2022 OMC 10/12

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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies

jlacosta

Apr 2, 2025

0 replies

the circle problem

miquelmas 0

an hour ago

Source: Olympiad of the city of Almaty (selection for the IZhO) 2010

The circle ω is centered at the quadrilateral ABCD. The lines AB and CD meet at point K, and the lines AD and BC meet at point L. A line passing through the center of ω and perpendicular to KL intersects KL, CD and AD at P, Q and R respectively. We need to prove that the lines QL, BP and KR all intersect at the same point.

0 replies

miquelmas

an hour ago

0 replies

How many approaches you got? (A lot)

IAmTheHazard 84

N 2 hours ago by User141208

Source: USAMO 2023/2

Let $\mathbb{R}^+$ be the set of positive real numbers. Find all functions $f \colon \mathbb{R}^+ \to \mathbb{R}^+$ such that, for all $x,y \in \mathbb{R}^+$ ,
$f(xy+f(x))=xf(y)+2.$

84 replies

IAmTheHazard

Mar 23, 2023

User141208

2 hours ago

MasterScholar North Carolina Math Camp

Ruegerbyrd 5

N 3 hours ago by ohiorizzler1434

Is this legit? Worth the cost ($6500)? Program Fees Cover: Tuition, course materials, field trip costs, and housing and meals at Saint Mary's School.

"Themes:

1. From Number Theory and Special Relativity to Game Theory
2. Applications to Economics

Subjects Covered:

Number Theory - Group Theory - RSA Encryption - Game Theory - Estimating Pi - Complex Numbers - Quaternions - Topology of Surfaces - Introduction to Differential Geometry - Collective Decision Making - Survey of Calculus - Applications to Economics - Statistics and the Central Limit Theorem - Special Relativity"

website(?): https://www.teenlife.com/l/summer/masterscholar-north-carolina-math-camp/

5 replies

Ruegerbyrd

Today at 3:15 AM

ohiorizzler1434

3 hours ago

9 Physical or online

wimpykid 0

4 hours ago

Do you think the AoPS print books or the online books are better?

0 replies

wimpykid

4 hours ago

0 replies

Three variables inequality

Headhunter 6

N 4 hours ago by lbh_qys

$\forall a\in R$ , $~\forall b\in R$ , $~\forall c \in R$
Prove that at least one of $(a-b)^{2}$ , $(b-c)^{2}$ , $(c-a)^{2}$ is not greater than $\frac{a^{2}+b^{2}+c^{2}}{2}$ .

I assume that all are greater than it, but can't go more.

6 replies

Headhunter

Apr 20, 2025

lbh_qys

4 hours ago

Sequence

lgx57 8

N 5 hours ago by Vivaandax

$a_1=1,a_{n+1}=a_n+\frac{1}{a_n}$ . Find the general term of $\{a_n\}$ .

8 replies

lgx57

Apr 27, 2025

Vivaandax

5 hours ago

AMSP Combo 2 and Alg 2.5

idk12345678 4

N 6 hours ago by Bread10

Im gonna be taking Geo 2 and i was deciding if to take combo 2, alg2.5, both, or neither.

My main goal is to qualify for JMO in 10th grade(next yr). Ive done aops int c+p but i didnt fully understand everything.

Would combo 2 and/or alg 2 be good for jmo qual?

