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

  • Your brilliant mind

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

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SAT1000 - P988

Given that e1^,e2^\hat{e_1}, {\hat{e_2}} are unit vectors, and b=xe1^+ye2^ (x,yR)\textbf b = x \hat{e_1} + y \hat{e_2}\ (x,y \in \mathbb R) is a nonzero vector.

If <e1^,e2^> =π6<\hat{e_1},\hat{e_2}>\ = \dfrac{\pi}{6}, then find the maximum value of xb\dfrac{|x|}{|\textbf b|}.

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

Alice Smith - 3 months, 1 week ago

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SAT1000 - P966

If function f(x)=x13sin2x+asinxf(x)=x-\dfrac{1}{3} \sin 2x + a \sin x is monotonic increasing for all xRx \in \mathbb R, what's the range of aa?

A. [1,1]A.\ [-1,1]

B. [1,13]B.\ [-1, \dfrac{1}{3}]

C. [13,13]C.\ [-\dfrac{1}{3}, \dfrac{1}{3}]

D. [1,13]D.\ [-1, -\dfrac{1}{3}]

Alice Smith - 3 months, 1 week ago

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SAT1000 - P965

In an acute triangle ABCABC, tanA=12\tan A = \dfrac{1}{2}, DD is a point on BCBC, [ABD]=2,[ACD]=4[ABD]=2, [ACD]=4.

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

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

Alice Smith - 3 months, 1 week ago

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SAT1000 - P964

Let function f(x)=2xcosxf(x)=2x - \cos x, {an}\{a_n\} is an arithmetic sequence whose common difference is π8\dfrac{\pi}{8}.

If k=15f(ak)=5π\displaystyle \sum_{k=1}^{5} f(a_k) = 5\pi, find f2(a3)a1a5f^2(a_3)-a_1 a_5.

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

Alice Smith - 3 months, 1 week ago

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SAT1000 - P941

As shown above, a small circle with diameter 11 is rotating counterclockwise along the interior side of the big circle with diameter 22 without slipping, M,NM,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,NM,N?

Alice Smith - 3 months, 1 week ago

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SAT1000 - P939

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

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

cosα13cosα2+α33sinα13sinα2+α33\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 AA denote the value. Submit 1000A\lfloor 1000A \rfloor.

Alice Smith - 3 months, 1 week ago

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SAT1000 - P937

66 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 66 classmates have performed 1313 rounds of exchange, what's the number of classmates that have received exactly 44 souvenirs?

A. 1 or 3A.\ \textup{1 or 3}

B. 1 or 4B.\ \textup{1 or 4}

C. 2 or 3C.\ \textup{2 or 3}

D. 2 or 4D.\ \textup{2 or 4}

Alice Smith - 3 months, 1 week ago

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SAT1000 - P936

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

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

Alice Smith - 3 months, 1 week ago

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SAT1000 - P935

There are three treasure chests which contains:

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

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

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

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

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

Alice Smith - 3 months, 1 week ago

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SAT1000 - P934

Four girls Alice,Betty,Cathy,Dora\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 22 of them wear black hats and the others wear red hats. After that, they decide to play hide-and-seek under the light.

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

Suddenly, the light goes out, and Alice\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. Betty can know the colors of all four girl’s hats.A.\ \textup{Betty can know the colors of all four girl's hats.}

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

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

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

Alice Smith - 3 months, 1 week ago

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SAT1000 - P933

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

Given that A1(3,20),A2(10,0),A3(14,0)A_1(3,20), A_2(-10,0), A_3(14,0), we want to find a point P(x0,y0) (y00)P(x_0,y_0)\ (y_0 \geq 0) so that k=13d(P,Ak)\displaystyle \sum_{k=1}^{3} d(P,A_k) has the minimum value.

However, the Taxicab path from PP to each point can't pass through the region: x2+y2<1x^2+y^2<1.

Then find the coordinates of PP.

If PP has coordinate (x0,y0)(x_0,y_0), submit 2y0x02y_0-x_0.

Alice Smith - 3 months, 1 week ago

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SAT1000 - P932

In the rectangular coordinate plane, the Taxicab distance from point P1(x1,y1)P_1(x_1,y_1) to P2(x2,y2)P_2(x_2,y_2) is defined as: d(P1,P2)=x1x2+y1y2d(P_1,P_2)=|x_1-x_2|+|y_1-y_2|

Given these integer points: A1(2,2),A2(3,1),A3(3,4),A4(2,3),A5(4,5)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(x0,y0)P(x_0,y_0) so that k=15d(P,Ak)\displaystyle \sum_{k=1}^{5} d(P,A_k) has the minimum value.

