# SAT 1000 problems series - Upcoming frequently

Now I think it's time to publish all of the 1000 problems of my problem set. They come from some particularly hard, popular or interesting Gaokao problems.

From then on, I will post all of the problems here, the order of the problem would be initially disarranged, but eventually I would complete all of my the problem set.

These problems will follow the same format so that it will be easier for the programs to index them and use them.

I'm thinking using the problem set for my games or other purposes.

Free to use the problem set as a dataset for machine learning or using the problems in your contest, games or programs, just indicate the source :)

Category:

P1-P159 - Functions

P160-P267 - Derivatives

P268-P370 - Trigonometry

P371-P436 - Vectors

P437-P503 - Sequences

P504-P562 - Inequalities

P563-P669 - Solid Geometry

P670-P846 - Analytic Geometry

P847-P861 - Modeling

P862-P941 - Math Insight

P942-P1000 - Miscellaneous

What do you have in Gaokao?

• A pen/pencil

• Draft paper

What can't you use in Gaokao?

• Calculator (Except for the submit process)

• Numerical method

• Mathematica, Geogebra or other related software/websites

Your pen/pencil, draft paper are the only things you can use. There you go!

Note by Alice Smith
4 months, 2 weeks ago

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

• Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
• Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
• Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$ ... $$ or $ ... $ to ensure proper formatting.
2 \times 3 $2 \times 3$
2^{34} $2^{34}$
a_{i-1} $a_{i-1}$
\frac{2}{3} $\frac{2}{3}$
\sqrt{2} $\sqrt{2}$
\sum_{i=1}^3 $\sum_{i=1}^3$
\sin \theta $\sin \theta$
\boxed{123} $\boxed{123}$

Sort by:

Given that $\hat{e_1}, {\hat{e_2}}$ are unit vectors, and $\textbf b = x \hat{e_1} + y \hat{e_2}\ (x,y \in \mathbb R)$ is a nonzero vector.

If $<\hat{e_1},\hat{e_2}>\ = \dfrac{\pi}{6}$, then find the maximum value of $\dfrac{|x|}{|\textbf b|}$.

Let $A$ deonte the answer. Submit $\lfloor 1000A \rfloor$.

- 3 months, 1 week ago

If function $f(x)=x-\dfrac{1}{3} \sin 2x + a \sin x$ is monotonic increasing for all $x \in \mathbb R$, what's the range of $a$?

$A.\ [-1,1]$

$B.\ [-1, \dfrac{1}{3}]$

$C.\ [-\dfrac{1}{3}, \dfrac{1}{3}]$

$D.\ [-1, -\dfrac{1}{3}]$

- 3 months, 1 week ago

In an acute triangle $ABC$, $\tan A = \dfrac{1}{2}$, $D$ is a point on $BC$, $[ABD]=2, [ACD]=4$.

If $E,F$ are points on $AB, AC$ respectively, and $DE \perp AB, DF \perp AC$, then find the value of $\overrightarrow{DE} \cdot \overrightarrow{DF}$.

Let $A$ denote the answer. Submit $\lfloor 1000A \rfloor$.

- 3 months, 1 week ago

Let function $f(x)=2x - \cos x$, $\{a_n\}$ is an arithmetic sequence whose common difference is $\dfrac{\pi}{8}$.

If $\displaystyle \sum_{k=1}^{5} f(a_k) = 5\pi$, find $f^2(a_3)-a_1 a_5$.

Let $A$ denote the answer. Submit $\lfloor 1000A \rfloor$.

- 3 months, 1 week ago

As shown above, a small circle with diameter $1$ is rotating counterclockwise along the interior side of the big circle with diameter $2$ without slipping, $M,N$ are two endpoints of a diameter of the small circle.

Then as it is rotating, which of the following is the locus of point $M,N$?

- 3 months, 1 week ago

As shown above, the enclosed curve $C$ is composed of three segments of arcs(solid line) and the circles that the arcs belong to pass through the same point $P$, and they have the same radius.