4 replies

idk12345678

Yesterday at 2:12 PM

Bread10

6 hours ago

Geometric inequality

ReticulatedPython 3

N Today at 4:27 AM by ItalianZebra

Let $A$ and $B$ be points on a plane such that $AB=n$ , where $n$ is a positive integer. Let $S$ be the set of all points $P$ such that $\frac{AP^2+BP^2}{(AP)(BP)}=c$ , where $c$ is a real number. The path that $S$ traces is continuous, and the value of $c$ is minimized. Prove that $c$ is rational for all positive integers $n.$

3 replies

ReticulatedPython

Apr 22, 2025

ItalianZebra

Today at 4:27 AM

Transformation of a cross product when multiplied by matrix A

Math-lover1 1

N Today at 1:02 AM by greenturtle3141

I was working through AoPS Volume 2 and this statement from Chapter 11: Cross Products/Determinants confused me.
[quote=AoPS Volume 2]A quick comparison of $|\underline{A}|$ to the cross product $(\underline{A}\vec{i}) \times (\underline{A}\vec{j})$ reveals that a negative determinant [of $\underline{A}$ ] corresponds to a matrix which reverses the direction of the cross product of two vectors.[/quote]
I understand that this is true for the unit vectors $\vec{i} = (1 \ 0)$ and $\vec{j} = (0 \ 1)$ , but am confused on how to prove this statement for general vectors $\vec{v}$ and $\vec{w}$ although its supposed to be a quick comparison.

How do I prove this statement easily with any two 2D vectors?

1 reply

Math-lover1

Yesterday at 10:29 PM

greenturtle3141

Today at 1:02 AM

Geometry books

T.Mousavidin 4

N Today at 12:10 AM by compoly2010

Hello, I wanted to ask if anybody knows some good books for geometry that has these topics in:
Desargues's Theorem, Projective geometry, 3D geometry,

4 replies

T.Mousavidin

Yesterday at 4:25 PM

compoly2010

Today at 12:10 AM

trigonometric functions

VivaanKam 3

N Yesterday at 10:08 PM by aok

Hi could someone explain the basic trigonometric functions to me like sin, cos, tan etc.
Thank you!

3 replies

VivaanKam

Yesterday at 8:29 PM

aok

Yesterday at 10:08 PM

Inequalities

sqing 16

N Yesterday at 5:25 PM by martianrunner

Let $a,b \in [0 ,1] .$ Prove that
$\frac{a}{ 1-ab+b }+\frac{b }{ 1-ab+a } \leq 2$ $\frac{a}{ 1+ab+b^2 }+\frac{b }{ 1+ab+a^2 }+\frac{ab }{2+ab } \leq 1$ $\frac{a}{ 1-ab+b^2 }+\frac{b }{ 1-ab+a^2 }+\frac{1 }{1+ab }\leq \frac{5}{2}$ $\frac{a}{ 1-ab+b^2 }+\frac{b }{ 1-ab+a^2 }+\frac{1 }{1+2ab }\leq \frac{7}{3}$ $\frac{a}{ 1+ab+b^2 }+\frac{b }{ 1+ab+a^2 } +\frac{ab }{1+ab }\leq \frac{7}{6 }$

16 replies

sqing

Apr 25, 2025

martianrunner

Yesterday at 5:25 PM

Geometry Angle Chasing

Sid-darth-vater 6

N Yesterday at 2:18 PM by sunken rock

Is there a way to do this without drawing obscure auxiliary lines? (the auxiliary lines might not be obscure I might just be calling them obscure)

For example I tried rotating triangle MBC 80 degrees around point C (so the BC line segment would now lie on segment AC) but I couldn't get any results. Any help would be appreciated!

6 replies

Sid-darth-vater

Apr 21, 2025

sunken rock

Yesterday at 2:18 PM

BABBAGE'S THEOREM EXTENSION

Mathgloggers 0

Yesterday at 12:18 PM

A few days ago I came across. this interesting result is someone interested in proving this.

$\boxed{\sum_{k=1}^{p-1} \frac{1}{k} \equiv \sum_{k=p+1}^{2p-1} \frac{1}{k} \equiv \sum_{k=2p+1}^{3p-1}\frac{1}{k} \equiv.....\sum_{k=p(p-1)+1}^{p^2-1}\frac{1}{k} \equiv 0(mod p^2)}$

0 replies

Mathgloggers

Yesterday at 12:18 PM

0 replies

2022 OMC 10/12

ilovemath04 99

N Nov 11, 2024 by jb2015007

Hello AoPS!

