[/quote]
,
satisfy,
Where 
}
such that the equality 
holds for all pairs of real numbers 
.
is an obvious solution. Now, let 
.
or 
.
.
and 
.
![\[ \begin{tabular}{c c c c}
{} & \textbf{Adams} & \textbf{Baker} & \textbf{Adams and Baker} \\
\textbf{Boys:} & 71 & 81 & 79 \\
\textbf{Girls:} & 76 & 90 & ? \\
\textbf{Boys and Girls:} & 74 & 84 & \\
\end{tabular}
\]](http://latex.artofproblemsolving.com/8/5/c/85c75770c3f90af49df0db600dead2a29e6674b3.png)
be real numbers . Prove that
They fill in real numbers to empty spaces (the coefficients of the terms other than 
and the constant term) in turn. If the resulting polynomial has no real root, Mr. Fat wins; otherwise, Mr. Taf wins. If Mr. Fat goes first, who has a winning strategy?
be a positive integer. Consider a 
array of square unit cells. Two corner cells that lie on the same longest diagonal are colored black, and the rest of the array is white. A move consists of choosing a row or a column and changing the color of every cell in the chosen row or column.
,
,
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 
.
are non-negative real numbers such that 
Prove that![\[\sqrt{a^2+6ab}+\sqrt{b^2+6bc}+\sqrt{c^2+6ca} \ge 5\sqrt{1 + \frac{153abc}{50(a+b+c)}}.\]](http://latex.artofproblemsolving.com/4/2/9/4295e20975f0a7c9902a73695bebb665385f01b3.png)
or 
of real numbers such that![\[ a_{m^2 + m + n} = a_{m}^2 + a_m + a_n\]](http://latex.artofproblemsolving.com/b/b/9/bb97053e86e7733fe5c26f7859c87566eb444b52.png)
for all positive integers 
and 
.
be a convex quadrilateral with 
and 
.
and 
be points on segments 
and 
,
intersects lines 
and 
at points 
and 
,
.
be 
and 
lie on a circle.
Y by eibc, ninjaforce, vrondoS, crazylizard, scannose
Hi AOPS community,
We are excited to announce that Westford Academy (located in Westford, MA) will be hosting its first ever math competition for middle school students (grades 5-8).
Based in Massachusetts, this tournament hosts ambitious and mathematically skilled students in grades 5–8 to compete against other middle school math teams while fostering their problem-solving skills and preparing them to continue enriching their STEM skills in high school and in the future.
This competition will be held on April 12, 2025 from 12:00 PM to 5:00 PM and will feature 3 rounds (team, speed, and accuracy). The problems will be of similar difficulty for AMC 8-10 and were written by USA(J)MO and AIME qualifiers.
If you are in the Massachusetts area and are curious about Mathematics, we cordially invite you to sign up by scanning the QR code on the attached flyer. Please note that teams consist of 4-6 competitors, but if you prefer to register as an individual competitor, you will be randomly placed on a team of other individual competitors. Feel free to refer the attached flyer and website as needed.
https://sites.google.com/westfordk12.us/wamt/home?authuser=2
We are excited to announce that Westford Academy (located in Westford, MA) will be hosting its first ever math competition for middle school students (grades 5-8).
Based in Massachusetts, this tournament hosts ambitious and mathematically skilled students in grades 5–8 to compete against other middle school math teams while fostering their problem-solving skills and preparing them to continue enriching their STEM skills in high school and in the future.
This competition will be held on April 12, 2025 from 12:00 PM to 5:00 PM and will feature 3 rounds (team, speed, and accuracy). The problems will be of similar difficulty for AMC 8-10 and were written by USA(J)MO and AIME qualifiers.
If you are in the Massachusetts area and are curious about Mathematics, we cordially invite you to sign up by scanning the QR code on the attached flyer. Please note that teams consist of 4-6 competitors, but if you prefer to register as an individual competitor, you will be randomly placed on a team of other individual competitors. Feel free to refer the attached flyer and website as needed.
https://sites.google.com/westfordk12.us/wamt/home?authuser=2
Attachments:
