Product of all even divisors

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0 replies

jlacosta

May 1, 2025

0 replies

进制问题1234

yyhloveu1314 1

N a few seconds ago by Mingkai

I want to represent a decimal number using an unconventional base (-4) numeral system, and I wish to prove the uniqueness and existence of this representation.

1 reply

yyhloveu1314

Yesterday at 2:28 PM

Mingkai

a few seconds ago

Inequalities

sqing 0

10 minutes ago

Let $a,b>0$ . Prove that
$\frac{a}{a^2+a +2b+1}+ \frac{b}{b^2+2a +b+1} \leq \frac{2}{5}$ $\frac{a}{a^2+2a +b+1}+ \frac{b}{b^2+a +2b+1} \leq \frac{2}{5}$

0 replies

sqing

10 minutes ago

0 replies

book/resource recommendations

walterboro 2

N 25 minutes ago by walterboro

hi guys, does anyone have book recs (or other resources) for like aime+ level alg, nt, geo, comb? i want to learn a lot of theory in depth
also does anyone know how otis or woot is like from experience?

2 replies

walterboro

Sunday at 8:57 PM

walterboro

25 minutes ago

Angle Formed by Points on the Sides of a Triangle

xeroxia 7

N 43 minutes ago by xeroxia

In triangle $ABC$ , points $D$ , $E$ , and $F$ lie on sides $BC$ , $CA$ , and $AB$ , respectively, such that
$BD = 20$ , $DC = 15$ , $CE = 13$ , $EA = 8$ , $AF = 6$ , $FB = 22$ .

What is the measure of $\angle EDF$ ?

7 replies

xeroxia

May 10, 2025

xeroxia

43 minutes ago

Inequalities

sqing 1

N an hour ago by sqing

Let $a, b, c >0, a^2 + \frac{b}{a} = 8$ and $3a + b + c \geq 9\sqrt{3} .$ Prove that $ab + c^2\geq 18$

1 reply

sqing

Yesterday at 9:08 AM

sqing

an hour ago

non-homogeneous inequality

BrocSoc 2

N 2 hours ago by sqing

$a+b+c+d=4$ . Prove that $\frac{4}{abcd} \ge \frac{a}{b} + \frac{b}{c} +\frac{c}{d} +\frac{d}{a}$
Source: Exercise 1.3.2 in https://web.williams.edu/Mathematics/sjmiller/public_html/161/articles/Riasat_BasicsOlympiadInequalities.pdf

2 replies

BrocSoc

Yesterday at 8:05 PM

sqing

2 hours ago

2 circles

macves 0

2 hours ago

Triangle ABC having AM as bisector of angle BAC. Let O và O’ be incenters of triangle ABM, ACM, respectively. (O) touchs AB at E, (O') touchs AC at F. EF intersects (O) at X and (O') at Y. Prove that EX = FY.

0 replies

macves

2 hours ago

0 replies

How many nonnegative integers

Darealzolt 5

N 5 hours ago by ilikemath247365

How many nonnegative integers can be written in the form
$a_7 \cdot 3^7 + a_6 \cdot 3^6 + a_5 \cdot 3^5 + a_4 \cdot 3^4 + a_3 \cdot 3^3 + a_2 \cdot 3^2 + a_1 \cdot 3^1 + a_0 \cdot 3^0$ where $a_i \in \{-1, 0, 1\}$ for $0 \le i \le 7$ ?

5 replies

Darealzolt

Yesterday at 4:58 AM

ilikemath247365

5 hours ago

Weird locus problem

Sedro 4

N Yesterday at 9:09 PM by ReticulatedPython

Points $A$ and $B$ are in the coordinate plane such that $AB=2$ . Let $\mathcal{H}$ denote the locus of all points $P$ in the coordinate plane satisfying $PA\cdot PB=2$ , and let $M$ be the midpoint of $AB$ . Points $X$ and $Y$ are on $\mathcal{H}$ such that $\angle XMY = 45^\circ$ and $MX\cdot MY=\sqrt{2}$ . The value of $MX^4 + MY^4$ can be expressed in the form $\tfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .

4 replies

Sedro

May 11, 2025

ReticulatedPython

Yesterday at 9:09 PM

How much sides does M and N have

Darealzolt 1

N Yesterday at 7:47 PM by Lankou

Two regular polygons have $m$ sides and $n$ sides, respectively. The total number of sides is 33, and the total number of diagonals is 243. What are the values of $m$ and $n$ ?

1 reply

Darealzolt

Yesterday at 5:00 AM

Lankou

Yesterday at 7:47 PM

Compilation of functions problems

Saucepan_man02 5

N Yesterday at 7:16 PM by Konigsberg

Could anyone post some handout/compilation of problems related to functions (difficulty similar to AIME/ARML/HMMT etc)?

Thanks..

