[/quote]
,
are positive real numbers such that 
,![\[ a^{b+c} b^{c+a} c^{a+b} \leq 1 . \]](http://latex.artofproblemsolving.com/0/c/e/0ce097b0b507403c959c2992cae1217ecdd7e581.png)
be a finite sequence of positive integers. Prove that there exist nonnegative integers 
and 
such that 
holds for all integers 
be a positive integer. Dominoes are placed on a 
board in such a way that every cell of the board is adjacent to exactly one cell covered by a domino. For each 
or 
.
of positive real numbers that satisfy the system:![\[
\begin{aligned}
11bc - 36b - 15c &= abc \\
12ca - 10c - 28a &= abc \\
13ab - 21a - 6b &= abc.
\end{aligned}
\]](http://latex.artofproblemsolving.com/1/7/0/170e1cfaee90f5c37428fa75192ac889261a8f83.png)
be positive reals. Prove that there is a cyclic permutation 
of 
such that the inequality:![\[
\frac{a}{xa + yb + zc} + \frac{b}{xb + yc + za} + \frac{c}{xc + ya + zb} \geq \frac{3}{x + y + z}
\]](http://latex.artofproblemsolving.com/5/5/3/553527fc40a1b7d29dbdfb4b5e63635308f46c85.png)
and 
.
be a scalene triangle with orthocenter 
and circumcenter 
.
,
,
.
with diameter 
,
at a point 
.
meets line 
at 
.
and 
intersect at a point 
on 
.
Prove that: 
is the sum of 
square numbers
Let 
and 
is the sum of 
is the sum of 
and incenter 
and let 
,
,
are selected on sides 
,
such that 
,
,
.
intersects 
other than 
.
and 
meet on 
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The Lexington High School Math Team is proud to announce LMT Spring 2025 and our inaugural Girls’ LMT 2025! LMT is a competition for middle school students interested in math. Students can participate individually, or on teams of 4-6 members. This announcement contains information for BOTH competitions.
LMT Spring 2025 will take place from 8:30 AM-5:00 PM on Saturday, May 3rd at Lexington High School, 251 Waltham St., Lexington, MA 02421.
The competition will include two individual rounds, a Team Round, and a Guts Round, with a break for lunch and mini-events. A detailed schedule is available at https://lhsmath.org/LMT/Schedule.
There is a $15 fee per participant, paid on the day of the competition. Pizza will be provided for lunch, at no additional cost.
Register for LMT at https://lhsmath.org/LMT/Registration/Home.
Girls’ LMT 2025 will be held ONLINE on MathDash from 11:00 AM-4:15 PM EST on Saturday, April 19th, 2025. Participation is open to middle school students who identify as female or non-binary. The competition will include an individual round and a team round with a break for lunch and mini-events. It is free to participate.
Register for GLMT at https://www.lhsmath.org/LMT/Girls_LMT.
More information is available on our website: https://lhsmath.org/LMT/Home. Email lmt.lhsmath@gmail.com with any questions.
LMT Spring 2025 will take place from 8:30 AM-5:00 PM on Saturday, May 3rd at Lexington High School, 251 Waltham St., Lexington, MA 02421.
The competition will include two individual rounds, a Team Round, and a Guts Round, with a break for lunch and mini-events. A detailed schedule is available at https://lhsmath.org/LMT/Schedule.
There is a $15 fee per participant, paid on the day of the competition. Pizza will be provided for lunch, at no additional cost.
Register for LMT at https://lhsmath.org/LMT/Registration/Home.
Girls’ LMT 2025 will be held ONLINE on MathDash from 11:00 AM-4:15 PM EST on Saturday, April 19th, 2025. Participation is open to middle school students who identify as female or non-binary. The competition will include an individual round and a team round with a break for lunch and mini-events. It is free to participate.
Register for GLMT at https://www.lhsmath.org/LMT/Girls_LMT.
More information is available on our website: https://lhsmath.org/LMT/Home. Email lmt.lhsmath@gmail.com with any questions.
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This post has been edited 1 time. Last edited by vrondoS, Mar 27, 2025, 5:18 AM
