Key Points: Normal Distribution –

- Mean
- Standard Deviation

Key Points: Normal Distribution –

- Mean
- Standard Deviation

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How likely is it that a mathematics student can’t solve IMO problems?

Is there a fear of embarrassment in being a math Ph.D. who can’t solve problems that high-school students can? by Cornelius Goh

**Problem A3**

A function f is defined on the positive integers by:

for all positive integers n,

Determine the number of positive integers n less than or equal to 1988 for which f(n) = n.

What is the explanation of the solution of problem 3 from IMO 1988? by Alon Amit

givdn any Triangle ABC, trisect each angle A,B,C, the 6 trisecting lines (2 each at A,B,C) meet at points M, N,Q inside.

Prove: Triangle MNQ is a regular triangle ?

Let N = 2n > 6

哥德巴赫猜想 Conjecture “1+1”: N = p1 + p2 (pj all primes)

陈景润 Chen Theorem “1+2”: N = p1+ p2.p3

Singapore is being mentioned around 4:54. Very nice video. The truth is that the classroom of today is still nearly the same as the classroom of 150 years ago. There needs to be a “Educational Revolution” parallel to that of the Industrial Revolution. Many children cannot fit into the single classroom model, leading to growth in diagnosis of behavioral “problems” such as ADHD in developed nations.

Americans who are tired of “Common Core” may want to check out Singapore Math for their kids, which is highly acclaimed in the educational realm.

**Malthus Model**

$latex displaystyle frac{dN}{dt}=BN-DN=kN$

$latex N$: Total population

$latex B$: Birth-rate per capita

$latex D$: Death-rate per capita

$latex k=B-D$

Solution to D.E.:

$latex displaystyle boxed{N(t)=widehat{N}e^{kt}},$

where $latex widehat{N}=N(0)$.

**Logistic Equation**

$latex begin{aligned}

D&=sN

frac{dN}{dt}&=BN-sN^2

widehat{N}&=N(0)

N_infty&=B/s

end{aligned}$

**Logistic Case 1: Increasing population ($latex widehat{N}<N_infty$)**

$latex begin{aligned}

N(t)&=frac{B}{s+(frac{B}{widehat{N}}-s)e^{-Bt}}

&=frac{N_infty}{1+(frac{N_infty}{widehat{N}}-1)e^{-Bt}}

end{aligned}$

The second expression can be derived from the first: divide by $latex s$ in both the numerator and denominator.

Logistic Case 2: Decreasing population ($latex widehat{N}>N_infty$)

$latex begin{aligned}

N(t)&=frac{B}{s-(s-frac{B}{widehat{N}})e^{-Bt}}

&=frac{N_infty}{1-(1-frac{N_infty}{widehat{N}})e^{-Bt}}

end{aligned}$

Logistic Case 3: Constant population ($latex widehat{N}=N_infty$)

$latex displaystyle N(t)=N_infty$

Harvesting

Basic Harvesting Model: $latex displaystyle boxed{frac{dN}{dt}=(B-sN)N-E}.$

$latex E$: Harvest rate (Amount harvested per unit time)

Maximum harvest rate without causing extinction: $latex boxed{dfrac{B^2}{4s}}$.

$latex displaystyle boxed{beta_1,beta_2=frac{Bmpsqrt{B^2-4Es}}{2s}}.$

$latex beta_1$: Unstable equilibrium population

$latex beta_2$: Stable equilibrium population

Extinction Time: $latex displaystyle boxed{T=int_{widehat{N}}^0frac{dN}{N(B-sN)-E}}.$

Laplace transform of $latex f$

$latex displaystyle F(s)=L(f)=int_0^infty e^{-st}f(t),dt$

Tip: Use this equation when the questions…

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