STEAM For All is excited to announce its third virtual

[center]OMC 10/12![/center]

The virtual Orange Mathematics Competition (OMC) 10/12, is a Mock American Mathematics Competition (AMC) 10/12 with questions written by volunteers with years of experience in competition math! Students may only take one exam. Like the AMCs, students in and below 10th grade can take the OMC 10 while students in and below 12th grade can take the OMC 12. Students will have a great opportunity to challenge themselves with new questions, listen to experienced competitors explain the most difficult questions, and be recognized in an awards ceremony!

Register here! Registration ends Friday, October 28, 2022, at 11:59 PDT.

The competition will be held on Saturday, October 29th, 2022 from 2 - 4:30 PM PDT. The general schedule will be as follows (note that all times below are in PDT):
2:00 - 2:15 PM: Introduction
2:15 - 3:30 PM: OMC 10/12 Competition
3:30 - 4:15 PM: Problem Review Session
4:15 - 4:30 PM: Awards and Closing Ceremony

FAQ

Who: All students in grades 12 or below!
When: Saturday, October 29th, 2022 from 2-4:30PM PST
Where: Via Zoom, link to be sent closer to the date!
Questions: ocmathcircle@gmail.com or in this thread

For sample problems, here are a few problems from past OMC 10/12s! Feel free to discuss any of them in this thread (or any other problems from past SFA contests).
Sample Problems

[quote=2021 Fall OMC 10] For how many even positive integers $n$ does the number $18^{18}$ leave a remainder of $\tfrac{n}{2}$ when divided by $n$ ?

$\textbf{(A)} 0\hspace{0.5cm}\textbf{(B)} 19 \hspace{0.5cm}\textbf{(C)} 37 \hspace{0.5cm}\textbf{(D)} 361 \hspace{0.5cm}\textbf{(E)} 703$ [/quote]

[quote=2021 Fall OMC 12] Let $ABCD$ be a trapezoid with $AB$ parallel to $CD$ , $BC=6$ , $DA=5$ , and $AB<CD$ . The angle bisectors of angles $DAB$ and $ABC$ intersect at a point $X$ on segment $CD$ . If $AB=AX$ , what is the perimeter of trapezoid $ABCD$ ?

$\textbf{(A)} 25 \hspace{0.5cm}\textbf{(B)} 26 \hspace{0.5cm}\textbf{(C)} 27 \hspace{0.5cm}\textbf{(D)} 28 \hspace{0.5cm}\textbf{(E)} 29$ [/quote]

[quote = 2021 Spring OMC 10] How many ordered triples $(a,b,c)$ of positive integers less than or equal to 20 are there such that $\displaystyle\frac{a-b}{c}$ , $\displaystyle\frac{b-c}{a}$ , and $\displaystyle\frac{c-a}{b}$ are all integers?

$\textbf{(A)} 20 \hspace{0.5cm}\textbf{(B)} 66 \hspace{0.5cm}\textbf{(C)} 101 \hspace{0.5cm}\textbf{(D)} 158 \hspace{0.5cm}\textbf{(E)} 198$

More Questions? Contact us at ocmathcircle@gmail.com or ask in this thread!

We hope that you join us for the upcoming competition!

99 replies

ilovemath04

Oct 16, 2022

jb2015007

Nov 11, 2024

2022 OMC 10/12

G H J

Reveal topic tags

G H BBookmark kLocked kLocked NReply

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ilovemath04

70 posts

ilovemath04

#1 h Oct 16, 2022, 10:37 PM • 15 Y

Y by Pleaseletmewin, CELLSecret, channing421, jacoporizzo, Random_Person_921, AT2005, SuperJJ, samrocksnature, ab456, megarnie, mathtiger6, M_2007, Lamboreghini, jmiao, aidan0626

Hello AoPS!

