mop tests recap (green)
by OronSH, Jul 17, 2024, 2:21 PM
heres the story of my second best performance lol
might contain isl spoilers, specifically, for problems A1, A2, A4, A5, C1, C4, G2, G3, G7, N2, N6, N7
i think about p1 for a bit before moving on and trying p3 since it looks like it would be fun to try. after about an hour i find some weird bounding argument that works and i write it up. then i work on p2 since its geo, it turns out to be a lot less scary then it looks and solve it in about 20 minutes. then i try p1, solve it and write it up. finally i work on p4, which ends up being really easy once u just test small cases, 10 minute solve. i then check my p3 sol since im pretty confident in my sols to the other problems. since i have nothing else to do i decide to leave the test 2 hours early. sweep????
no sweep
at dinner i realize i fakesolved p1 
anyway i failed isl guessing because p4 is n7 somehow.
at least now i can say i solved an n7 in 10 minutes 
final score: 2777
i start with p3 since its geo and solve it in 15 minutes. really cute problem, probably my favorite of all the mop test problems. then i try p2, its pretty routine weighted amgm, the writeup is slightly annoying though. then i solve p1 pretty quickly. i spend the rest of my time trying p4, construction is immediate but i dont make any other progress for the next 2 hours. decide to submit my construction and a bunch of messy fake progress.
after the test i couldnt help but feel like ive seen p2 before... my suspicions were confirmed when peace09 found the problem on alcumus
anyway isl guessing went pretty well. i got 2nd... or technically a mysterious person by the name of "skibidi kibidi ibidi bidi idi di i" got 2nd
final score: 7771
my first thought when seeing this test was something like "what are these problems bruh". i start by drawing the diagram for p4 cuz its geo. i notice the two triangles look similar but cant prove it. then i try p2 and p3 for a while and dont get anywhere, mainly because p2 i thought the inequality being strict meant equality holding
and because for p3 i forget to do anything nice and just try and list out all kawaii numbers hoping that it would work. finally i try p1 and i solve it using ham sandwich theorem lol 
finish the writeup 3 hours in. soon after i notice my mistake on p2 and solve it pretty quickly. now with 1 hour left i decide to work on p4 instead of p3 cuz its geo. i find a lot of cyclic quads and prove the claim, then i try some weird spiral similarity stuff to try and finish. with 10 minutes left i reduce it to a single claim. with 5 minutes left, i prove the claim!!!!
surprisingly this wasnt even my first time at mop clutching a geo in the last 5 minutes. maybe it'll happen again... maybe for a nongeo....
also isl guessing was pretty mid and the leaderboard did not contain skibidis rip
final score: 7717
start with p1 and solve it quickly. then i try p3, its really bashy and annoying but i solve it. then p2 is just a standard grid combo with a standard long writeup. i have about 2 hours left by now to try p4. i try to find the construction first since that feels easier, then fail to do so and make no progress for the next hour and a half. suddenly, with 30 minutes left, i find the main idea, and prove the bound in a very rushed writeup. once im done with the writeup, i have 10 minutes left, and still havent made progress on the second part of the problem ... and just then i suddenly find the construction!!! i put my pen down with 2 minutes left. sweep????
yes!!!!! somehow had no docks (rip jatloe)
also isl guessing was mid again
final score: 7777
Finally here's one last fun exercise:
might contain isl spoilers, specifically, for problems A1, A2, A4, A5, C1, C4, G2, G3, G7, N2, N6, N7
with the following property: for any positive real numbers 
with sum 
there exist nonnegative integers 
with sum 
such that ![\[|a_1-b_1|+\dots+|a_{100}-b_{100}|\le D.\]](http://latex.artofproblemsolving.com/3/a/9/3a99f29d5900baed357972c0e8c38af574483aa6.png)
be a triangle with 
let 
be the circumcircle of 
and let 
be its radius. Point 
is chosen on 
such that 
and point 
is the foot of the perpendicular from 
Ray 
meets 
Point 
is chosen on line 
such that 
and 
lie on a line in that order. Finally, let 
be a point satisfying 
and 
Prove that 
denote the set of positive real numbers. Determine all functions 
such that, for all positive real 
and 
we have ![\[x(f(x)+f(y))\ge(f(f(x))+y)f(y).\]](http://latex.artofproblemsolving.com/0/1/e/01e2d241b79df9603bf4642368b2a5455c087cb5.png)
are positive integers satisfying ![\[\frac{ab}{a+b}+\frac{cd}{c+d}=\frac{(a+b)(c+d)}{a+b+c+d}.\]](http://latex.artofproblemsolving.com/b/8/c/b8c3fe14640a2dcb30b755b2347721da5d080a78.png)
Determine all possible values of 
i think about p1 for a bit before moving on and trying p3 since it looks like it would be fun to try. after about an hour i find some weird bounding argument that works and i write it up. then i work on p2 since its geo, it turns out to be a lot less scary then it looks and solve it in about 20 minutes. then i try p1, solve it and write it up. finally i work on p4, which ends up being really easy once u just test small cases, 10 minute solve. i then check my p3 sol since im pretty confident in my sols to the other problems. since i have nothing else to do i decide to leave the test 2 hours early. sweep????
