[/quote]
,
be a triangle. On side 
take points 
and 
such that 
take points 
and 
such that 
and on side 
take points 
and 
such that 
Let 
and 
are concurrent.
of occurring, Person A wins.
of occurring, Person A wins.
of occurring, Person A wins.
of occurring. Now, note that there is no possible way for someone to win with less than 
points at this point, so we can ignore that condition.
-
Solving yields 
.
(as a percent) as a blind guess.
be a convex quadrilateral with ![\[\angle ADC = 90^\circ, \ \ \angle BCD = \angle ABC > 90^\circ, \mbox{ and } AB = 2CD.\]](http://latex.artofproblemsolving.com/b/4/1/b416d5c0b29c3a69a3e87cd22b6000aa8e70d456.png)
The line through 
,
,
at point 
.
,
,
,
,
can be divided into two groups, so that the angles in each group have a sum of 
.
cards on a table, flipped face down. Madina knows that on each card a single number is written and that the numbers are different integers from 
to 
cards, and she is told a number that is written on one of the chosen cards, but not which specific card it is on. After several moves, Madina must determine the written numbers on as many cards as possible. What is the maximum number of cards Madina can ensure to determine?
and 
of the acute-angled triangle 
intersect at point 
,
intersect at point 
,
.
.
on 
.
points on the 
,
units of effort to gain 
points on the 
,
.
points on 
and 
points on 
to make 
be positive real numbers. Prove that there exists a cyclic permutation 
of 
such that for all positive real numbers 
the following holds:![\[
\frac{a}{x\,a + y\,b + z\,c}
\;+\;
\frac{b}{x\,b + y\,c + z\,a}
\;+\;
\frac{c}{x\,c + y\,a + z\,b}
\;\ge\;
\frac{3}{x + y + z}.
\]](http://latex.artofproblemsolving.com/1/5/2/15261b3f712c2cc8ffbeb92a9711e06ad0992b5d.png)
.
be distinct primes and let 
be nonnegative integers. Define![\[
m \;=\;
\frac12
\Bigl(\prod_{i=2}^k p_i^{a_i}\Bigr)
\Bigl(\prod_{i=1}^k(p_i+1)\;+\;\sum_{i=1}^k(p_i-1)\Bigr),
\]](http://latex.artofproblemsolving.com/7/9/a/79af85375957986298d37c00d288b6d402f40106.png)
![\[
n \;=\;
\frac12
\Bigl(\prod_{i=2}^k p_i^{a_i}\Bigr)
\Bigl(\prod_{i=1}^k(p_i+1)\;-\;\sum_{i=1}^k(p_i-1)\Bigr).
\]](http://latex.artofproblemsolving.com/e/2/3/e23799ff03400d5eade3731bb83456fcd702a5e1.png)
Prove that![\[
p^2-1 \;\bigm|\; p\,m \;-\; n.
\]](http://latex.artofproblemsolving.com/3/7/e/37e4535b59ef183908730946213429e9ebc05dd7.png)
satisfying
Prove
is satisfied. Point 
is taken on the circumcircle of 
.
which passes from 
and 
.
Y by
Hello, I am going to be a 7th grader next year and I really want to qualify for USAJMO in 8th grade, so far I have these goals reached
1. AMC 10 Honor Roll A and B 2025
2. AMC 8 DHR and HR
3. AIME 3
This year on AIME something happened and I got a 3 on the AMC's I got a 105 on AMC 10 A and I got a 114 on AMC 10 B. I want to improve mostly on AIME but since the AMC 10 is coming up quicker what would you guys recommend for getting 110+ on both of the AMC 10's and getting a 6+ on AIME? So far I am only doing Alcumus and have no books so far.... Checking the table of contents on the books Alcumus provides the same topics. I was thinking to take WOOT 1 and AMC 10 Problem Series.
1. AMC 10 Honor Roll A and B 2025
2. AMC 8 DHR and HR
3. AIME 3
This year on AIME something happened and I got a 3
on the AMC's I got a 105 on AMC 10 A and I got a 114 on AMC 10 B. I want to improve mostly on AIME but since the AMC 10 is coming up quicker what would you guys recommend for getting 110+ on both of the AMC 10's and getting a 6+ on AIME? So far I am only doing Alcumus and have no books so far.... Checking the table of contents on the books Alcumus provides the same topics. I was thinking to take WOOT 1 and AMC 10 Problem Series.