STEAM For All is excited to announce its third virtual

OMC 10/12!

The virtual Orange Mathematics Competition (OMC) 10/12, is a Mock American Mathematics Competition (AMC) 10/12 with questions written by volunteers with years of experience in competition math! Students may only take one exam. Like the AMCs, students in and below 10th grade can take the OMC 10 while students in and below 12th grade can take the OMC 12. Students will have a great opportunity to challenge themselves with new questions, listen to experienced competitors explain the most difficult questions, and be recognized in an awards ceremony!

Register here! Registration ends Friday, October 28, 2022, at 11:59 PDT.

The competition will be held on Saturday, October 29th, 2022 from 2 - 4:30 PM PDT. The general schedule will be as follows (note that all times below are in PDT):
2:00 - 2:15 PM: Introduction
2:15 - 3:30 PM: OMC 10/12 Competition
3:30 - 4:15 PM: Problem Review Session
4:15 - 4:30 PM: Awards and Closing Ceremony

FAQ

Who: All students in grades 12 or below!
When: Saturday, October 29th, 2022 from 2-4:30PM PST
Where: Via Zoom, link to be sent closer to the date!
Questions: ocmathcircle@gmail.com or in this thread

For sample problems, here are a few problems from past OMC 10/12s! Feel free to discuss any of them in this thread (or any other problems from past SFA contests).
Sample Problems

2021 Fall OMC 10 wrote:

For how many even positive integers $n$ does the number $18^{18}$ leave a remainder of $\tfrac{n}{2}$ when divided by $n$ ?

$\textbf{(A)} 0\hspace{0.5cm}\textbf{(B)} 19 \hspace{0.5cm}\textbf{(C)} 37 \hspace{0.5cm}\textbf{(D)} 361 \hspace{0.5cm}\textbf{(E)} 703$

2021 Fall OMC 12 wrote:

Let $ABCD$ be a trapezoid with $AB$ parallel to $CD$ , $BC=6$ , $DA=5$ , and $AB<CD$ . The angle bisectors of angles $DAB$ and $ABC$ intersect at a point $X$ on segment $CD$ . If $AB=AX$ , what is the perimeter of trapezoid $ABCD$ ?

$\textbf{(A)} 25 \hspace{0.5cm}\textbf{(B)} 26 \hspace{0.5cm}\textbf{(C)} 27 \hspace{0.5cm}\textbf{(D)} 28 \hspace{0.5cm}\textbf{(E)} 29$

2021 Spring OMC 10 wrote:

How many ordered triples $(a,b,c)$ of positive integers less than or equal to 20 are there such that $\displaystyle\frac{a-b}{c}$ , $\displaystyle\frac{b-c}{a}$ , and $\displaystyle\frac{c-a}{b}$ are all integers?

$\textbf{(A)} 20 \hspace{0.5cm}\textbf{(B)} 66 \hspace{0.5cm}\textbf{(C)} 101 \hspace{0.5cm}\textbf{(D)} 158 \hspace{0.5cm}\textbf{(E)} 198$

More Questions? Contact us at ocmathcircle@gmail.com or ask in this thread!

We hope that you join us for the upcoming competition!

Z K Y

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Pleaseletmewin

1574 posts

Pleaseletmewin

#2 h Oct 16, 2022, 10:41 PM • 4 Y

Y by ilovemath04, jacoporizzo, Mango247, Mango247

I took this test last year and it helped me get a 21 on USAJMO. Please sign up!

Z K Y

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pi271828

3368 posts

pi271828

#3 h Oct 16, 2022, 10:53 PM

Y by

this is quite the suspicious mongoose

Z K Y

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suvamkonar

3092 posts

suvamkonar

#4 h Oct 17, 2022, 12:06 AM

Y by

could you post one of the past tests?