Submit 2y0x02y_0-x_0.

Alice Smith - 3 months, 1 week ago

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SAT1000 - P931

As shown above, from top and bottom, l1,l2,l3l_1, l_2, l_3 are three parallel lines on the same plane, and the distance from l1l_1 to l2l_2 is 11, the distance from l2l_2 to l3l_3 is 22, and point A,B,CA,B,C are on l1,l2,l3l_1, l_2, l_3 respectively.

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

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

Alice Smith - 3 months, 2 weeks ago

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SAT1000 - P929

Given that Sn=k=1nsinkπ7 (nN+)S_n=\displaystyle \sum_{k=1}^{n} \sin \dfrac{k \pi}{7} \ (n \in \mathbb N^+)

Then how many positive numbers are there for S1,S2,,S100S_1,S_2, \cdots, S_{100}?

Alice Smith - 3 months, 2 weeks ago

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SAT1000 - P928

Five girls Alice, Betty, Cathy, Dora, Emily\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 11, the second next to the right hand side yells the number 11, too, from then on, the girl next to the right hand side of the fromer yells the sum of the numbers of the last 22 students.

  • If the number is divisible by 33, the girl who yells it needs to clap her hands once.

Given that Alice\textup{Alice} is the first to yell, how many times should she clap when they have yelled the 100 th100\ th number?

Alice Smith - 3 months, 2 weeks ago

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SAT1000 - P926

Let {bn}\{b_n\} be the sequence of all triangular numbers that are divisible by 55, ordering from smallest to largest.

Then compute b2k1b_{2k-1} at k=2020k=2020.

Alice Smith - 3 months, 2 weeks ago

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SAT1000 - P925

Given that: cos10α=a1cos10α+a2cos8α+a3cos6α+a4cos4α+a5cos2α+a6\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 a1a4+a5a_1-a_4+a_5.

Alice Smith - 3 months, 2 weeks ago

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SAT1000 - P924

Given that:

i=1nik=ak+1nk+1+aknk+ak1nk1+ak2nk2++a1n+a0\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 107(ak+1+ak+ak1+ak2)\lfloor 10^7 (a_{k+1}+a_k+a_{k-1}+a_{k-2}) \rfloor at k=2020k=2020.

Alice Smith - 3 months, 2 weeks ago

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SAT1000 - P923

Find the value of k=12n(sinkπ2n+1)2\displaystyle \sum_{k=1}^{2n} (\sin \dfrac{k \pi}{2n+1})^{-2} at n=2020n=2020.

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

Alice Smith - 3 months, 2 weeks ago

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SAT1000 - P915

Let N=2n (nN+,n2)N=2^n\ (n \in \mathbb N^+, n \geq 2), NN distinct numbers denoted as x1,x2,,xNx_1,x_2,\cdots,x_N are put into NN positions labeled 1,2,,N1,2,\cdots,N, then we will get the permutation P0=x1x2xNP_0=x_1 x_2 \cdots x_N.

The transform CC on the permutation PP is as follows:

  • Separate the terms on the odd and even positions, and put them to the first N2\dfrac{N}{2}, last N2\dfrac{N}{2} positions respectively, keeping the original order.

If we apply CC to P0P_0 once, we will get P1=x1x3xN1x2x4xNP_1=x_1 x_3 \cdots x_{N-1} x_2 x_4 \cdots x_N.

After that, we divide P1P_1 into 22 consecutive segments consisting of N2\dfrac{N}{2} numbers, and apply CC to each segment, we will get P2P_2.

For example, if n=3, N=8n=3,\ N=8, we will get P2=x1x5x3x7x2x6x4x8P_2=x_1 x_5 x_3 x_7 x_2 x_6 x_4 x_8, at this time, x7x_7 is on 4 th4\ th position of P2P_2.

From then on, when 2in22 \leq i \leq n-2, we divide PiP_i into 2i2^i consecutive segments consisting of N2i\dfrac{N}{2^i} numbers, and apply CC to each segment to get Pi+1P_{i+1}.

Then when n=32, N=232n=32,\ N=2^{32}, x173x_{173} is on the M thM\ th position of P4P_4. Submit MM.

Alice Smith - 3 months, 3 weeks ago

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SAT1000 - P914

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

  • {an}\{a_n\} has 2m2m terms (mN+)(m \in \mathbb N^+).

  • Exactly mm terms are 00 and mm terms are 11.

  • k2m (kN+)\forall k \geq 2m\ (k \in \mathbb N^+), the number of 00's is always greater or equal to the number of 11's for subsequence a1,a2,,aka_1,a_2,\cdots,a_k.