If the $k\ th$ segment of arc corresponds the center angle $\alpha_k\ (k=1,2,3)$, then find the value of:

$\cos \dfrac{\alpha_1}{3} \cos \dfrac{\alpha_2+\alpha_3}{3} - \sin \dfrac{\alpha_1}{3} \sin \dfrac{\alpha_2+\alpha_3}{3}$

Let $A$ denote the value. Submit $\lfloor 1000A \rfloor$.

- 3 months, 1 week ago

$6$ classmates are exchanging souvenirs among each other in the graduation party after Gaokao. For each pair of two students, they can exchange once at most. For each round of exchange, the two students send each other a souvenir.

Given that these $6$ classmates have performed $13$ rounds of exchange, what's the number of classmates that have received exactly $4$ souvenirs?

$A.\ \textup{1 or 3}$

$B.\ \textup{1 or 4}$

$C.\ \textup{2 or 3}$

$D.\ \textup{2 or 4}$

- 3 months, 1 week ago

The torch relay of the Guangzhou 2010 Asian Games was held in cities $A,B,C,D,E$, and the distance of two cities are shown in the picture above.

If $A$ is the starting point and $E$ is the goal, each city is passed by and only once, what's the minimum total distance travelled of the torch relay?

- 3 months, 1 week ago

There are three treasure chests which contains:

$A.\ \textup{A gold coin and a silver coin.}$

$B.\ \textup{A gold coin and a rock.}$

$C.\ \textup{A silver coin and a rock.}$

respectively. $\textup{Alice, Betty, Cathy}$ took each one of them, and $\textup{Alice}$ looked at $\textup{Betty's}$ chest and said: "My chest and hers don't both contain a silver coin." $\textup{Betty}$ looked at $\textup{Cathy's}$ chest and said: "My chest and hers don't both contain a gold coin." $\textup{Cathy}$ said: "What's in my chest isn't a silver coin and a rock."

Then what's in $\textup{Alice's}$ treasure chest?

- 3 months, 1 week ago

Four girls $\textup{Alice,Betty,Cathy,Dora}$ are called to wear black or red hats, and none of them can see the colors of their own hats.

They are informed that exactly $2$ of them wear black hats and the others wear red hats. After that, they decide to play hide-and-seek under the light.

Then $\textup{Alice}$ has seen $\textup{Betty's}$ and $\textup{Cathy's}$ hats, $\textup{Betty}$ has seen $\textup{Cathy's}$ hat, and $\textup{Dora}$ has seen $\textup{Alice's}$ hat, but not vise versa.

Suddenly, the light goes out, and $\textup{Alice}$ says: "I still don't know what color my hat is".

If what she says and what they are informed are true, which of the statement is true?

$A.\ \textup{Betty can know the colors of all four girl's hats.}$

$B.\ \textup{Dora can know the colors of all four girl's hats.}$

$C.\ \textup{Betty can know the color of Dora's hat, and vise versa.}$

$D.\ \textup{Betty, Dora can know the color of their own hats.}$

- 3 months, 1 week ago

In the rectangular coordinate plane, the Taxicab path from point $M$ and $N$ is such path that moves only horizontally or vertically from $M$ to $N$. And $d(M,N)$ denotes the length of the path.

Given that $A_1(3,20), A_2(-10,0), A_3(14,0)$, we want to find a point $P(x_0,y_0)\ (y_0 \geq 0)$ so that $\displaystyle \sum_{k=1}^{3} d(P,A_k)$ has the minimum value.

However, the Taxicab path from $P$ to each point can't pass through the region: $x^2+y^2<1$.

Then find the coordinates of $P$.

If $P$ has coordinate $(x_0,y_0)$, submit $2y_0-x_0$.

- 3 months, 1 week ago

In the rectangular coordinate plane, the Taxicab distance from point $P_1(x_1,y_1)$ to $P_2(x_2,y_2)$ is defined as: $d(P_1,P_2)=|x_1-x_2|+|y_1-y_2|$

Given these integer points: $A_1(-2,2), A_2(3,1), A_3(3,4), A_4(-2,3), A_5(4,5)$, then find the integer point $P(x_0,y_0)$ so that $\displaystyle \sum_{k=1}^{5} d(P,A_k)$ has the minimum value.

Submit $2y_0-x_0$.

