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Middle School Math Grades 5-8, Ages 10-13, MATHCOUNTS, AMC 8

Grades 5-8, Ages 10-13, MATHCOUNTS, AMC 8

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Middle School Math Grades 5-8, Ages 10-13, MATHCOUNTS, AMC 8

Grades 5-8, Ages 10-13, MATHCOUNTS, AMC 8

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Area of Polygon

AIME15 43

N 4 hours ago by EthanNg6

The area of polygon $ABCDEF$ , in square units, is

IMAGE

$\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 46 \qquad \textbf{(D)}\ 66 \qquad \textbf{(E)}\ 74$

43 replies

AIME15

Jan 12, 2009

EthanNg6

4 hours ago

2025 OMOUS Problem 6

enter16180 2

N Yesterday at 9:06 PM by loup blanc

Source: Open Mathematical Olympiad for University Students (OMOUS-2025)

Let $A=\left(a_{i j}\right)_{i, j=1}^{n} \in M_{n}(\mathbb{R})$ be a positive semi-definite matrix. Prove that the matrix $B=\left(b_{i j}\right)_{i, j=1}^{n} \text {, where }$ $b_{i j}=\arcsin \left(x^{i+j}\right) \cdot a_{i j}$ , is also positive semi-definite for all $x \in(0,1)$ .

2 replies

enter16180

Apr 18, 2025

loup blanc

Yesterday at 9:06 PM

Sum of multinomial in sublinear time

programjames1 0

Yesterday at 7:45 PM

Source: Own

A frog begins at the origin, and makes a sequence of hops either two to the right, two up, or one to the right and one up, all with equal probability.

1. What is the probability the frog eventually lands on $(a, b)$ ?

2. Find an algorithm to compute this in sublinear time.

0 replies

programjames1

Yesterday at 7:45 PM

0 replies

Find the answer

JetFire008 1

N Yesterday at 6:42 PM by Filipjack

Source: Putnam and Beyond

Find all pairs of real numbers $(a,b)$ such that $a\lfloor bn \rfloor = b\lfloor an \rfloor$ for all positive integers $n$ .

1 reply

JetFire008

Yesterday at 12:31 PM

Filipjack

Yesterday at 6:42 PM

Pyramid packing in sphere

smartvong 2

N Yesterday at 4:23 PM by smartvong

Source: own

Let $A_1$ and $B$ be two points that are diametrically opposite to each other on a unit sphere. $n$ right square pyramids are fitted along the line segment $\overline{A_1B}$ , such that the apex and altitude of each pyramid $i$ , where $1\le i\le n$ , are $A_i$ and $\overline{A_iA_{i+1}}$ respectively, and the points $A_1, A_2, \dots, A_n, A_{n+1}, B$ are collinear.

(a) Find the maximum total volume of $n$ pyramids, with altitudes of equal length, that can be fitted in the sphere, in terms of $n$ .

(b) Find the maximum total volume of $n$ pyramids that can be fitted in the sphere, in terms of $n$ .

(c) Find the maximum total volume of the pyramids that can be fitted in the sphere as $n$ tends to infinity.

Note: The altitudes of the pyramids are not necessarily equal in length for (b) and (c).

2 replies

smartvong

Apr 13, 2025

smartvong

Yesterday at 4:23 PM

Interesting Limit

Riptide1901 1

N Yesterday at 1:45 PM by Svyatoslav

Find $\displaystyle\lim_{x\to\infty}\left|f(x)-\Gamma^{-1}(x)\right|$ where $\Gamma^{-1}(x)$ is the inverse gamma function, and $f^{-1}$ is the inverse of $f(x)=x^x.$

1 reply

Riptide1901

Apr 18, 2025

Svyatoslav

Yesterday at 1:45 PM

2022 Putnam B1

giginori 25

N Yesterday at 12:13 PM by cursed_tangent1434

Suppose that $P(x)=a_1x+a_2x^2+\ldots+a_nx^n$ is a polynomial with integer coefficients, with $a_1$ odd. Suppose that $e^{P(x)}=b_0+b_1x+b_2x^2+\ldots$ for all $x.$ Prove that $b_k$ is nonzero for all $k \geq 0.$

