traveling from town and ending at town ? (You may pass through town in the middle).IMAGE
2 replies
A Projection Theorem
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Source: VN Math Olympiad For High School Students P1 - 2025
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Y by buratinogigle
too lazy to pull up a diagram
but you consider the cases where the triangle is acute, and where either angle B or angle C is obtuse
if the triangle is acute, dropping an altitude from A basically directly shows it
if the triangle is obtuse (let's say C is, other case basically the same), let the intersection of the altitude from A and the line BC be D, BC=c*cos(B), CD=b*cos(pi-C)=-b*cos(C), a=BC=BD-CD=c*cos(B)-(-b*cos(C))=c*cos(B)+b*cos(C)
but you consider the cases where the triangle is acute, and where either angle B or angle C is obtuse
if the triangle is acute, dropping an altitude from A basically directly shows it
if the triangle is obtuse (let's say C is, other case basically the same), let the intersection of the altitude from A and the line BC be D, BC=c*cos(B), CD=b*cos(pi-C)=-b*cos(C), a=BC=BD-CD=c*cos(B)-(-b*cos(C))=c*cos(B)+b*cos(C)
Z
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Y
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N Quick Reply
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