- 3 months, 1 week ago

As shown above, from top and bottom, $l_1, l_2, l_3$ are three parallel lines on the same plane, and the distance from $l_1$ to $l_2$ is $1$, the distance from $l_2$ to $l_3$ is $2$, and point $A,B,C$ are on $l_1, l_2, l_3$ respectively.

If $\triangle ABC$ is an equilateral triangle, then find the side length of $\triangle ABC$.

Let $l$ denote the side length. Submit $\lfloor 1000l \rfloor$.

- 3 months, 2 weeks ago

Given that $S_n=\displaystyle \sum_{k=1}^{n} \sin \dfrac{k \pi}{7} \ (n \in \mathbb N^+)$

Then how many positive numbers are there for $S_1,S_2, \cdots, S_{100}$?

- 3 months, 2 weeks ago

Five girls $\textup{Alice, Betty, Cathy, Dora, Emily}$ are sitting around a circle table clockwise. They are playing a game whose rules are as follows:

• The first girl yells the number $1$, the second next to the right hand side yells the number $1$, too, from then on, the girl next to the right hand side of the fromer yells the sum of the numbers of the last $2$ students.

• If the number is divisible by $3$, the girl who yells it needs to clap her hands once.

Given that $\textup{Alice}$ is the first to yell, how many times should she clap when they have yelled the $100\ th$ number?

- 3 months, 2 weeks ago

Let $\{b_n\}$ be the sequence of all triangular numbers that are divisible by $5$, ordering from smallest to largest.

Then compute $b_{2k-1}$ at $k=2020$.

- 3 months, 2 weeks ago

Given that: $\cos 10\alpha = a_1 \cos^{10} \alpha + a_2 \cos^8 \alpha + a_3 \cos^6 \alpha + a_4 \cos^4 \alpha + a_5 \cos^2 \alpha + a_6$

Then find $a_1-a_4+a_5$.

- 3 months, 2 weeks ago

Given that:

$\displaystyle \sum_{i=1}^{n} i^k = a_{k+1} n^{k+1} + a_k n^k + a_{k-1} n^{k-1} + a_{k-2} n^{k-2} + \cdots + a_1 n + a_0$

Then find the value of $\lfloor 10^7 (a_{k+1}+a_k+a_{k-1}+a_{k-2}) \rfloor$ at $k=2020$.

- 3 months, 2 weeks ago

Find the value of $\displaystyle \sum_{k=1}^{2n} (\sin \dfrac{k \pi}{2n+1})^{-2}$ at $n=2020$.

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

- 3 months, 2 weeks ago

Let $N=2^n\ (n \in \mathbb N^+, n \geq 2)$, $N$ distinct numbers denoted as $x_1,x_2,\cdots,x_N$ are put into $N$ positions labeled $1,2,\cdots,N$, then we will get the permutation $P_0=x_1 x_2 \cdots x_N$.

The transform $C$ on the permutation $P$ is as follows:

• Separate the terms on the odd and even positions, and put them to the first $\dfrac{N}{2}$, last $\dfrac{N}{2}$ positions respectively, keeping the original order.

If we apply $C$ to $P_0$ once, we will get $P_1=x_1 x_3 \cdots x_{N-1} x_2 x_4 \cdots x_N$.

After that, we divide $P_1$ into $2$ consecutive segments consisting of $\dfrac{N}{2}$ numbers, and apply $C$ to each segment, we will get $P_2$.

For example, if $n=3,\ N=8$, we will get $P_2=x_1 x_5 x_3 x_7 x_2 x_6 x_4 x_8$, at this time, $x_7$ is on $4\ th$ position of $P_2$.

From then on, when $2 \leq i \leq n-2$, we divide $P_i$ into $2^i$ consecutive segments consisting of $\dfrac{N}{2^i}$ numbers, and apply $C$ to each segment to get $P_{i+1}$.

Then when $n=32,\ N=2^{32}$, $x_{173}$ is on the $M\ th$ position of $P_4$. Submit $M$.

- 3 months, 3 weeks ago

The 0-1 regular sequence $\{a_n\}$ is defined as follows:

• $\{a_n\}$ has $2m$ terms $(m \in \mathbb N^+)$.

