[/quote]
,
such that ![\[ f(x^2 + yf(z)) = xf(x) + zf(y) , \]](http://latex.artofproblemsolving.com/2/b/a/2baee1d8464992fc38a16561b48b18f4b35807bd.png)
for all 
.
,
,
be an infinite sequence of positive integers. Suppose that there is an integer 
such that, for each 
,
is an integer. Prove that there is a positive integer 
such that 
for all 
.
,
,
be positive real numbers such that 
.![$$\sqrt[3]{(a^2+1)(b^2+1)(c^2+1)} \le \left(\frac{a+b+c}{3}\right)^2 + 1.$$](http://latex.artofproblemsolving.com/2/0/7/207934bca53047858a3820f37705171ff0abfd45.png)
and 
and 
Prove that![\[\frac{a}{(b+c-a)^2} + \frac{b}{(c+a-b)^2} + \frac{c}{(a+b-c)^2} \geq \frac{3}{(abc)^2}.\]](http://latex.artofproblemsolving.com/0/a/b/0ab4b4e6c28a1ae67788525642661dbcd04d0bd8.png)
and in 
steps and has to reach 
is valid but 
is not valid.
,
lie on a line in this order and satisfy 
.
be the midpoints of 
and 
,
is acute, and let 
be its orthocentre. Points 
lie on lines 
and 
,
and 
and 
and 
such for any 
.
satisfy,
Where 
}
be an integer. We are given a balance and 
weights of weight 
.
to satisfy that the equation 
has at least 1 roots 
of the equation![\[n^3=p^2-p-1\]](http://latex.artofproblemsolving.com/f/f/8/ff88093bb0aa93afcfdff3ae25149ec5fbacb61f.png)
is a prime number and 
?
be an integer. In a configuration of an 
board, each of the 
cells contains an arrow, either pointing up, down, left, or right. Given a starting configuration, Turbo the snail starts in one of the cells of the board and travels from cell to cell. In each move, Turbo moves one square unit in the direction indicated by the arrow in her cell (possibly leaving the board). After each move, the arrows in all of the cells rotate 
counterclockwise. We call a cell good if, starting from that cell, Turbo visits each cell of the board exactly once, without leaving the board, and returns to her initial cell at the end. Determine, in terms of 
Y by
In an acute triangle 
, an arbitrary point 
is chosen on the altitude 
. The points 
and 
are the midpoints of 
and 
, respectively. The perpendiculars from on 
and from on 
intersect at the point 
. Show that 
.
, an arbitrary point 
is chosen on the altitude 
. The points 
and 
are the midpoints of 
and 
, respectively. The perpendiculars from on 
and from on 
intersect at the point 
. Show that 
.This post has been edited 1 time. Last edited by parmenides51, Nov 14, 2021, 8:22 AM
,
,
the circuncenter of 
.
and both lines are perpendicular to 
using the Intercept theorem.
, 
are parallel to 
,
we conclude that 
is perpendicular to ![[asy]
import olympiad;
size(7 cm);
defaultpen(fontsize(10pt));
pair A, B, C, H, E, F, P, M, G, K, O1, O2, O3;
A = (2,5); B = (6,0); C = (0,0); E= (1, 2.5); F= (4,2.5); P= (2,3);
O1 = (P+C)/2;
O2 = (P+B)/2;
M = orthocenter(C,E,P);
H=orthocenter(A,B,C);
G= orthocenter(P,B,F);
K= extension(E,M,F,G);
O3 = (A+P)/2;
draw(circumcircle(A,B,C), dashed+blue);
draw(A--B--C--A);
draw(B--K);
draw(C--K);
draw(E--O1, lightblue);
draw(F--O2, lightblue);
draw(A--O3, lightblue);
dot("$A$",A);dot("$B$",B);dot("$C$",C);dot("$F$" ,F, blue);dot("$E$" ,E, blue);dot("$P$",P, purple);dot("$K$", K, red);dot("$B'$", O1, orange); dot("$F'$", O2, orange);dot(O3);
[/asy]](http://latex.artofproblemsolving.com/2/e/4/2e466132fcfa3c193ce288a6f2abb6312baa3cdf.png)