For m=2020m=2020, the number of such 0-1 regular sequences is MM. Submit log2M\lfloor \log_2 M \rfloor.

Alice Smith - 3 months, 3 weeks ago

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SAT1000 - P911

For set E={a1,a2,,a100}E=\{a_1,a_2,\cdots,a_{100}\}, whose subset is X={ai1,ai2,,aik}X=\{a_{i_1},a_{i_2},\cdots,a_{i_k}\}, let‘s define the characteristic sequence of XX as: x1,x2,,x100x_1,x_2,\cdots,x_{100}, where xi1=xi2==xik=1x_{i_1}=x_{i_2}=\cdots=x_{i_k}=1, and the other terms are all 00.

For instance, the characteristic sequence of {a2,a3}\{a_2,a_3\} is 0,1,1,0,0,,00,1,1,0,0,\cdots,0.

If P,QP,Q are subsets of EE, and PP has characteristic sequence: p1,p2,,p100p_1,p_2,\cdots,p_{100} such that p1=1, pi+pi+1=1, 1i99p_1=1,\ p_i+p_{i+1}=1,\ 1 \leq i \leq 99, QQ has characteristic sequence: q1,q2,,q100q_1,q_2,\cdots,q_{100} such that q1=1, qj+qj+1+qj+2=1, 1j98q_1=1,\ q_{j}+q_{j+1}+q_{j+2}=1,\ 1 \leq j \leq 98, then find the cardinality of PQP \cap Q.

Alice Smith - 3 months, 3 weeks ago

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SAT1000 - P909

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

  • T={f(x)xS}T= \{f(x)|x \in S\}.

  • x1,x2S\forall x_1, x_2 \in S, when x1<x2x_1<x_2, f(x1)<f(x2)f(x_1)<f(x_2).

Then SS and TT are Order Isomorphic.

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

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

B. A={x1x3},B={xx=80<x10}B.\ A=\{x|-1 \leq x \leq 3\}, B=\{x|x=-8 \vee 0 < x \leq 10\}

C. A={x0<x<1},B=RC.\ A=\{x|0<x<1\}, B=\mathbb R

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

Alice Smith - 3 months, 3 weeks ago

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SAT1000 - P904

If we define S={al1,al2,,aln}S=\{a_{l_1},a_{l_2}, \cdots ,a_{l_n}\} as the k thk\ th subset for set E={a1,a2,,a10}E=\{a_1,a_2,\cdots,a_{10}\}, where k=i=1n2li1k=\displaystyle \sum_{i=1}^{n} 2^{l_{i}-1}.

Then what's the 211 th211\ th subset for EE?

How to submit:

Sort the indices of the elements of SS from smallest to largest and put them together. For example, if S={a1,a2,a3}S=\{a_1,a_2,a_3\}, then submit 123123, and when S={a1,a2,a10}S=\{a_1,a_2,a_{10}\}, then submit 12101210.

Alice Smith - 3 months, 3 weeks ago

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SAT1000 - P907

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?

Alice Smith - 3 months, 3 weeks ago

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SAT1000 - P901

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 τ\tau.

As shown above, τ1,τ2,τ3,τ4\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 τ2<τ1<τ3<τ4\tau_2<\tau_1<\tau_3<\tau_4, submit 21342134.

Alice Smith - 3 months, 3 weeks ago

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SAT1000 - P900

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 thk\ th tree is planted at Pk(xk,yk)P_k(x_k,y_k), where x1=1,y1=1x_1=1, y_1=1, and when k2k \geq 2:

{xk=xk1+15(k15k25)yk=yk1+k15k25\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 th2008\ th tree?

Let the coordinate be x0,y0x_0,y_0. Submit 2y0x02y_0-x_0.

Alice Smith - 3 months, 3 weeks ago

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SAT1000 - P897

In the coordinate plane, the Hamilton Distance from point P1(x1,y1)P_1(x_1,y_1) to P2(x2,y2)P_2(x_2,y_2) is defined as: d(P1,P2)=x1x2+y1y2d(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 CC is on segment ABAB, then d(A,C)+d(C,B)=d(A,B)d(A,C)+d(C,B)=d(A,B).

2.In ABC\triangle ABC, if C=90°\angle C=90 \degree, then d2(A,C)+d2(C,B)=d2(A,B)d^2(A,C)+d^2(C,B)=d^2(A,B) (d2(A,B)=(d(A,B))2)(d^2(A,B)=(d(A,B))^2).

