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such that every collection of consecutive positive integers contains an integer whose sum of digits is divisible by 
. for which he can do this, or prove that none exist.
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for which there exist infinitely many positive integers 
such that ![\[ \frac{a^m+a-1}{a^n+a^2-1} \]](http://latex.artofproblemsolving.com/2/8/a/28a68e409451bbdcaee7c1286341f6aeea9ffdad.png)
is itself an integer.
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for which there exists a sequence of 
pairwise distinct points 
in the plane satisfying the following four conditions: [list=i] [*]no three of the points are collinear;
for all 
, where 
;
for all 
, where 
; and
for all and 
.[/list]
be a positive integer. Find the largest real number 
as a function of 
such that the inequality![\[\max\left(|a_1+a_2|, |a_2+a_3|, \dots ,|a_{n-1}+a_{n}| , |a_n+a_1|\right) \ge t_n \cdot \max(|a_1|,|a_2|, \dots ,|a_n|)\]](http://latex.artofproblemsolving.com/2/8/2/2828c2c73c3d6d7094d50bfcc2b114f492bb4f14.png)
holds for all real numbers 
.
, 
and the hypotenuse 
of a right triangle 
, points 
and 
, respectively, were chosen so that the angles 
and 
are equal. Prove that 
.
, and the hypotenuse of a right triangle , points and , respectively, were chosen so that the angles and are equal. Prove that .
plane let us take 
. If the hypothenuse 
is represented by the line 
, then let 
(for 
) with points 
such that 
(for ![$\theta \in (0,\pi/2]$](http://latex.artofproblemsolving.com/9/0/9/909367977736e40266f47582e9c94c10e45faa6a.png)
).![$\overline{KM} = \frac{t}{\cos[\arctan(b/a)-\theta]}$](http://latex.artofproblemsolving.com/b/b/8/bb8c84aa87eb8e11c44581f8236bed6dbca8cbd8.png)
(i);![$\overline{ML} = \frac{b-\frac{b}{a}t}{\cos[\arctan(a/b)-\theta]}$](http://latex.artofproblemsolving.com/0/d/3/0d3b1d987389e74a0bf92c6cd74e1a11b9db8a67.png)
(ii)![$\overline{KM}+\overline{ML} \le \overline{AB} \Rightarrow \frac{t}{\cos[\arctan(b/a)-\theta]} + \frac{b-\frac{b}{a}t}{\cos[\arctan(a/b)-\theta]} \le \overline{AB}$](http://latex.artofproblemsolving.com/a/c/e/acecbc0c6b8b5e35ffd30bf52b266d5a8d4ce421.png)
;
;
(iii).
such that 
:
(iv);![$f_{\theta} = t\left[\frac{a\sin\theta-b\cos\theta}{(a\cos\theta+b\sin\theta)^2}\right] + \left(\frac{b}{a}t-b\right)\left[\frac{a\cos\theta-b\sin\theta}{(a\sin\theta+b\cos\theta)^2}\right]=0$](http://latex.artofproblemsolving.com/6/e/4/6e4ebe72ee0cab7231575a84327f52faec8a6430.png)
(v)
. These in turn yield for (iii)
![$\left[\frac{1}{\sqrt{2}} \cdot \frac{t+a-t}{a}\right]\cdot \sqrt{a^2+b^2} \le \sqrt{a^2+b^2}$](http://latex.artofproblemsolving.com/7/d/0/7d0a04044e45a6295a86e5bfce03eabad0c61e12.png)
;
TRUE.
(which occurs when 
) 
and 
, we obtain equality 
TRUE.