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Math4Life7

1703 posts

Math4Life7

#5 h Oct 17, 2022, 1:26 AM • 1 Y

Y by NegativeZeroPlusOne

Pleaseletmewin wrote:

I took this test last year and it helped me get a 21 on USAJMO. Please sign up!

bruhbruh noway omc 10 helped u on usajmo STOP THE CAP

Z K Y

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samrocksnature

8791 posts

samrocksnature

#6 h Oct 17, 2022, 1:34 AM • 2 Y

Y by Mango247, Mango247

Math4Life7 wrote:

Pleaseletmewin wrote:

I took this test last year and it helped me get a 21 on USAJMO. Please sign up!

bruhbruh noway omc 10 helped u on usajmo STOP THE CAP

Don't accuse my favorite 8th grade girl like that.

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Pleaseletmewin

1574 posts

Pleaseletmewin

#7 h Oct 17, 2022, 1:37 AM

Y by

Math4Life7 wrote:

Pleaseletmewin wrote:

I took this test last year and it helped me get a 21 on USAJMO. Please sign up!

bruhbruh noway omc 10 helped u on usajmo STOP THE CAP

If I hadn't taken the OMC 10, I doubt I would gotten over 100 on the real AMC 10. Because I took the high-quality OMC 10, I proceeded to get a 130.5 on the AMC 10B. If I had gotten less than 100 on the AMC 10, my JMO index would have been too low to even qualify for JMO. Thanks to the OMC 10/12, I was able to qualify for JMO.

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ilovepizza2020

12156 posts

ilovepizza2020

#8 h Oct 17, 2022, 1:41 AM

Y by

Pleaseletmewin wrote:

Math4Life7 wrote:

Pleaseletmewin wrote:

I took this test last year and it helped me get a 21 on USAJMO. Please sign up!

bruhbruh noway omc 10 helped u on usajmo STOP THE CAP

I expect a full research essay on the effects of taking the OMC ma'am

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ilovemath04

70 posts

ilovemath04

#9 h Oct 17, 2022, 1:42 AM

Y by

suvamkonar wrote:

could you post one of the past tests?

Yes, past threads can be found here (2021 Spring) and here (2021 Fall).

This post has been edited 1 time. Last edited by ilovemath04, Oct 17, 2022, 1:42 AM

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Math4Life7

1703 posts

Math4Life7

#10 h Oct 17, 2022, 1:48 AM • 2 Y

Y by NegativeZeroPlusOne, Mango247

Pleaseletmewin wrote:

ur still capping mebbe omc is v gud but it can't raise scores by 30.5 points by itself (wait 30.5 raise isn't possible :thonk:). also qualing for jmo is different than 21 so specify. and you can take past tests which prolly have close to that same effect as omc.

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asdf334

7585 posts

asdf334

#11 h Oct 17, 2022, 1:51 AM • 3 Y

Y by Mango247, Mango247, Mango247

bro now this is two contests im skipping for an alabama local math comp
sad

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TheEvilMathematician

303 posts

TheEvilMathematician

#12 h Oct 17, 2022, 2:11 AM

Y by

\signup i want to do it

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Math4Life7

1703 posts

Math4Life7

#13 h Oct 17, 2022, 2:33 AM • 1 Y

Y by NegativeZeroPlusOne

asdf334 wrote:

bro now this is two contests im skipping for an alabama local math comp
sad

alabama local math comp orz tho

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ilovepizza2020

12156 posts

ilovepizza2020

#15 h Oct 18, 2022, 9:29 PM

Y by

asdf334 wrote:

bro now this is two contests im skipping for an alabama local math comp
sad

Umm *blushes*

will you see significant there? tell him I said hai :blush:

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TheEvilMathematician

303 posts

TheEvilMathematician

#16 h Oct 19, 2022, 11:11 AM

Y by

OMC or the OMMC or the MOMMY

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P_Groudon

871 posts

P_Groudon

#92 h Nov 24, 2022, 5:38 PM • 2 Y

Y by samrocksnature, Mango247

I forgot about this until now, but the last two problems I don't have solutions for, I know for sure are incorrect.

