Reach for the Summit problem set - Mathematics

Reach for the Summit - M-S1-A1

Let set $A=\{x|x=a^2+b^2, a,b \in \mathbb Z\}$.

If $x_1,x_2 \in A$, is it always true that $x_1 \cdot x_2 \in A$?

Reach for the Summit - M-S1-A2

Given that $a,b \in \mathbb R,\ a>0$, if $4^a-3a^b=16,\ \log_{2}a=\dfrac{a+1}{b}$, find the sum of all possible value(s) of $a^b$.

Reach for the Summit - M-S1-A3

Find the monotonic increasing interval for function $y=\log_{0.5}(x^2+4x+4)$.

Reach for the Summit - M-S1-A4

If $a>0,\ f(x)=\sqrt{x}-\ln(x+a)$ is monotonic increasing for $x \in (0,+\infty)$, find the minimum value of $a$.

Reach for the Summit - M-S2-A1

Compare $\cos(\sin x)$ and $\sin(\cos x)$ for $x \in [0,\pi]$.

Reach for the Summit - M-S2-A2

Without using the calculator, find the value of:

$(\sec 50 \degree + \tan 10 \degree)(\cos \dfrac{2 \pi}{7}+\cos \dfrac{4 \pi}{7}+ \cos \dfrac{6 \pi}{7})(\tan 6 \degree \tan 42 \degree \tan 66 \degree \tan 78 \degree)$

Let $A$ denote the value. Submit $\lfloor 10000A \rfloor$.

Reach for the Summit - M-S3-A1

Given a positive infinite sequence $\{a_n\}$, $S_n = \displaystyle \sum_{k=1}^{n} a_k$.

If $\forall n \in \mathbb N^+$, the arithmetic mean of $a_n$ and $2$ is equal to the geometric mean of $S_n$ and $2$, then find $a_{2020}$.

Reach for the Summit - M-S3-A2

In sequence $\{a_n\}$, $a_1=1,a_2=3,a_3=6$, for $n \geq 4,\ a_n=3a_{n-1}-a_{n-2}-2a_{n-3}$.

Then $\forall n \geq 4$, $a_n>\lambda \times 2^{n-2}\ (\lambda \in \mathbb N^+)$, then find the maximum value of $\lambda$.

Reach for the Summit - M-S4-A1

If $z$ is one of the $7^{th}$ roots of unity, $z \neq 1$, find $(|z+z^2+z^4|)^2$.

Reach for the Summit - M-S4-A2

If three complex roots of the equation: $x^3+px+1=0$ form an equilateral triangle on the complex plane, then find the area of that triangle.

Let $S$ be the area. Submit $\lfloor 1000S \rfloor$.

Reach for the Summit - M-S4-A3

Find the value of:

$\displaystyle (\sum_{k=0}^{1010} (-1)^k {2020 \choose 2k} )^2 + (\sum_{k=0}^{1009} (-1)^k {2020 \choose 2k+1} )^2 \pmod {998244353}$

Reach for the Summit - M-S5-A1

As shown above, in $\triangle ABC$, $A(3,4), B(-5,0), N(2,0)$, $C$ is a point on the positive x-axis such that $\angle BAM = \angle CAN$.

If the circumcenter of $\triangle AMN$ is $D(x_1,y_1)$, the circumcenter of $\triangle ABC$ is $E(x_2,y_2)$,

Find $\lfloor 1000(x_1+2y_1+3x_2+4y_2)\rfloor$.

Hint: Find the relationship of $AD$ and $AE$.

Reach for the Summit - M-S5-A2

Find the smallest and largest inscribed equilateral triangle in an $1 \times 1$ square.

If the smallest triangle has area $S_{min}$, the largest has area $S_{max}$, submit $\lfloor 10^6 (S_{max}-S_{min})\rfloor$.

Reach for the Summit - M-S5-A3

As shown above, a copied version of this image of Alice has been scaled uniformly, rotated and put inside the original one.

Then how many points are there correspond to the same relative position of the copy and the original one?

Reach for the Summit - M-S6-A1

Given that $A(1,-1), B(-4,5)$, $C$ is on line $AB$, and $|AC|=3|AB|$, then find all possible coordinates of point $C$.

How to submit:

• First, find the number of all possible solutions $(x,y)$. Let $N$ denote the number of solutions.
• Then sort the solutions by $x$ from smallest to largest, if $x$ is the same, then sort by $y$ from smallest to largest.
• Let the sorted solutions be: $(x_1,y_1), (x_2,y_2), (x_3,y_3), \cdots ,(x_N,y_N)$, then $M=\displaystyle \sum_{k=1}^N k(x_k+y_k)$.

For instance, if the solution is $(-1,2), (-1,1), (1,3), (0,4)$, the sorted solution will be: $(-1,1), (-1,2), (0,4), (1,3)$, then $N=4$ and $\\ M=\displaystyle \sum_{k=1}^4 k(x_k+y_k)= 1 \times (-1+1) + 2 \times (-1+2) + 3 \times (0+4) + 4 \times (1+3) =30$.

For this problem, submit $\lfloor M+N \rfloor$.

Reach for the Summit - M-S6-A2

As shown above, line $l$ is tangent to curve $C:x^2+y^2-2x-2y+1=0$ and it intersects with y-axis at point $A$, x-axis at point $B$, $O$ is the origin, $|OA|>2, |OB|>2$. Then find the minimum area for $\triangle AOB$.

Let $S$ denote the minimum area. Submit $\lfloor 1000S \rfloor$.

Reach for the Summit - M-S7-A1

If point $A,B,C,D$ are in the $3D$ space, $\angle ABC = \angle BCD = \angle CDA = \angle DAB = 90 \degree$, is it always true that $A,B,C,D$ are on the same plane?