• Exactly $m$ terms are $0$ and $m$ terms are $1$.

• $\forall k \geq 2m\ (k \in \mathbb N^+)$, the number of $0$'s is always greater or equal to the number of $1$'s for subsequence $a_1,a_2,\cdots,a_k$.

For $m=2020$, the number of such 0-1 regular sequences is $M$. Submit $\lfloor \log_2 M \rfloor$.

- 3 months, 3 weeks ago

For set $E=\{a_1,a_2,\cdots,a_{100}\}$, whose subset is $X=\{a_{i_1},a_{i_2},\cdots,a_{i_k}\}$, let‘s define the characteristic sequence of $X$ as: $x_1,x_2,\cdots,x_{100}$, where $x_{i_1}=x_{i_2}=\cdots=x_{i_k}=1$, and the other terms are all $0$.

For instance, the characteristic sequence of $\{a_2,a_3\}$ is $0,1,1,0,0,\cdots,0$.

If $P,Q$ are subsets of $E$, and $P$ has characteristic sequence: $p_1,p_2,\cdots,p_{100}$ such that $p_1=1,\ p_i+p_{i+1}=1,\ 1 \leq i \leq 99$, $Q$ has characteristic sequence: $q_1,q_2,\cdots,q_{100}$ such that $q_1=1,\ q_{j}+q_{j+1}+q_{j+2}=1,\ 1 \leq j \leq 98$, then find the cardinality of $P \cap Q$.

- 3 months, 3 weeks ago

Given that set $S,T$ are two non-empty subsets of $\mathbb R$, if there exists a function $f$ from $S$ to $T$ such that:

• $T= \{f(x)|x \in S\}$.

• $\forall x_1, x_2 \in S$, when $x_1, $f(x_1).

Then $S$ and $T$ are Order Isomorphic.

Which pair of the set $A,B$ is not Order Isomorphic?

$A.\ A=\mathbb N^+, B=\mathbb N$

$B.\ A=\{x|-1 \leq x \leq 3\}, B=\{x|x=-8 \vee 0 < x \leq 10\}$

$C.\ A=\{x|0

$D.\ A=\mathbb Z, B=\mathbb Q$

- 3 months, 3 weeks ago

If we define $S=\{a_{l_1},a_{l_2}, \cdots ,a_{l_n}\}$ as the $k\ th$ subset for set $E=\{a_1,a_2,\cdots,a_{10}\}$, where $k=\displaystyle \sum_{i=1}^{n} 2^{l_{i}-1}$.

Then what's the $211\ th$ subset for $E$?

How to submit:

Sort the indices of the elements of $S$ from smallest to largest and put them together. For example, if $S=\{a_1,a_2,a_3\}$, then submit $123$, and when $S=\{a_1,a_2,a_{10}\}$, then submit $1210$.

- 3 months, 3 weeks ago

The image shows all of the available roads for road construction, where the letters represent cities and the numbers denote the corresponding cost for certain road.

What's the minimum total cost to construct roads so that one can travel from any city to every other one?

- 3 months, 3 weeks ago

For a closed region, the maximum distance of the two points in the region is called the diameter of the region, and the ratio of the perimeter to the diameter is denoted by $\tau$.

As shown above, $\tau_1, \tau_2, \tau_3, \tau_4$ denotes the ratio of the perimeter to the diameter of the four regions, from left to right. Then compare them from smallest to largest.

For example , if $\tau_2<\tau_1<\tau_3<\tau_4$, submit $2134$.

- 3 months, 3 weeks ago

A group of members from brilliant.org decide to plant trees on the playground defined by coordinate plane, and the schedule is as follows:

The $k\ th$ tree is planted at $P_k(x_k,y_k)$, where $x_1=1, y_1=1$, and when $k \geq 2$:

\begin{cases} \begin{aligned} x_k & = x_{k-1}+1-5(\lfloor \dfrac{k-1}{5} \rfloor - \lfloor \dfrac{k-2}{5} \rfloor) \\ y_k & = y_{k-1} + \lfloor \dfrac{k-1}{5} \rfloor - \lfloor \dfrac{k-2}{5} \rfloor \end{aligned} \end{cases}

What's the coordinate of the $2008\ th$ tree?