#22 on OMC 10 (which I discussed before)

I need to write up a casework solution but here's all the quadruples that work (along with the common value of all the stuff in the long equation)

Quadruples

[1, 0, 0, 0, 1]
[1, 0, 0, 1, 1]
[1, 0, 0, 2, 1]
[1, 0, 0, 3, 1]
[1, 0, 0, 4, 1]
[1, 0, 0, 5, 1]
[1, 0, 0, 6, 1]
[1, 0, 0, 7, 1]
[1, 0, 0, 8, 1]
[1, 0, 0, 9, 1]
[1, 0, 1, 0, 1]
[1, 0, 2, 0, 1]
[1, 0, 3, 0, 1]
[1, 0, 4, 0, 1]
[1, 0, 5, 0, 1]
[1, 0, 6, 0, 1]
[1, 0, 7, 0, 1]
[1, 0, 8, 0, 1]
[1, 0, 9, 0, 1]
[1, 1, 0, 0, 1]
[1, 2, 0, 0, 1]
[1, 3, 0, 0, 1]
[1, 4, 0, 0, 1]
[1, 5, 0, 0, 1]
[1, 6, 0, 0, 1]
[1, 7, 0, 0, 1]
[1, 8, 0, 0, 1]
[1, 9, 0, 0, 1]
[2, 0, 0, 0, 1]
[2, 0, 0, 1, 2]
[2, 0, 0, 2, 4]
[2, 0, 0, 3, 8]
[2, 0, 0, 4, 16]
[2, 0, 0, 5, 32]
[2, 0, 0, 6, 64]
[2, 0, 0, 7, 128]
[2, 0, 0, 8, 256]
[2, 0, 0, 9, 512]
[2, 0, 2, 2, 16]
[2, 0, 6, 3, 512]
[2, 2, 0, 2, 16]
[2, 6, 0, 3, 512]
[3, 0, 0, 0, 1]
[3, 0, 0, 1, 3]
[3, 0, 0, 2, 9]
[3, 0, 0, 3, 27]
[3, 0, 0, 4, 81]
[3, 0, 0, 5, 243]
[3, 0, 0, 6, 729]
[3, 0, 0, 7, 2187]
[3, 0, 0, 8, 6561]
[3, 0, 0, 9, 19683]
[3, 0, 6, 6, 531441]
[3, 6, 0, 6, 531441]
[4, 0, 0, 0, 1]
[4, 0, 0, 1, 4]
[4, 0, 0, 2, 16]
[4, 0, 0, 3, 64]
[4, 0, 0, 4, 256]
[4, 0, 0, 5, 1024]
[4, 0, 0, 6, 4096]
[4, 0, 0, 7, 16384]
[4, 0, 0, 8, 65536]
[4, 0, 0, 9, 262144]
[4, 0, 4, 8, 16777216]
[4, 4, 0, 8, 16777216]
[5, 0, 0, 0, 1]
[5, 0, 0, 1, 5]
[5, 0, 0, 2, 25]
[5, 0, 0, 3, 125]
[5, 0, 0, 4, 625]
[5, 0, 0, 5, 3125]
[5, 0, 0, 6, 15625]
[5, 0, 0, 7, 78125]
[5, 0, 0, 8, 390625]
[5, 0, 0, 9, 1953125]
[6, 0, 0, 0, 1]
[6, 0, 0, 1, 6]
[6, 0, 0, 2, 36]
[6, 0, 0, 3, 216]
[6, 0, 0, 4, 1296]
[6, 0, 0, 5, 7776]
[6, 0, 0, 6, 46656]
[6, 0, 0, 7, 279936]
[6, 0, 0, 8, 1679616]
[6, 0, 0, 9, 10077696]
[7, 0, 0, 0, 1]
[7, 0, 0, 1, 7]
[7, 0, 0, 2, 49]
[7, 0, 0, 3, 343]
[7, 0, 0, 4, 2401]
[7, 0, 0, 5, 16807]
[7, 0, 0, 6, 117649]
[7, 0, 0, 7, 823543]
[7, 0, 0, 8, 5764801]
[7, 0, 0, 9, 40353607]
[8, 0, 0, 0, 1]
[8, 0, 0, 1, 8]
[8, 0, 0, 2, 64]
[8, 0, 0, 3, 512]
[8, 0, 0, 4, 4096]
[8, 0, 0, 5, 32768]
[8, 0, 0, 6, 262144]
[8, 0, 0, 7, 2097152]
[8, 0, 0, 8, 16777216]
[8, 0, 0, 9, 134217728]
[9, 0, 0, 0, 1]
[9, 0, 0, 1, 9]
[9, 0, 0, 2, 81]
[9, 0, 0, 3, 729]
[9, 0, 0, 4, 6561]
[9, 0, 0, 5, 59049]
[9, 0, 0, 6, 531441]
[9, 0, 0, 7, 4782969]
[9, 0, 0, 8, 43046721]
[9, 0, 0, 9, 387420489]