25 replies

giginori

Dec 4, 2022

cursed_tangent1434

Yesterday at 12:13 PM

Number of A^2=I3

EthanWYX2009 1

N Yesterday at 11:21 AM by loup blanc

Source: 2025 taca-14

Determine the number of $A\in\mathbb F_5^{3\times 3}$ , such that $A^2=I_3.$

1 reply

EthanWYX2009

Yesterday at 7:51 AM

loup blanc

Yesterday at 11:21 AM

Prove this recursion!

Entrepreneur 3

N Yesterday at 11:06 AM by quasar_lord

Source: Amit Agarwal

Let $I_n=\int z^n e^{\frac 1z}dz.$ Prove that $\color{blue}{I_n=(n+1)!I_0+e^{\frac 1z}\sum_{n=1}^n n! z^{n+1}.}$

3 replies

Entrepreneur

Jul 31, 2024

quasar_lord

Yesterday at 11:06 AM

Pove or disprove

Butterfly 1

N Yesterday at 10:05 AM by Filipjack

Denote $y_n=\max(x_n,x_{n+1},x_{n+2})$ . Prove or disprove that if $\{y_n\}$ converges then so does $\{x_n\}.$

1 reply

Butterfly

Yesterday at 9:34 AM

Filipjack

Yesterday at 10:05 AM

fractional binomial limit sum

Levieee 3

N Yesterday at 9:44 AM by Levieee

this was given to me by a friend

$\lim_{n \to \infty} \sum_{k=1}^{n}{\frac{1}{\binom{n}{k}}}$

a nice solution using sandwich is
$\frac{1}{n} + \frac{1}{n} + 1 + \frac{n-3}{\binom{n}{2}} \ge \frac{1}{n} + \sum_{k=2}^{n-2}{\frac{1}{\binom{n}{k}}}+ \frac{1}{n} + 1 \ge \frac{1}{n} + + \frac{1}{n} + 1$

therefore $\lim_{n \to \infty} \sum_{k=1}^{n}{\frac{1}{\binom{n}{k}}}$ = $1$

ALSO ANOTHER SOLUTION WHICH I WAS THINKING OF WITHOUT SANDWICH BUT I CANT COMPLETE WAS TO USE THE GAMMA FUNCTION

we know

$B(x, y) = \int_0^1 t^{x - 1} (1 - t)^{y - 1} \, dt$

$B(x, y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x + y)}$

and $\Gamma(n) = (n-1)!$ for integers,

$\frac{1}{\binom{n}{k}}$ = $\frac{k! (n-k)!}{n!}$

therefore from the gamma function we get

$(n+1) \int_{0}^{1} x^k (1-x)^{n-k} dx$ = $\frac{1}{\binom{n}{k}}$ = $\frac{k! (n-k)!}{n!}$
$\Rightarrow$ $\lim_{n \to \infty} (n+1) \int_{0}^{1} \sum_{k=1}^{n} x^k (1-x)^{n-k} dx$ $=\lim_{n \to \infty} \sum_{k=1}^{n}{\frac{1}{\binom{n}{k}}}$

somehow im supposed to show that

$\lim_{n \to \infty} (n+1) \int_{0}^{1} \sum_{k=1}^{n} x^k (1-x)^{n-k} dx$ $= 1$

all i could observe was if we do L'hopital (which i hate to do as much as you do)

i get $\frac{ \int_{0}^{1} \sum_{k=1}^{n} x^k (1-x)^{n-k} dx}{1/n+1}$

now since $x \in (0,1)$ , as $n \to \infty$ the $(1-x)^{n-k} \to 0$ which gets us the $\frac{0}{0}$ form therefore L'hopital came to my mind , which might be a completely wrong intuition, anyway what should i do to find that limit

:noo: :pilot:

3 replies

Levieee

Saturday at 9:51 PM

Levieee

Yesterday at 9:44 AM

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