Let the coordinate be $x_0,y_0$. Submit $2y_0-x_0$.

- 3 months, 3 weeks ago

In the coordinate plane, the Hamilton Distance from point $P_1(x_1,y_1)$ to $P_2(x_2,y_2)$ is defined as: $d(P_1,P_2)=|x_1-x_2|+|y_1-y_2|$

Then given these three statements, which of them are true?

1.If point $C$ is on segment $AB$, then $d(A,C)+d(C,B)=d(A,B)$.

2.In $\triangle ABC$, if $\angle C=90 \degree$, then $d^2(A,C)+d^2(C,B)=d^2(A,B)$ $(d^2(A,B)=(d(A,B))^2)$.

3.In $\triangle ABC$, $d(A,C)+d(C,B)>d(A,B)$.

How to submit:

Let $p_1, p_2,\cdots,p_n$ be the boolean value of statement $1,2,\cdots,n$, if statement $k$ is true, $p_k=1$, else $p_k=0$.

Then submit $\displaystyle \sum_{k=1}^n p_k \cdot 2^{k-1}$.

- 3 months, 3 weeks ago

In the coordinate plane, the Hamilton Distance from point $P_1(x_1,y_1)$ to $P_2(x_2,y_2)$ is defined as: $d(P_1,P_2)=|x_1-x_2|+|y_1-y_2|$ If $F_1,F_2$ are two point on the x-axis and they are symmetric about the y-axis, Which choice may be the locus of the point $P$ such that $d(P,F_1)+d(P,F_2)=C$ ($C$ is constant and $C>|F_1 F_2|$)?

$A.$

$B.$

$C.$

$D.$

- 3 months, 3 weeks ago

Imagine an equilateral $\triangle ABC$ divided into $n^2 (n \geq 2, n \in \mathbb N^+)$ congruent equilateral pieces. (The image above shows $n=4$ case).

Let's put each number on every vertex of all triangles, so that the numbers on the same side of $\triangle ABC$ or on the same line parallel to the sides of $\triangle ABC$ (if more than $3$ numbers) form Arithmetic Progressions.

$a,b,c$ denote the number on vertices $A,B,C$ respectively, $a,b,c$ are not equal to each other and $a+b+c=1$.

Let $f(n)$ denote the sum of all numbers on the vertices, find $f(2020)$.

- 3 months, 4 weeks ago

If sequence $\{a_n\}$ satisfies the property: $\forall n \in \mathbb N^+$, there exists finite amounts of positive integer $m$ such that $a_m. Then for any given $n$, if we count the corresponding number of solutions for $m$, and take note of the value at the same indices, then we will generate a new sequence $\{(a_n)^*\}$.

For example, if $\{a_n\}$ is $1,2,3,\cdots,n$, then $\{(a_n)^*\}$ will be: $0,1,2,\cdots,n-1$.

Given that $\forall n \in \mathbb N^+$, $a_n=n^2$. If we define sequence $\{b_n\}$ as $\{((a_{n})^*)^*\}$, find $b_{2020}$.

- 3 months, 4 weeks ago

Let $P$ be an arbitrary point in $\triangle ABC$, $\lambda_1 = \dfrac{S_{\triangle PBC}}{S_{\triangle ABC}}$,$\lambda_2 = \dfrac{S_{\triangle PCA}}{S_{\triangle ABC}}$, $\lambda_3 = \dfrac{S_{\triangle PAB}}{S_{\triangle ABC}}$.

Define $f: \mathbb R^2 \mapsto \mathbb R^3$, $f(P)=(\lambda_1,\lambda_2,\lambda_3)$.

If point $G$ is the centroid of $\triangle ABC$, $f(Q)=(\dfrac{1}{2},\dfrac{1}{3},\dfrac{1}{6})$, then which choice is true?

$A.\ \textup{Q is always in}\ \triangle GAB.$

$B.\ \textup{Q is always in}\ \triangle GBC.$

$C.\ \textup{Q is always in}\ \triangle GCA.$

$D.\ \textup{Q is concurrent with G}.$

Note: $S_{\triangle ABC}$ denotes the area of $\triangle ABC$.

