JMO winner cutoffs

by dolphinday, Mar 21, 2025, 5:46 PM

What are JMO winner cutoffs typically?
are they the same as top 24 on https://web.evanchen.cc/exams/posted-usamo-statistics.pdf
L

0 on jmo

by Rong0625, Mar 21, 2025, 12:14 PM

How many people actually get a flat 0/42 on jmo? I took it for the first time this year and I had never done oly math before so I really only had 2 weeks to figure it out since I didn’t think I would qual. I went in not expecting much but I didn’t think I wouldn’t be able to get ANYTHING. So I’m pretty sure I got 0/42 (unless i get pity points for writing incorrect solutions). Is that bad, am I sped, and should I be embarrassed? Or do other people actually also get 0?
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funny title placeholder

by pikapika007, Mar 21, 2025, 12:10 PM

Let $S$ be a set of integers with the following properties:
  • $\{ 1, 2, \dots, 2025 \} \subseteq S$.
  • If $a, b \in S$ and $\gcd(a, b) = 1$, then $ab \in S$.
  • If for some $s \in S$, $s + 1$ is composite, then all positive divisors of $s + 1$ are in $S$.
Prove that $S$ contains all positive integers.
This post has been edited 1 time. Last edited by pikapika007, Today at 12:12 PM
Reason: wrong year

usamOOK geometry

by KevinYang2.71, Mar 21, 2025, 12:00 PM

Let $H$ be the orthocenter of acute triangle $ABC$, let $F$ be the foot of the altitude from $C$ to $AB$, and let $P$ be the reflection of $H$ across $BC$. Suppose that the circumcircle of triangle $AFP$ intersects line $BC$ at two distinct points $X$ and $Y$. Prove that $C$ is the midpoint of $XY$.

Distributing cupcakes

by KevinYang2.71, Mar 21, 2025, 12:00 PM

Let $m$ and $n$ be positive integers with $m\geq n$. There are $m$ cupcakes of different flavors arranged around a circle and $n$ people who like cupcakes. Each person assigns a nonnegative real number score to each cupcake, depending on how much they like the cupcake. Suppose that for each person $P$, it is possible to partition the circle of $m$ cupcakes into $n$ groups of consecutive cupcakes so that the sum of $P$'s scores of the cupcakes in each group is at least $1$. Prove that it is possible to distribute the $m$ cupcakes to the $n$ people so that each person $P$ receives cupcakes of total score at least $1$ with respect to $P$.

Scary Binomial Coefficient Sum

by EpicBird08, Mar 21, 2025, 11:59 AM

Determine, with proof, all positive integers $k$ such that $$\frac{1}{n+1} \sum_{i=0}^n \binom{n}{i}^k$$is an integer for every positive integer $n.$
This post has been edited 2 times. Last edited by EpicBird08, Today at 12:06 PM

MOHS for Day 1

by MajesticCheese, Mar 20, 2025, 3:15 PM

What is your opinion for MOHS for day 1?

JMO 1:
JMO 2/AMO 1:
JMO 3:
AMO 2:
AMO 3:
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high tech FE as J1?!

by imagien_bad, Mar 20, 2025, 12:00 PM

Let $\mathbb Z$ be the set of integers, and let $f\colon \mathbb Z \to \mathbb Z$ be a function. Prove that there are infinitely many integers $c$ such that the function $g\colon \mathbb Z \to \mathbb Z$ defined by $g(x) = f(x) + cx$ is not bijective.
Note: A function $g\colon \mathbb Z \to \mathbb Z$ is bijective if for every integer $b$, there exists exactly one integer $a$ such that $g(a) = b$.
This post has been edited 1 time. Last edited by imagien_bad, Yesterday at 12:09 PM

AMC 10.........

by BAM10, Mar 2, 2025, 8:02 PM

I'm in 8th grade and have never taken the AMC 10. I am currently in alg2. I have scored 20 on AMC 8 this year and 34 on the chapter math counts last year. Can I qualify for AIME. Also what should I practice AMC 10 next year?

On a^4+b^4=c^4+d^4=e^5

by v_Enhance, Apr 29, 2015, 9:13 PM

Let $a$, $b$, $c$, $d$, $e$ be distinct positive integers such that $a^4+b^4=c^4+d^4=e^5$. Show that $ac+bd$ is a composite number.

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