no sweep
at dinner i realize i fakesolved p1 anyway i failed isl guessing because p4 is n7 somehow.
at least now i can say i solved an n7 in 10 minutes final score: 2777
for which one can fill in the cells of an 
grid with the numbers 
such that, when calculating the sum of the numbers in each row and each column, the numbers 
are obtained in some order.![\[\sqrt{a+\sqrt{b+\sqrt c}}\ge(abc)^r\]](http://latex.artofproblemsolving.com/9/b/a/9ba699c43532ce6235fc18605a88516ea76aa1cb.png)
holds for all positive real numbers 
and 
be a quadrilateral, with 
inscribed in circle 
be the midpoint of arc 
on 
Suppose that point 
and 
Prove that lines 
and 
are concurrent.
let 
denote the unique integer 
satisfying 
for which 
is a multiple of 
strip of fabric, where the 
cell is labeled with 
for all 
They wish to cut the strip into several pieces (respecting gridlines) and translate (without rotating or flipping) them to obtain an 
column is labeled with 
for all 
and 
i start with p3 since its geo and solve it in 15 minutes. really cute problem, probably my favorite of all the mop test problems. then i try p2, its pretty routine weighted amgm, the writeup is slightly annoying though. then i solve p1 pretty quickly. i spend the rest of my time trying p4, construction is immediate but i dont make any other progress for the next 2 hours. decide to submit my construction and a bunch of messy fake progress.
after the test i couldnt help but feel like ive seen p2 before... my suspicions were confirmed when peace09 found the problem on alcumus
anyway isl guessing went pretty well. i got 2nd... or technically a mysterious person by the name of "skibidi kibidi ibidi bidi idi di i" got 2nd final score: 7771
contestants, who each got a score in 
The bronze cutoff was 
points, and every score at or below the bronze cutoff was achieved by at least one student. Moreover, the average score was 
students each, such that the average score of each group is also 
be a function such that ![\[f(x+y)f(x-y)\ge f(x)^2-f(y)^2\]](http://latex.artofproblemsolving.com/a/2/6/a267d78a6706ced5b738339410ea54a393b4dbce.png)
for all real 
Prove that either 
always or 
always.
is called kawaii if 
and ![\[(a_{n+2}-3a_{n+1}+2a_n)(a_{n+2}-4a_{n+1}+3a_n)=0\]](http://latex.artofproblemsolving.com/9/5/0/95042645bea21b8178fbd7c3784d49b793423bfb.png)
for all integers 
An integer is called kawaii if it belongs to some kawaii sequence.
and 
are both kawaii (not necessarily belonging to the same kawaii sequence). Prove that 
and that 
is also kawaii.
Let 
be the line through the reflection of 
in 
and the reflection of 
in 
Lines 
and 
are defined similarly.
bound a triangle 
Prove that the orthocenter of 
the circumcenter of 
are collinear.my first thought when seeing this test was something like "what are these problems bruh". i start by drawing the diagram for p4 cuz its geo. i notice the two triangles look similar but cant prove it. then i try p2 and p3 for a while and dont get anywhere, mainly because p2 i thought the inequality being strict meant equality holding
and because for p3 i forget to do anything nice and just try and list out all kawaii numbers hoping that it would work. finally i try p1 and i solve it using ham sandwich theorem lol finish the writeup 3 hours in. soon after i notice my mistake on p2 and solve it pretty quickly. now with 1 hour left i decide to work on p4 instead of p3 cuz its geo. i find a lot of cyclic quads and prove the claim, then i try some weird spiral similarity stuff to try and finish. with 10 minutes left i reduce it to a single claim. with 5 minutes left, i prove the claim!!!! surprisingly this wasnt even my first time at mop clutching a geo in the last 5 minutes. maybe it'll happen again... maybe for a nongeo....
also isl guessing was pretty mid and the leaderboard did not contain skibidis rip
final score: 7717
Choose point 
such that 
and 
Prove that either 
or 
(or both).
and 
square grid has a coin with tails up. A move consists of the following steps:
square in the grid.
for which the configuration of all coins heads up is reachable.
find the number of reachable configurations.
of positive integers, with 
prime, such that 
is a perfect square.
are positive integers such that
and
is a permutation of 
and give an example where equality occurs.start with p1 and solve it quickly. then i try p3, its really bashy and annoying but i solve it. then p2 is just a standard grid combo with a standard long writeup. i have about 2 hours left by now to try p4. i try to find the construction first since that feels easier, then fail to do so and make no progress for the next hour and a half. suddenly, with 30 minutes left, i find the main idea, and prove the bound in a very rushed writeup. once im done with the writeup, i have 10 minutes left, and still havent made progress on the second part of the problem ... and just then i suddenly find the construction!!! i put my pen down with 2 minutes left. sweep????
yes!!!!!
somehow had no docks (rip jatloe)also isl guessing was mid again
final score: 7777
Finally here's one last fun exercise:
This post has been edited 2 times. Last edited by OronSH, Jul 17, 2024, 10:49 PM
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