So it is clear anything in the form $(a, 0, 0, d)$ works. But then you have the sneakier cases to work with, and that's where I think things can get messy and easy to miss stuff (reminiscent of 2018 AIME I #9).

Quadruples where (b, c) isn't (0, 0)

[1, 0, 1, 0]
[1, 0, 2, 0]
[1, 0, 3, 0]
[1, 0, 4, 0]
[1, 0, 5, 0]
[1, 0, 6, 0]
[1, 0, 7, 0]
[1, 0, 8, 0]
[1, 0, 9, 0]
[1, 1, 0, 0]
[1, 2, 0, 0]
[1, 3, 0, 0]
[1, 4, 0, 0]
[1, 5, 0, 0]
[1, 6, 0, 0]
[1, 7, 0, 0]
[1, 8, 0, 0]
[1, 9, 0, 0]
[2, 0, 2, 2]
[2, 0, 6, 3]
[2, 2, 0, 2]
[2, 6, 0, 3]
[3, 0, 6, 6]
[3, 6, 0, 6]
[4, 0, 4, 8]
[4, 4, 0, 8]

#16 on OMC 12

It seems that what was intended was to use greedy algorithm (place the largest squares possible as you go) so that it comes out to $D(1) = 1$ , $D(2) = D(3) = 3$ , $D(4) = D(5) = ... = D (7) = 7$ , $D(8) = ... = D(15) = 15$ , that's incorrect. The thing with greedy algorithm is while bigger tiles may cover more area, they can sometimes restrict covering the rest of the area with smaller but fewer tiles. I have a counterexample for $n = 5$ , where it uses 6 tiles instead of 7.

Put a 2x2 in the bottom left corner. Then put a 2x2 directly above and a 2x2 directly right. Then, fill in the rest with 1x1s.

So thus, the logic with greedy algorithm is not rigorous for justifying the minimum, unfortunately.

This post has been edited 2 times. Last edited by P_Groudon, Nov 24, 2022, 6:56 PM

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ab456

223 posts

ab456

#94 h May 18, 2023, 9:50 PM

Y by

Lol I did this mock and got score and dist

94.5, with dist 11111/110B1/11111/BBBBB/BBBBB

I'd say this is a decent test, I'd probably barely miss/make AIME given todays standards for MAAs tests.

I think that 14-16 are fairly decent problems that require minimum computation and have a clever idea within them, while not being too hard. However, a lot of earlier problems require quite a bit of computation. Pretty on par with MAA's tests.