- 3 months, 4 weeks ago

As shown above, in cuboid $ABCD-A_1B_1C_1D_1$, $AB=11, AD=7, AA_1=12$. The faces of the cuboid are all mirrors.

A ray is emitted from $A(0,0,0)$ to $E(4,3,12)$ and reflected when meeting the faces of the cuboid following the reflection law. Let $L_i$ denote the length of the ray between the $(i-1)\ th$ reflection to the $i\ th$ reflection $(i=2,3,4)$, $L_1=AE$.

Compare $L_1,L_2,L_3,L_4$.

$A.\ L_1=L_2>L_3=L_4$

$B.\ L_1=L_2=L_3=L_4$

$C.\ L_1=L_2>L_3

$D.\ L_1=L_2>L_3>L_4$

- 3 months, 4 weeks ago

As shown above, in $\triangle ABC$, $A=90 \degree$, $AB=AC=4$, $P$ is a point on segment $AB$ (excluding point $A,B$).

A ray is emitted from $P$ and it is reflected at $Q$ on $BC$, $R$ on $AC$ and returned to point $P$ again.

If ray $QR$ passes through the centroid of $\triangle ABC$, find the length of $AP$.

Submit $\lfloor 1000|AP| \rfloor$.

- 3 months, 4 weeks ago

As shown above, the four vertices of the mirror rectangle are $A(0,0), B(2,0), C(2,1), D(0,1)$.

A ray is emitted from $P_0(1,0)$ at angle $\theta$ with $AB$ and reflected at $P_1$ on $BC$, $P_2$ on $CD$, $P_3$ on $DA$, $P_4$ on $AB$ following the law of reflection.

If $P_4$ has coordinate $(x_4,0)$ and $x_4 \in (1,2)$, find the range of $\tan \theta$.

The range can be expressed as $(l,r)$. Submit $\lfloor 1000(2r-l) \rfloor$.

- 3 months, 4 weeks ago

If set $S$ is a non-empty subset of integer set $\mathbb Z$, if $\forall a,b \in S, ab \in S$, then $S$ is closed under multiplication.

Given that for set $T,U$: $T \subseteq \mathbb Z, V \subseteq \mathbb Z, T \cap V = \emptyset , T \cup V = \mathbb Z$, and $\forall a,b,c \in T, abc \in T$, $\forall x,y,z \in V, xyz \in V$, then which of the choices is true?

$A.\ \textup{At least one of T,V is closed under multiplication.}$

$B.\ \textup{At most one of T,V is closed under multiplication.}$

$C.\ \textup{Only one of T,V is closed under multiplication.}$

$D.\ \textup{Both T,V are closed under multiplication.}$

- 4 months ago

For function $f(x), g(x)$ which have the same domain $D$, if there exists $h(x)=kx+b$ ($k,b$ are constant), so that:

$\forall m \in (0,+\infty), \exists x_0 \in D, \forall x \in D \wedge x>x_0, 0

Then line $l$: $y=kx+b$ is called the bipartite asymptote for curve $y=f(x)$ and $y=g(x)$.

Here are four groups of functions which are defined at $(1,+\infty)$:

1. $f(x)=x^2, g(x)=\sqrt{x}$.

2. $f(x)=10^{-x}+2, g(x)=\dfrac{2x-3}{x}$.

3. $f(x)=\dfrac{x^2+1}{x}, g(x)=\dfrac{x \ln x+1}{\ln x}$.

4. $f(x)=\dfrac{2x^2}{x+1}, g(x)=2(x-1-e^{-x})$.

Which groups of curve $y=f(x)$ and $y=g(x)$ has a bipartite asymptote?

How to submit:

Let $p_1, p_2, p_3, p_4$ be the boolean value of the group $1,2,3,4$, if group $k$ has a bipartite asymptote, $p_k=1$, else $p_k=0$.

Then submit $\displaystyle \sum_{k=1}^4 p_k \cdot 2^{k-1}$.

- 4 months ago

Let $A$ be the set of all functions whose range is $\mathbb R$, $B$ is the set of all functions $\phi(x)$ which has the following properties:

• For $\phi(x)$, let $R_0$ denote the range of $\phi(x)$, $\exists M \in (0,+\infty), R_0 \subseteq [-M,M]$.

