2025 ELMOCOUNTS - Mock MATHCOUNTS Nationals
by vincentwant, Apr 20, 2025, 6:29 PM
text totally not copied over from wmc (thanks jason <3)
Quick Links:
Eddison Chen (KS '22 '24), Aarush Goradia (CO '24), Ethan Imanuel (NJ '24), Benjamin Jiang (FL '23 '24), Rayoon Kim (PA '23 '24), Jason Lee (NC '23 '24), Puranjay Madupu (AZ '23 '24), Andy Mo (OH '23 '24), George Paret (FL '24), Arjun Raman (IN '24), Vincent Wang (TX '24), Channing Yang (TX '23 '24), and Jefferson Zhou (MN '23 '24) present:
2024 MATHCOUNTS Nationals alumni from all across the nation have come together to administer the first-ever ELMOCOUNTS Competition, a mock written by the 2024 Nationals alumni given to the 2025 Nationals participants. By providing the next generation of mathletes with free, high quality practice, we're here to boast how strong of an alumni community MATHCOUNTS has, as well as foster interest in the beautiful art that is problem writing!
The tests and their corresponding submissions forms will be released here, on this thread, on Monday, April 21, 2025. The deadline is May 10, 2025. Tests can be administered asynchronously at your home or school, and your answers should be submitted to the corresponding submission form. If you include your AoPS username in your submission, you will be granted access to the private discussion forum on AoPS, where you can discuss the tests even before the deadline.
Quick Links:
- National: (Sprint) (Target) (Team) (Sprint + Target Submission) (Team Submission)
- Miscellaneous: (Leaderboard) (Sprint + Target Private Discussion Forum) (Team Discussion Forum)
Eddison Chen (KS '22 '24), Aarush Goradia (CO '24), Ethan Imanuel (NJ '24), Benjamin Jiang (FL '23 '24), Rayoon Kim (PA '23 '24), Jason Lee (NC '23 '24), Puranjay Madupu (AZ '23 '24), Andy Mo (OH '23 '24), George Paret (FL '24), Arjun Raman (IN '24), Vincent Wang (TX '24), Channing Yang (TX '23 '24), and Jefferson Zhou (MN '23 '24) present:
Image credits to Simon Joeng.
2024 MATHCOUNTS Nationals alumni from all across the nation have come together to administer the first-ever ELMOCOUNTS Competition, a mock written by the 2024 Nationals alumni given to the 2025 Nationals participants. By providing the next generation of mathletes with free, high quality practice, we're here to boast how strong of an alumni community MATHCOUNTS has, as well as foster interest in the beautiful art that is problem writing!
The tests and their corresponding submissions forms will be released here, on this thread, on Monday, April 21, 2025. The deadline is May 10, 2025. Tests can be administered asynchronously at your home or school, and your answers should be submitted to the corresponding submission form. If you include your AoPS username in your submission, you will be granted access to the private discussion forum on AoPS, where you can discuss the tests even before the deadline.
This post has been edited 6 times. Last edited by vincentwant, Apr 22, 2025, 3:25 PM
Mathpath vs. AMSP
by FuturePanda, Jan 30, 2025, 3:15 AM
Loading poll details...
This post has been edited 1 time. Last edited by FuturePanda, Jan 30, 2025, 3:16 AM
sussy baka stop intersecting in my lattice points
by Spectator, Nov 11, 2022, 5:00 PM
Let 
, 
, and 
be squares that have vertices at lattice points (i.e., points whose coordinates are both integers) in the coordinate plane, together with their interiors. The bottom edge of each square is on the x-axis. The left edge of and the right edge of are on the 
-axis, and contains 
as many lattice points as does . The top two vertices of are in 
, and contains 
of the lattice points contained in . See the figure (not drawn to scale).