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

How to submit:

Let p1,p2,,pnp_1, p_2,\cdots,p_n be the boolean value of statement 1,2,,n1,2,\cdots,n, if statement kk is true, pk=1p_k=1, else pk=0p_k=0.

Then submit k=1npk2k1\displaystyle \sum_{k=1}^n p_k \cdot 2^{k-1}.

Alice Smith - 3 months, 3 weeks ago

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SAT1000 - P896

In the coordinate plane, the Hamilton Distance from point P1(x1,y1)P_1(x_1,y_1) to P2(x2,y2)P_2(x_2,y_2) is defined as: d(P1,P2)=x1x2+y1y2d(P_1,P_2)=|x_1-x_2|+|y_1-y_2| If F1,F2F_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 PP such that d(P,F1)+d(P,F2)=Cd(P,F_1)+d(P,F_2)=C (CC is constant and C>F1F2C>|F_1 F_2|)?

A.A.

B.B.

C.C.

D.D.

Alice Smith - 3 months, 3 weeks ago

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SAT1000 - P894

Imagine an equilateral ABC\triangle ABC divided into n2(n2,nN+)n^2 (n \geq 2, n \in \mathbb N^+) congruent equilateral pieces. (The image above shows n=4n=4 case).

Let's put each number on every vertex of all triangles, so that the numbers on the same side of ABC\triangle ABC or on the same line parallel to the sides of ABC\triangle ABC (if more than 33 numbers) form Arithmetic Progressions.

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

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

Alice Smith - 3 months, 4 weeks ago

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SAT1000 - P893

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

For example, if {an}\{a_n\} is 1,2,3,,n1,2,3,\cdots,n, then {(an)}\{(a_n)^*\} will be: 0,1,2,,n10,1,2,\cdots,n-1.

Given that nN+\forall n \in \mathbb N^+, an=n2a_n=n^2. If we define sequence {bn}\{b_n\} as {((an))}\{((a_{n})^*)^*\}, find b2020b_{2020}.

Alice Smith - 3 months, 4 weeks ago

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SAT1000 - P889

Let PP be an arbitrary point in ABC\triangle ABC, λ1=SPBCSABC\lambda_1 = \dfrac{S_{\triangle PBC}}{S_{\triangle ABC}},λ2=SPCASABC\lambda_2 = \dfrac{S_{\triangle PCA}}{S_{\triangle ABC}}, λ3=SPABSABC\lambda_3 = \dfrac{S_{\triangle PAB}}{S_{\triangle ABC}}.

Define f:R2R3f: \mathbb R^2 \mapsto \mathbb R^3, f(P)=(λ1,λ2,λ3)f(P)=(\lambda_1,\lambda_2,\lambda_3).

If point GG is the centroid of ABC\triangle ABC, f(Q)=(12,13,16)f(Q)=(\dfrac{1}{2},\dfrac{1}{3},\dfrac{1}{6}), then which choice is true?

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

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

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

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

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

Alice Smith - 3 months, 4 weeks ago

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SAT1000 - P888

As shown above, in cuboid ABCDA1B1C1D1ABCD-A_1B_1C_1D_1, AB=11,AD=7,AA1=12AB=11, AD=7, AA_1=12. The faces of the cuboid are all mirrors.

A ray is emitted from A(0,0,0)A(0,0,0) to E(4,3,12)E(4,3,12) and reflected when meeting the faces of the cuboid following the reflection law. Let LiL_i denote the length of the ray between the (i1) th(i-1)\ th reflection to the i thi\ th reflection (i=2,3,4)(i=2,3,4), L1=AEL_1=AE.

Compare L1,L2,L3,L4L_1,L_2,L_3,L_4.

A. L1=L2>L3=L4A.\ L_1=L_2>L_3=L_4

B. L1=L2=L3=L4B.\ L_1=L_2=L_3=L_4

C. L1=L2>L3<L4C.\ L_1=L_2>L_3<L_4

D. L1=L2>L3>L4D.\ L_1=L_2>L_3>L_4

Alice Smith - 3 months, 4 weeks ago

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SAT1000 - P887

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

A ray is emitted from PP and it is reflected at QQ on BCBC, RR on ACAC and returned to point PP again.

If ray QRQR passes through the centroid of ABC\triangle ABC, find the length of APAP.

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

Alice Smith - 3 months, 4 weeks ago

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SAT1000 - P886

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

A ray is emitted from P0(1,0)P_0(1,0) at angle θ\theta with ABAB and reflected at P1P_1 on BCBC, P2P_2 on CDCD, P3P_3 on DADA, P4P_4 on ABAB following the law of reflection.