It's easy to prove that for $\phi_1(x)=x^3$, $\phi_2(x)=\sin x$, $\phi_1(x) \in A$, $\phi_2(x) \in B$.

Here are the following statements:

1. Let $D$ be the domain of $f(x)$, then the necessary and sufficient condition for $f(x) \in A$ is: $\forall b \in \mathbb R, \exists a \in D, f(a)=b$.

2. The necessary and sufficient condition for $f(x) \in B$ is $f(x)$ has the maximum and minimum value.

3. If $f(x), g(x)$ have the same domain, then if $f(x) \in A, g(x) \in B$, then $f(x)+g(x) \notin B$.

4. If $f(x)=a\ln(x+2)+\dfrac{x}{x^2+1}\ (x>-2, a \in \mathbb R)$ has the maximum value, then $f(x) \in B$.

Which statements are true?

How to submit:

Let $p_1, p_2,\cdots,p_n$ be the boolean value of statement $1,2,\cdots,n$, if statement $k$ is true, $p_k=1$, else $p_k=0$.

Then submit $\displaystyle \sum_{k=1}^n p_k \cdot 2^{k-1}$.

- 4 months ago

Given the function $y=f(x)\ (x \in \mathbb R)$, for function $y=g(x)\ (x \in I)$, let's define symmetric function of $g(x)$ respect to $f(x)$ as $y=h(x)\ (x \in I)$, $y=h(x)$ is such that $\forall x \in I$, point $(x,h(x)), (x,g(x))$ are symmetric about point $(x,f(x))$.

GIven that $h(x)$ is the symmetric function of $g(x)=\sqrt{4-x^2}$ respect to $f(x)=3x+b\ (b \in \mathbb R)$, $h(x)>g(x)$ is always true for all $x$ on the domain of $g(x)$, then find the range of $b$.

The range can be expressed as $(L,+\infty)$, submit $L^2$.

- 4 months ago

Here's the definition of harmonically bisect: Given that $A_1, A_2, A_3, A_4$ are four distinct points on the coordinate plane, if $\overrightarrow{A_1A_3}=\lambda \overrightarrow{A_1A_2}$, $\overrightarrow{A_1A_4}=\mu \overrightarrow{A_1A_2}$, $\dfrac{1}{\lambda}+\dfrac{1}{\mu}=2$, then $A_3, A_4$ harmonically bisect $A_1, A_2$.

Given that $C(c,0), D(d,0)\ (c,d \in \mathbb R)$ harmonically bisect $A(0,0), B(1,0)$, which choice is true?

$A.\ \textup{C could be the midpoint of AB.}$

$B.\ \textup{D could be the midpoint of AB.}$

$C.\ \textup{C, D could be both on segment AB.}$

$D.\ \textup{C, D can't be both on the extension line of AB.}$

- 4 months ago

Given that an infinite sequence $\{a_n\}$ consists of $k$ distinct values, $S_n=\displaystyle \sum_{i=1}^n a_i$.

If $\forall n \in \mathbb N^+$, $S_n \in \{2,3\}$, then find the maximum of $k$.

- 4 months ago

If there exists $t \in \mathbb R$, so that the following system of $n$ equations all holds:

$\begin{cases} \lfloor t \rfloor = 1 \\ \lfloor t^2 \rfloor = 2 \\ \lfloor t^3 \rfloor = 3 \\ \cdots \\ \lfloor t^n \rfloor = n \end{cases}$

Then find the maximum of positive integer $n$.

- 4 months ago

For all real number $x,y$, which is always true?

$A.\ \lfloor -x \rfloor=-\lfloor x \rfloor$

$B.\ \lfloor 2x \rfloor=2\lfloor x \rfloor$

$C.\ \lfloor x+y \rfloor \leq \lfloor x \rfloor + \lfloor y \rfloor$

$D.\ \lfloor x-y \rfloor \leq \lfloor x \rfloor - \lfloor y \rfloor$

- 4 months ago

Without using calculator, Find $4 \cos 50 \degree - \tan 40 \degree$.