![[asy]
//kaaaaaaaaaante314
size(8cm);
import olympiad;
label(scale(.8)*"$y$", (0,60), N);
label(scale(.8)*"$x$", (60,0), E);
filldraw((0,0)--(55,0)--(55,55)--(0,55)--cycle, yellow+orange+white+white);
label(scale(1.3)*"$R$", (55/2,55/2));
filldraw((0,0)--(0,28)--(-28,28)--(-28,0)--cycle, green+white+white);
label(scale(1.3)*"$S$",(-14,14));
filldraw((-10,0)--(15,0)--(15,25)--(-10,25)--cycle, red+white+white);
label(scale(1.3)*"$T$",(3.5,25/2));
draw((0,-10)--(0,60),EndArrow(TeXHead));
draw((-34,0)--(60,0),EndArrow(TeXHead));[/asy]](//latex.artofproblemsolving.com/2/4/d/24da341df5acb1143cb0f2ca5f29ded541559594.png)
The fraction of lattice points in that are in 
is 27 times the fraction of lattice points in that are in 
. What is the minimum possible value of the edge length of plus the edge length of plus the edge length of ?
, 
, and 
be squares that have vertices at lattice points (i.e., points whose coordinates are both integers) in the coordinate plane, together with their interiors. The bottom edge of each square is on the x-axis. The left edge of and the right edge of are on the 
-axis, and contains 
as many lattice points as does . The top two vertices of are in 
, and contains 
of the lattice points contained in . See the figure (not drawn to scale).![[asy]
//kaaaaaaaaaante314
size(8cm);
import olympiad;
label(scale(.8)*"$y$", (0,60), N);
label(scale(.8)*"$x$", (60,0), E);
filldraw((0,0)--(55,0)--(55,55)--(0,55)--cycle, yellow+orange+white+white);
label(scale(1.3)*"$R$", (55/2,55/2));
filldraw((0,0)--(0,28)--(-28,28)--(-28,0)--cycle, green+white+white);
label(scale(1.3)*"$S$",(-14,14));
filldraw((-10,0)--(15,0)--(15,25)--(-10,25)--cycle, red+white+white);
label(scale(1.3)*"$T$",(3.5,25/2));
draw((0,-10)--(0,60),EndArrow(TeXHead));
draw((-34,0)--(60,0),EndArrow(TeXHead));[/asy]](http://latex.artofproblemsolving.com/2/4/d/24da341df5acb1143cb0f2ca5f29ded541559594.png)
The fraction of lattice points in
that are in 
is 27 times the fraction of lattice points in that are in 
. What is the minimum possible value of the edge length of plus the edge length of plus the edge length of ?
This post has been edited 3 times. Last edited by Spectator, Dec 24, 2022, 12:54 PM
Reason: Asy
Reason: Asy
Jumping on Lily Pads to Avoid a Snake
by brandbest1, Feb 20, 2014, 3:52 PM
In a small pond there are eleven lily pads in a row labeled 
through 
. A frog is sitting on pad 
. When the frog is on pad 
, 
, it will jump to pad 
with probability 
and to pad 
with probability 
. Each jump is independent of the previous jumps. If the frog reaches pad it will be eaten by a patiently waiting snake. If the frog reaches pad it will exit the pond, never to return. What is the probability that the frog will escape being eaten by the snake?
through 
. A frog is sitting on pad 
. When the frog is on pad 
, 
, it will jump to pad 
with probability 
and to pad 
with probability 
. Each jump is independent of the previous jumps. If the frog reaches pad it will be eaten by a patiently waiting snake. If the frog reaches pad it will exit the pond, never to return. What is the probability that the frog will escape being eaten by the snake?
This post has been edited 2 times. Last edited by LauraZed, Dec 27, 2018, 4:41 PM
Reason: fixed capitalization
Reason: fixed capitalization
System
by worthawholebean, Apr 3, 2008, 3:55 PM
Let 
and 
be positive real numbers with 
. Let 
be the maximum possible value of 
for which the system of equations
![\[ a^2+y^2=b^2+x^2=(a-x)^2+(b-y)^2\]](//latex.artofproblemsolving.com/1/3/4/13432a7d6e5c1465e08d513ee64c6c7ce99d75f9.png)
has a solution in 
satisfying 
and 
. Then 
can be expressed as a fraction 
, where 
and 
are relatively prime positive integers. Find 
.
and 
be positive real numbers with 
. Let 
be the maximum possible value of 
for which the system of equations![\[ a^2+y^2=b^2+x^2=(a-x)^2+(b-y)^2\]](http://latex.artofproblemsolving.com/1/3/4/13432a7d6e5c1465e08d513ee64c6c7ce99d75f9.png)
has a solution in 
satisfying 
and 
. Then 
can be expressed as a fraction 
, where 
and 
are relatively prime positive integers. Find 
.
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