prove that a chord is tangent to the incircle
by ihategeo_1969, Mar 23, 2025, 9:01 PM
Let 
be a triangle with incenter 
and intouch triangle 
. Let 
be the foot of the perpendicular from 
onto 
. Assume that 
, 
intersect the sides 
, 
in 
respectively. Finally, let the rays 
, 
meet the circumcircle of 
in 
, 
respectively. Prove that the tangent from to the incircle and the line 
meet on the circumcircle of .
Proposed by Aditya Khurmi
be a triangle with incenter 
and intouch triangle 
. Let 
be the foot of the perpendicular from 
onto 
. Assume that 
, 
intersect the sides 
, 
in 
respectively. Finally, let the rays 
, 
meet the circumcircle of 
in 
, 
respectively. Prove that the tangent from to the incircle and the line 
meet on the circumcircle of .Proposed by Aditya Khurmi
This post has been edited 1 time. Last edited by ihategeo_1969, 4 hours ago
such that![\[
f(xy) = f(x)f(y) \;-\; f(x + y) \;+\; 1,
\quad \forall x, y \in \mathbb{R}.
\]](http://latex.artofproblemsolving.com/7/a/5/7a51a7be2306d18f4e80db1f5aac6e9bbf25c25c.png)
satisfying 
and their sum is an even number, consider the quadratic polynomial:![\[
P(x) = x^2 - (m^2 - m + 1)x + (m^2 - n^2 - m)(n^2 + 1).
\]](http://latex.artofproblemsolving.com/d/5/1/d51aaddfddfe06cc91abfdd3476a155770a8a5fd.png)
are positive integers but are not perfect squares.
such that for any positive integer 
,
.
be a positive integer. For every positive integer 
the sequence 
is defined, where 
are integers. Among these 
such that![\[
ad = bc, \quad a + d = 2^k, \quad b + c = 2^m
\]](http://latex.artofproblemsolving.com/1/1/d/11d16732e3a77af46ad17adecf104a442a5d7f67.png)
for some positive integers 
and 
.
be a sequence of real numbers with 
.
of a permutation 
of 
to be the minima of the sum 
over all real numbers 
.
,
,
lie respectively on sides 
.
and 
and 
cross again at point 
different from 
.
is clean if it has a unique representation as a sum of an odd number of distinct elements from 
denote the time it takes for car 
to get to the first flag (choose a starting time arbitrarily). Suppose car 
when in and out of town, respectively; let 
(not both zero) be the distances from flag 
to flag 
along in and out of town routes. Then car 
to get to flag 
for some 
,
for all 
.
pairwise distinct cars (WLOG cars 
)
for all 
and 
.
iff![\[(x_k,y_k)\in S_{i,j}=\{(x,y)\mid L_{i,j}(x,y)=(a_i-a_j)x+(b_i-b_j)y+(t_i-t_j)>0\},\]](http://latex.artofproblemsolving.com/f/9/f/f9f56603a6e3049e2438f5de19aa67b1403e9a91.png)
and 
iff 
.
in the Cartesian plane for 
and 
)
(where 
is the number of lines) distinct regions (well-known, e.g. 
and 
are in the same region iff the orderings at flags 
are equal, so we're done by pigeonhole.
(in reduced form) is infinitely often not a prime power.
and primes from 
to 
for 
,
,
as well. Subtracting dudes, get a contradiction.
July 2014
June 2014