Let $A$ denote the answer. Submit $\lfloor 1000A \rfloor$.

- 4 months ago

If $\tan \alpha=2 \tan \dfrac{\pi}{5}$, find $\dfrac{\cos(\alpha-\dfrac{3\pi}{10})}{\sin(\alpha-\dfrac{\pi}{5})}$.

Let $A$ denote the answer. Submit $\lfloor 1000A \rfloor$.

- 4 months ago

Without using calculator, Find $\dfrac{\cos 10 \degree}{\tan 20 \degree}+\sqrt{3} \sin 10 \degree \tan 70 \degree - 2 \cos 40 \degree$.

Let $A$ denote the answer. Submit $\lfloor 1000A \rfloor$.

- 4 months ago

Given that $f(x)=\cos 2x, g(x)=\sin x$.

Find all possible real number $a$ and positive integer $n$, so that $f(x)+a g(x)=0$ has exactly $2013$ roots on the interval $(0,n \pi)$.

How to submit:

• First, find the number of all possible solutions $(a,n)$. Let $N$ denote the number of solutions.

• Then sort the solutions by $a$ from smallest to largest, if $a$ is the same, then sort by $n$ from smallest to largest.

• Let the sorted solutions be: $(a_1,n_1), (a_2,n_2), (a_3,n_3), \cdots ,(a_N,n_N)$, then $M=\displaystyle \sum_{k=1}^N k(a_k+n_k)$.

For instance, if the solution is: $(-1,2), (-1,1), (1,3), (0,4)$

Then the sorted solution will be: $(-1,1), (-1,2), (0,4), (1,3)$

Then $N=4$, $M=\displaystyle \sum_{k=1}^4 k(a_k+n_k)= 1 \times (-1+1) + 2 \times (-1+2) + 3 \times (0+4) + 4 \times (1+3) =30$.

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

- 4 months ago

As shown above, the circle has radius $r=1\ m$, $O$ is its center. At $t=0$, $O$ is at $(0,-1)$, and it is moving at $v=1\ m/s$ upwards along the y-axis. Let $x(t)$ be the length of the arc above the x-axis, $f(t)=\cos (x(t))$.

For $0 \leq t \leq 1$, what is the best graph for $f(t)$?

- 4 months ago

$f(x)$ is a function defined at $[0,1]$ such that:

• $f(0)=f(1)=0$.

• $\forall x,y \in [0,1]\ (x \neq y), |f(x)-f(y)| < \dfrac{1}{2} |x-y|$.

If $\forall x,y \in [0,1], |f(x)-f(y)|, find the minimum value of $k$.

Let $K$ be the minimum value. Submit $\lfloor 1000K \rfloor$.

- 4 months ago

Given that $\{a_n\}$ is a Geometric Sequence, its common ratio $q=\sqrt{2}$, $S_n=\displaystyle \sum_{k=1}^n a_k$.

Let $T_n=\dfrac{17 S_n - S_{2n}}{a_{n+1}}\ (n \in \mathbb N^+)$. If $T_{m}$ is the maximum term of sequence $\{T_n\}$, find $m$.

- 4 months ago

$\{a_n\}$ is a sequence such that $a_1=m\ (m \in \mathbb N^+)$, $a_{n+1}=\begin{cases} \dfrac{a_n}{2} ,\ a_n \equiv 0 \pmod{2} \\ 3 a_n+1 ,\ a_n \equiv 1 \pmod{2} \end{cases}$ If $a_6=1$, find the sum of all possible value(s) for $m$.

- 4 months ago

Given that $a_n=n^2(\cos^2 \dfrac{n\pi}{3}-\sin^2 \dfrac{n\pi}{3})\ (n \in \mathbb N^+)$, let $S_n=\displaystyle \sum_{k=1}^{n} a_k$.
$b_n=\dfrac{S_{3n}}{n \cdot 4^n}\ (n \in \mathbb N^+)$, $T_n=\displaystyle \sum_{k=1}^{n} b_k$.
$T_{10} = \dfrac{p}{q}$, where $p,q$ are positive coprime integers.
Submit $\lfloor p-q+2(S_{100}+S_{201}+S_{302}) \rfloor$