[/quote]
,
.
s
such that 
be the divisors of 
.
.
a
that satisfy 
by ![\[\displaystyle f(x) = \left\{ \begin{array}{lr} x+\frac 12 & \text{if}\ \ x < \frac 12\\ x^2 & \text{if}\ \ x \ge \frac 12 \end{array} \right.\]](http://latex.artofproblemsolving.com/c/b/d/cbd9b412893774283a438296cdeb05b80cc878bf.png)
Let 
and 
be two real numbers such that 
.
and 
by 
,
,
for 
.![\[(a_n - a_{n-1})(b_n-b_{n-1})<0.\]](http://latex.artofproblemsolving.com/a/5/c/a5c8844d595bb1186d4804beadd69a3bc1d40ea3.png)
is defined recursively by ![\[a_0=-1,\qquad\sum_{k=0}^n\dfrac{a_{n-k}}{k+1}=0\quad\text{for}\quad n\geq 1.\]](http://latex.artofproblemsolving.com/f/4/b/f4b8f95e49de7274cccd8ee498e0b496f31ddfea.png)
Show that 
for all 
.
be positive integers satisfying the conditions 
and 
Show that there exists a real number 
with the property that all the three numbers 
have their fractional parts lying in the interval ![$ \left(\frac {1}{3}, \frac {2}{3} \right].$](http://latex.artofproblemsolving.com/4/2/b/42ba58ddd6032a3176122f1c9b2015cb6f4ca925.png)
table can be filled with one of the letters 
and 
in such a way that:
,
and one third are 
to 
with 
.
,
diagonals of two types. A diagonal of first type consists all cells 
is a constant, and the diagonal of this second type consists all cells 
is constant.
such that if any 
elements are removed from the set 
,
Y by Benq, MSTang, brianzhang, dantx5, rjiang16, champion999, Adventure10
Welcome to the BMC!
The Blitz Math Challenge's problems are fairly easy to solve (early-to-mid AMC10 level), but the true challenge will be to solve all twenty problems within ten minutes. Are you up to the challenge?
Please sign up below, updating the following list:
UPDATE: THE TEST HAS BEEN RELEASED!
The test is attached below. Please MAKE SURE to read all rules and formatting notes before you start! Breaks are FORBIDDEN as the contest is only ten minutes long. Submit your answers at tinyurl.com/blitzmath when you are done. Ties will be broken mathcounts-style (hardest question answered), followed by a tiebreaker round if needed. Good luck!
ERRATA: V1.1 HAS A FIXED VERSION OF #16, V1.3 HAS A FIXED VERSION OF #19
---
UPDATE 2 - And we're done!
http://www.artofproblemsolving.com/community/c5h1101725p5025122 has all of the end-of-test info, like solutions and results!
The Blitz Math Challenge's problems are fairly easy to solve (early-to-mid AMC10 level), but the true challenge will be to solve all twenty problems within ten minutes. Are you up to the challenge?
Please sign up below, updating the following list:
UPDATE: THE TEST HAS BEEN RELEASED!
The test is attached below. Please MAKE SURE to read all rules and formatting notes before you start! Breaks are FORBIDDEN as the contest is only ten minutes long. Submit your answers at tinyurl.com/blitzmath when you are done. Ties will be broken mathcounts-style (hardest question answered), followed by a tiebreaker round if needed. Good luck!
ERRATA: V1.1 HAS A FIXED VERSION OF #16, V1.3 HAS A FIXED VERSION OF #19
---
UPDATE 2 - And we're done!
http://www.artofproblemsolving.com/community/c5h1101725p5025122 has all of the end-of-test info, like solutions and results!
This post has been edited 7 times. Last edited by DeathLlama9, Jun 27, 2015, 9:38 PM
