[Signups Now!] - Inaugural Academy Math Tournament
by elements2015, May 12, 2025, 8:13 PM
Hello!
Pace Academy, from Atlanta, Georgia, is thrilled to host our Inaugural Academy Math Tournament online through Saturday, May 31.
AOPS students are welcome to participate online, as teams or as individuals (results will be reported separately for AOPS and Georgia competitors). The difficulty of the competition ranges from early AMC to mid-late AIME, and is 2 hours long with multiple sections. The format is explained in more detail below. If you just want to sign up, here's the link:
https://forms.gle/ih548axqQ9qLz3pk7
If participating as a team, each competitor must sign up individually and coordinate team names!
Detailed information below:
Divisions & Teams
Both sections—Sprint and Challenge—will be administered consecutively in a single, individually completed 120-minute test. Students may allocate time between the sections however they wish to.
Awards & Scoring
Please have EACH STUDENT INDIVIDUALLY reserve a 120-minute window for your team's online test in THIS GOOGLE FORM:
https://forms.gle/ih548axqQ9qLz3pk7
EACH STUDENT MUST REPLY INDIVIDUALLY TO THE GOOGLE FORM.
You may select any slot from now through May 31, weekdays or weekends. You will receive an email with the questions and a form for answers at the time you receive the competition. There will be a 15-minute grace period for entering answers after the competition.
Pace Academy, from Atlanta, Georgia, is thrilled to host our Inaugural Academy Math Tournament online through Saturday, May 31.
AOPS students are welcome to participate online, as teams or as individuals (results will be reported separately for AOPS and Georgia competitors). The difficulty of the competition ranges from early AMC to mid-late AIME, and is 2 hours long with multiple sections. The format is explained in more detail below. If you just want to sign up, here's the link:
https://forms.gle/ih548axqQ9qLz3pk7
If participating as a team, each competitor must sign up individually and coordinate team names!
Detailed information below:
Divisions & Teams
- Junior Varsity: Students in 10th grade or below who are enrolled in Algebra 2 or below.
- Varsity: All other students.
- Teams of up to four students compete together in the same division.
- (If you have two JV‑eligible and two Varsity‑eligible students, you may enter either two teams of two or one four‑student team in Varsity.)
- You may enter multiple teams from your school in either division.
- Teams need not compete at the same time. Each individual will complete the test alone, and team scores will be the sum of individual scores.
Both sections—Sprint and Challenge—will be administered consecutively in a single, individually completed 120-minute test. Students may allocate time between the sections however they wish to.
- Sprint Section
- 25 multiple‑choice questions (five choices each)
- recommended 2 minutes per question
- 6 points per correct answer; no penalty for guessing
- Challenge Section
- 18 open‑ended questions
- answers are integers between 1 and 10,000
- recommended 3 or 4 minutes per question
- 8 points each
Awards & Scoring
- There are no cash prizes.
- Team Awards: Based on the sum of individual scores (four‑student teams have the advantage). Top 8 teams in each division will be recognized.
- Individual Awards: Top 8 individuals in each division, determined by combined Sprint + Challenge scores, will receive recognition.
Please have EACH STUDENT INDIVIDUALLY reserve a 120-minute window for your team's online test in THIS GOOGLE FORM:
https://forms.gle/ih548axqQ9qLz3pk7
EACH STUDENT MUST REPLY INDIVIDUALLY TO THE GOOGLE FORM.
You may select any slot from now through May 31, weekdays or weekends. You will receive an email with the questions and a form for answers at the time you receive the competition. There will be a 15-minute grace period for entering answers after the competition.
be a parallelogram with 
.
,
,
intersects diagonal 
at points 
and 
with 
,
,
,
.
,
and 
are positive integers, and 
.![[asy]
defaultpen(linewidth(0.6)+fontsize(11));
size(8cm);
pair A,B,C,D,P,Q;
A=(0,0);
label("$A$", A, SW);
B=(6,15);
label("$B$", B, NW);
C=(30,15);
label("$C$", C, NE);
D=(24,0);
label("$D$", D, SE);
P=(5.2,2.6);
label("$P$", (5.8,2.6), N);
Q=(18.3,9.1);
label("$Q$", (18.1,9.7), W);
draw(A--B--C--D--cycle);
draw(C--A);
draw(Circle((10.95,7.45), 7.45));
dot(A^^B^^C^^D^^P^^Q);
[/asy]](http://latex.artofproblemsolving.com/9/4/7/9471215d85465568eba3e615c0538a62e755bcf8.png)
and 
are written on a blackboard, where 
and 
are relatively prime positive integers. At any point, Evan may pick two of the numbers 
and 
written on the board and write either their arithmetic mean 
or their harmonic mean 
on the board as well. Find all pairs 
such that Evan can write 1 on the board in finitely many steps.
and 
intersect at 
,
and 
intersect at 
,
and 
intersect at 
.
intersects line 
at 
and 
intersects line 
and 
and 
are collinear in that order. Prove that if lines 
and 
intersect at 
,
.
and 
intersect at points 
and 
.
is tangent to 
and 
passes through 
and intersecting 
.
,
,
,
.
.![\[ \dots, \; P_{-3}, \; P_{-2},\; P_{-1},\; P_0,\; P_1,\; P_2,\; P_3,\; \dots \]](http://latex.artofproblemsolving.com/a/c/c/acc097f524dc9fe9bd85e8eb659ea4a75aec475d.png)
in the plane with the following property: For any three distinct integers 
and 
,
,
,
are collinear if and only if 
.
be an acute, scalene triangle, and let 
,
,
be the midpoints of 
,
,
,
intersect ray 
in points 
and 
respectively, and let lines 
and 
intersect in point 
,
,
March 2009
February 2009