[/quote]
,
,
,
is not greater than 
.
be positive reals with 
.![\[\sum_{i=1}^n \frac{x_i}{2-x_i} \ge \frac{n}{2n-1}\]](http://latex.artofproblemsolving.com/f/6/5/f65f6fce4465125dfacf002f0659c898df7ae150.png)
.
.
and 
Prove that
and 
are the incenter and the 
-
,
be the midpoint of arc 
on the circumcircle, and 
be the foot of the
.
intersects the circumcircle again at 
. For
on the circumcircle of triangle 
intersect 
. Prove
are concyclic.
squares, we cover the area with several S-Tetrominos (=Z-Tetrominos) along the square so that in every square, there are two or fewer tiles covering that (tiles can be overlap). Find the maximum possible number of squares covered by at least one tile.
satisfies
for any non-negative integers 
and 
.
.
be a function from the Euclidean plane to the real numbers such that 
for any acute triangle 
with circumcenter 
,
and orthocenter 
.
is constant.
be the positive real number. Prove that
,
,
grid. Alice goes first, and they take turns drawing a black point from the coordinate set![\[\{(i, j) \mid i, j \in \mathbb{N}, 1 \leq i, j \leq n\}\]](http://latex.artofproblemsolving.com/4/3/f/43f7493d1dff43079eb5481ee17bc8f16d7b9553.png)
There is a constraint that the distance between any two black points cannot be an integer. The player who cannot draw a black point loses. Find all integers 
Y by yao981113, MathRook7817, yrnsmurf, Sedro, aidan0626
Some of the discussion in the Contests & Programs SMT thread reminded me of Scorecards puzzles from the US Puzzle Championship. They behave similarly to "24 game" puzzles, but the allowable operations are slightly expanded.
As an example, the puzzle 
would have answer 
.
These have appeared on the USPC three years: 2018, 2022, and 2023. Try your hand at these! Some of them are much more devious than they first appear.
Scorecards rules wrote:
Operations are limited to addition ("+"), subtraction ("-"), multiplication ("x"), division ("/"), and exponentiation ("^").
Decimal points may be used; ...; use minus sign ("-") to indicate negative values. Use parentheses if needed to disambiguate operator precedence.
Decimal points may be used; ...; use minus sign ("-") to indicate negative values. Use parentheses if needed to disambiguate operator precedence.
would have answer 
.These have appeared on the USPC three years: 2018, 2022, and 2023. Try your hand at these! Some of them are much more devious than they first appear.
- 2018
- 2022
- 2023
This post has been edited 1 time. Last edited by djmathman, Apr 14, 2025, 5:54 PM