5 replies

Saucepan_man02

May 7, 2025

Konigsberg

Yesterday at 7:16 PM

polynomials book recs

sunshine_12 2

N Yesterday at 7:14 PM by Konigsberg

hi all! I know pretty much all of the basic high school algebra upto 11th grade- quadratics, solving equations, matrices nd determinants, etc. I was looking for book recs or handouts on polynomials, but pls know that I have no previous experience whatsoever in olympiad algebra. I did try from an excursion in mathematics but couldn't really approach the problems. any help would be rlly appreciated.
xx

2 replies

sunshine_12

Yesterday at 4:07 PM

Konigsberg

Yesterday at 7:14 PM

Continued fraction

ReticulatedPython 4

N Yesterday at 6:04 PM by jasperE3

Find the exact value of the continued fraction $1^2+\frac{1}{2^2+\frac{1}{3^2+\frac{1}{4^2+\frac{1}{5^2+\cdots}}}}$
I know that it is approximately $1.2432$ but I am looking for the exact value. Does anyone know how to solve this problem?

4 replies

ReticulatedPython

Yesterday at 4:03 PM

jasperE3

Yesterday at 6:04 PM

Can someone catch my mistake?

jL56L06B9 3

N Yesterday at 4:12 PM by RegalSparrow

$4$ knights and $4$ jokers are to be seated around a round table. They randomly pick the particular seats they want to sit at. What's the probability they'd be seated alternately?
sol

$\frac{1}{35}$

what I did

$\frac{\frac{4!4!}{8}}{\frac{8!}{8}}=\frac{4!4!}{8!}=\frac{1}{70}$
The problem was solved with the idea that $1$ knight was already seated at the table so we don't have to take into account of the rotations if they were seated alternately. Though my solution does take into account of the different rotations of the knights and jokers.
Is there something wrong with how I solved the problem with my method or is it that we can't use my method at all (if so why)?

3 replies

jL56L06B9

Aug 2, 2020

RegalSparrow

Yesterday at 4:12 PM

Product of all even divisors

girishpimoli 4

N Apr 29, 2025 by williamxiao

$(1)$ Product of all even divisors of $9000$

$(2)$ If $4$ dice are rolled, Then number of ways of getting sum at least $13$ is

4 replies

girishpimoli

Apr 28, 2025

williamxiao

Apr 29, 2025

Product of all even divisors

G H J

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girishpimoli

216 posts

girishpimoli

#1 h Apr 28, 2025, 2:13 PM

Y by

$(1)$ Product of all even divisors of $9000$

$(2)$ If $4$ dice are rolled, Then number of ways of getting sum at least $13$ is

Z K Y

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hellohi321

63 posts

hellohi321

#2 h Apr 28, 2025, 9:55 PM

Y by

Solution to problem 1

We notice that the product of all the even divisors of 9000 is equal to the product of all the divisors of 9000 divided by all the odd divisors of 9000.
$9000=2^3\cdot3^2\cdot5^3$ meaning that it has 48 total divisors which gives a total product of $9000^{24}$
Now, if we want to find the product of all the odd divisors, we have to "remove" the twos from the product and find the product of all the divisors of $3^2\cdot5^3=1125$
This has 12 total divisors, giving a total product of $1125^{6}$
Thus, the answer is $\boxed{\frac{9000^{24}}{1125^{6}}}$

This post has been edited 4 times. Last edited by hellohi321, Apr 29, 2025, 2:08 AM

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imbadatmath1233

21 posts

imbadatmath1233

#3 h Apr 28, 2025, 11:41 PM

Y by

should be 1125^6 not 1125^12 for @above

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hellohi321

63 posts

hellohi321

#4 h Apr 29, 2025, 2:08 AM

Y by

Thanks for catching my mistake

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williamxiao

2516 posts

williamxiao

#5 h Apr 29, 2025, 2:29 AM

Y by

There are $6^4=1296$ possible dice rolls with 4 die. Observe that if our sequence of dice rolls $a,b,c,d$ with sum $S$ , there is a corresponding set of dice rolls $7-a, 7-b, 7-c, 7-d$ with sum $28-S$ . This means that the number of ways to get a sum strictly less than 14 is the same as the number of ways to get a sum strictly more than 14.

We then solve as follows:
Number of ways to get 13:
1156: 12 ways to arrange
1246: 24
1255: 12
1336: 12
1345: 24
1444: 4
2236: 12
2245: 12
2335: 12
2344: 12
3334: 4
Total: 140

Sum=14: 1166: 6
1256: 24
1346:24
1355: 12
1445: 12
2246:12
2255: 6
2336:12
2345:24
2444:4
3335:4
3344:6
Total: 146

Sum >14: By symmetry, this is equal to (6^4-146)/2 = 575

sum together and get $\boxed{861}$

This post has been edited 1 time. Last edited by williamxiao, Apr 29, 2025, 2:29 AM

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