Aime ll 2022 problem 5

by Rook567, May 8, 2025, 9:08 PM

I don’t understand the solution. I got 220 as answer. Why does it insist, for example two primes must add to the third, when you can take 2,19,19 or 2,7,11 which for drawing purposes is equivalent to 1,1,2 and 2,7,9?

AIME

AMC

Can I make the IMO team next year?

by aopslover08, May 8, 2025, 7:46 PM

Hi everyone,

I am a current 11th grader living in Orange, Texas. I recently started doing competition math and I think I am pretty good at it. Recently I did a mock AMC8 and achieved a score of 21/25, which falls in the top 1% DHR. I also talked to my math teacher and she says I am an above average student.

Given my natural talent and the fact that I am willing to work ~3.5 hours a week studying competition math, do you think I will be able to make IMO next year? I am aware of the difficulty of this task but my mom says that I can achieve whatever I put my mind to, as long as I work hard.

Here is my plan for the next few months:

month 1-2: finish studying pre-algebra and learn geometry
month 3-4: learn pre-calculus
month 5-6: start doing IMO shortlist problems
month 7+: keep doing ISL/IMO problems.

Is this a feasible task? I am a girl btw.

26 Comments

Past USAMO Medals

by sdpandit, May 8, 2025, 7:44 PM

Does anyone know where to find lists of USAMO medalists from past years? I can find the 2025 list on their website, but they don't seem to keep lists from previous years and I can't find it anywhere else. Thanks!

AMC

USA(J)MO

USAMO

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 $XY$ .

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, Mar 20, 2025, 12:09 PM

Will I make JMO?

by EaZ_Shadow, Feb 7, 2025, 4:20 PM

Loading poll details...

will I be able to make it... will the cutoffs will be pre-2024

poll

An FE. Who woulda thunk it?

by nikenissan, Apr 15, 2021, 5:13 PM

Let $\mathbb{N}$ denote the set of positive integers. Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for positive integers $a$ and $b,$ $f(a^2 + b^2) = f(a)f(b) \text{ and } f(a^2) = f(a)^2.$

function

USAJMO

functional equation

Isosceles everywhere

by reallyasian, Mar 12, 2020, 4:12 PM

In $\triangle ABC$ with $AB=AC$ , point $D$ lies strictly between $A$ and $C$ on side $\overline{AC}$ , and point $E$ lies strictly between $A$ and $B$ on side $\overline{AB}$ such that $AE=ED=DB=BC$ . The degree measure of $\angle ABC$ is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .

This post has been edited 4 times. Last edited by djmathman, Apr 10, 2020, 3:54 PM
Reason: \Delta -> \triangle

Geo #3 EQuals FReak out

by Th3Numb3rThr33, Apr 18, 2018, 11:00 PM

Let $ABCD$ be a quadrilateral inscribed in circle $\omega$ with $\overline{AC} \perp \overline{BD}$ . Let $E$ and $F$ be the reflections of $D$ over lines $BA$ and $BC$ , respectively, and let $P$ be the intersection of lines $BD$ and $EF$ . Suppose that the circumcircle of $\triangle EPD$ meets $\omega$ at $D$ and $Q$ , and the circumcircle of $\triangle FPD$ meets $\omega$ at $D$ and $R$ . Show that $EQ = FR$ .

This post has been edited 2 times. Last edited by Th3Numb3rThr33, Apr 18, 2018, 11:44 PM

USAJMO

2018 USAJMO Problem 3

USAJMO problem 2: Side lengths of an acute triangle

by BOGTRO, Apr 24, 2012, 9:57 PM

Find all integers $n \geq 3$ such that among any $n$ positive real numbers $a_1, a_2, \hdots, a_n$ with $\text{max}(a_1,a_2,\hdots,a_n) \leq n \cdot \text{min}(a_1,a_2,\hdots,a_n)$ , there exist three that are the side lengths of an acute triangle.