This post has been edited 1 time. Last edited by ab456, May 18, 2023, 9:55 PM

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smartatmath

768 posts

smartatmath

#95 h Jul 5, 2023, 8:49 PM

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OMC 12 #5 is like 2021 AMC 12B #21

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P_Groudon

871 posts

P_Groudon

#96 h Jul 13, 2023, 12:57 PM • 2 Y

Y by peace09, Sedro

Many months late on this, but fixed the 2 other problems so that their intended solutions work

Attachments:

2022_Fall_OMC_12_Solutions.pdf (267kb)

2022_Fall_OMC_10_Solutions.pdf (243kb)

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heheman

999 posts

heheman

#97 h Jul 13, 2023, 3:36 PM

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Pleaseletmewin wrote:

Math4Life7 wrote:

Pleaseletmewin wrote:

I took this test last year and it helped me get a 21 on USAJMO. Please sign up!

bruhbruh noway omc 10 helped u on usajmo STOP THE CAP

Poll: Moronic ravings or factual statement????

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Opsecretpro

145 posts

Opsecretpro

#98 h Jul 13, 2023, 3:39 PM

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nuni how are there 89 posts when its like posted 10 mins ago

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joshualiu315

2533 posts

joshualiu315

#99 h Jul 13, 2023, 6:33 PM

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@2above i think they were being factual bc i didn't do the high quality OMC10 and bombed it with 2 sillies each

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r00tsOfUnity

695 posts

r00tsOfUnity

#100 h Oct 16, 2023, 4:59 PM

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Is the competition still happening this year? I would love to participate in it as preparation for the official AMC 12s!

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pi271828

3368 posts

pi271828

#101 h Oct 16, 2023, 10:57 PM

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pi271828 wrote:

this is quite the suspicious mongoose

hi me from 1 yr ago

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Grolarbear

348 posts

Grolarbear

#102 h Oct 17, 2023, 1:11 AM

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Last year I did not get my results back and I got ghosted after 2 messages to steam for all :mad:

Did anyone actually get their scores back?

Edit: Don't get me wrong, I think these tests are well made

This post has been edited 1 time. Last edited by Grolarbear, Oct 17, 2023, 1:12 AM

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Lunarmoon123

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Lunarmoon123

#103 h Jul 1, 2024, 8:53 PM

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ilovemath04 wrote:

For sample problems, here are a few problems from past OMC 10/12s! Feel free to discuss any of them in this thread (or any other problems from past SFA contests).
Sample Problems

2021 Fall OMC 10 wrote:

2021 Fall OMC 12 wrote:

2021 Spring OMC 10 wrote:

Could someone please give answers to these sample problems? What level would these be on the actual AMC 10?

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Lunarmoon123

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Lunarmoon123

#104 h Jul 1, 2024, 8:54 PM

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Could someone please give answers to these sample problems? What level would these be on the actual AMC 10? (Accidentally edited the question into the quote)

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PaperMath

957 posts

PaperMath

#105 h Nov 11, 2024, 1:24 AM

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predict aime cutoff on this please

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PaperMath

957 posts

PaperMath

#106 h Nov 11, 2024, 2:28 AM

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P_Groudon wrote:

Many months late on this, but fixed the 2 other problems so that their intended solutions work

in the solutions why is question 22
How many positive three-digit integers abc with a ̸= 0 satisfy ab+c =(a+b)c?
(For example, 222 works as 22+2 = (2 + 2)2.)
(A) 92 (B) 96 (C) 99 (D) 103 (E) 106

and not

How many positive four-digit integers abcd with a ̸= 0 satisfy
ab+c+d = (a+b)c+d = (a+b+c)d?
(For example, 2022 works as 20+2+2 = (2 + 0)2+2 = (2 + 0 + 2)2.)
(A) 92 (B) 94 (C) 96 (D) 98 (E) 100

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jb2015007

1926 posts

jb2015007

#107 h Nov 11, 2024, 2:35 AM

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Pleaseletmewin wrote:

I took this test last year and it helped me get a 21 on USAJMO. Please sign up!

Hes glazing trust

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