Any tips for IOQM(indian olympiad qualifier for math)

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0 replies

jlacosta

May 1, 2025

0 replies

Infinite polynomial's value

Kempu33334 3

N 25 minutes ago by Kempu33334

Source: Me

Let us define $P(x)$ to be an infinite polynomial where $P(x) = \lim_{n\to \infty}\prod_{k=0}^n \left(x-\left(\cos\left(\dfrac{2\pi k}{n}\right)+i\sin\left(\dfrac{2\pi k}{n}\right)\right)\right).$
a) Find $P(0)$ , $P(\pi)$ . Hint

Weierstrass factorization theorem

b) Find the coefficient of the $x^2$ , and $x^4$ terms. Hint

Taylor series

@below, yep thank you!

3 replies

Kempu33334

Today at 2:30 PM

Kempu33334

25 minutes ago

[PMO25 Areas, I.13] log + cubic equation

jdcuber13 2

N an hour ago by Vivaandax

Let $a,b,c$ be real numbers with $1 < a < b < c$ that satisfy the equations
$\begin{align*} \log_{a} b + \log_{b} c + \log_{c} a &= 6.5 \\\\ \log_{b} a + \log_{c} b + \log_{a} c &= 5 \end{align*}$
Then max{ $\log_{a} b, \log_{b} c, \log_{c} a$ } can be written in the form $\sqrt{x} + \sqrt{y}$ , where $x$ , and $y$ are positive integers. What is $x+y$ ?

Answer Confirmation

$16, (\sqrt{9} + \sqrt{7})$

2 replies

+1 w

jdcuber13

Today at 12:44 PM

Vivaandax

an hour ago

Easy expected value problem

Kempu33334 6

N 3 hours ago by aidan0626

Source: Me

Bob takes two real numbers, $x$ and $y$ , independently at random from the range $0$ to $1000$ . He then selects an operation out of $+$ , $-$ , and $\times$ and performs that operation on them. Find the expected value of the answer.

6 replies

Kempu33334

Today at 2:17 PM

aidan0626

3 hours ago

Challenge Problem in Similar Triangles

yes45 0

6 hours ago

$\triangle ABC$ has a right angle at $B$ . $AB = 9$ and $AC = 15$ . Point $D$ lies on $AB$ such that $AD = 2$ and point $E$ lies on $AC$ such that $AE$ = $\frac{12}{5}$ . Find the perimeter of $BCED$ .
Answer Confirmation

$\frac{168}{5}$

Solution

The perimeter of $BCED$ = $BC + CE + ED + BD$ .

$BC = 12$ .

$CE = 15 - \frac{12}{5} = \frac{63}{5}$ .

$BD = 9 - 2 = 7$ .

Now, to get the perimeter, we only need to solve for $ED$ .

Draw the altitude of $\triangle ADE$ from base $AE$ , and you will get two right triangles. Let point F be the point where that altitude intersects with $AE$ . $\triangle AFD$ shares angles $\angle{A}$ with $\triangle ABC$ and also has a right angle, so we can prove that $\triangle AFD \sim \triangle ABC$ through AA Postulate. $\frac{AB}{AC} = \frac {AF}{AD} = \frac{3}{5}$ , so $FA = 2 \cdot \frac{3}{5} = \frac{6}{5}$ and $FE = \frac{12}{5} - \frac{6}{5} = \frac{6}{5}$ . Since $FE \cong AF$ , $\angle{EFD} \cong \angle{DFA}$ (as both are right angles), and $DF \cong DF$ , we can prove $\triangle AFD \cong \triangle EFD$ through SAS postulate. $ED$ corresponds with $AD$ , so $ED = AD = 2$ .

$12 + \frac{63}{5} + 7 + 2 = \boxed{\frac{168}{5}}$ .

0 replies

yes45

6 hours ago

0 replies

No more topics!

Any tips for IOQM(indian olympiad qualifier for math)

Indianmathenthusiast1234 4

N Apr 17, 2025 by Indianmathenthusiast1234

Guys I'm preparing for the IOQM, and i feel like the linear congruences, modular arithmetic and number bases are a bit hard to grasp. Any tips?

4 replies

Indianmathenthusiast1234

Apr 17, 2025

Indianmathenthusiast1234

Apr 17, 2025

Any tips for IOQM(indian olympiad qualifier for math)

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Indianmathenthusiast1234

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Indianmathenthusiast1234

#1 h Apr 17, 2025, 3:49 PM

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Guys I'm preparing for the IOQM, and i feel like the linear congruences, modular arithmetic and number bases are a bit hard to grasp. Any tips?

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Alex-131

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Alex-131

#2 h Apr 17, 2025, 3:51 PM

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intro to number theory

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Indianmathenthusiast1234

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Indianmathenthusiast1234

#3 h Apr 17, 2025, 3:53 PM

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Done with that book. Any general ones?
Btw thanks for the advice

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Alex-131

5395 posts

Alex-131

#4 h Apr 17, 2025, 3:55 PM

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You can take a look at MONT (modern olympiad number theory), maybe the first two chapters. It's structured for more olympiads, but may be helpful if you've done intro to nt before.

You could also mess around with OTIS Excerpts, specifically the nt part.

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Indianmathenthusiast1234

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Indianmathenthusiast1234

#5 h Apr 17, 2025, 4:08 PM

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Thank you so much
I will definitely look into it

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