If P4P_4 has coordinate (x4,0)(x_4,0) and x4(1,2)x_4 \in (1,2), find the range of tanθ\tan \theta.

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

Alice Smith - 3 months, 4 weeks ago

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SAT1000 - P884

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

Given that for set T,UT,U: TZ,VZ,TV=,TV=ZT \subseteq \mathbb Z, V \subseteq \mathbb Z, T \cap V = \emptyset , T \cup V = \mathbb Z, and a,b,cT,abcT\forall a,b,c \in T, abc \in T, x,y,zV,xyzV\forall x,y,z \in V, xyz \in V, then which of the choices is true?

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

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

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

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

Alice Smith - 4 months ago

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SAT1000 - P880

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

m(0,+),x0D,xDx>x0,0<f(x)h(x)<m,0<h(x)g(x)<m,\forall m \in (0,+\infty), \exists x_0 \in D, \forall x \in D \wedge x>x_0, 0<f(x)-h(x)<m, 0<h(x)-g(x)<m,

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

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

  1. f(x)=x2,g(x)=xf(x)=x^2, g(x)=\sqrt{x}.

  2. f(x)=10x+2,g(x)=2x3xf(x)=10^{-x}+2, g(x)=\dfrac{2x-3}{x}.

  3. f(x)=x2+1x,g(x)=xlnx+1lnxf(x)=\dfrac{x^2+1}{x}, g(x)=\dfrac{x \ln x+1}{\ln x}.

  4. f(x)=2x2x+1,g(x)=2(x1ex)f(x)=\dfrac{2x^2}{x+1}, g(x)=2(x-1-e^{-x}).

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

How to submit:

Let p1,p2,p3,p4p_1, p_2, p_3, p_4 be the boolean value of the group 1,2,3,41,2,3,4, if group kk has a bipartite asymptote, pk=1p_k=1, else pk=0p_k=0.

Then submit k=14pk2k1\displaystyle \sum_{k=1}^4 p_k \cdot 2^{k-1}.

Alice Smith - 4 months ago

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SAT1000 - P879

Let AA be the set of all functions whose range is R\mathbb R, BB is the set of all functions ϕ(x)\phi(x) which has the following properties:

  • For ϕ(x)\phi(x), let R0R_0 denote the range of ϕ(x)\phi(x), M(0,+),R0[M,M]\exists M \in (0,+\infty), R_0 \subseteq [-M,M].

It's easy to prove that for ϕ1(x)=x3\phi_1(x)=x^3, ϕ2(x)=sinx\phi_2(x)=\sin x, ϕ1(x)A\phi_1(x) \in A, ϕ2(x)B\phi_2(x) \in B.

Here are the following statements:

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

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

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

  4. If f(x)=aln(x+2)+xx2+1 (x>2,aR)f(x)=a\ln(x+2)+\dfrac{x}{x^2+1}\ (x>-2, a \in \mathbb R) has the maximum value, then f(x)Bf(x) \in B.

Which statements are true?

How to submit:

Let p1,p2,,pnp_1, p_2,\cdots,p_n be the boolean value of statement 1,2,,n1,2,\cdots,n, if statement kk is true, pk=1p_k=1, else pk=0p_k=0.

Then submit k=1npk2k1\displaystyle \sum_{k=1}^n p_k \cdot 2^{k-1}.

Alice Smith - 4 months ago

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SAT1000 - P878

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

GIven that h(x)h(x) is the symmetric function of g(x)=4x2g(x)=\sqrt{4-x^2} respect to f(x)=3x+b (bR)f(x)=3x+b\ (b \in \mathbb R), h(x)>g(x)h(x)>g(x) is always true for all xx on the domain of g(x)g(x), then find the range of bb.

The range can be expressed as (L,+)(L,+\infty), submit L2L^2.

Alice Smith - 4 months ago

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SAT1000 - P877

Here's the definition of harmonically bisect: Given that A1,A2,A3,A4A_1, A_2, A_3, A_4 are four distinct points on the coordinate plane, if A1A3=λA1A2\overrightarrow{A_1A_3}=\lambda \overrightarrow{A_1A_2}, A1A4=μA1A2\overrightarrow{A_1A_4}=\mu \overrightarrow{A_1A_2}, 1λ+1μ=2\dfrac{1}{\lambda}+\dfrac{1}{\mu}=2, then A3,A4A_3, A_4 harmonically bisect A1,A2A_1, A_2.

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

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

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

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

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

Alice Smith - 4 months ago

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SAT1000 - P876

Given that an infinite sequence {an}\{a_n\} consists of kk distinct values, Sn=i=1naiS_n=\displaystyle \sum_{i=1}^n a_i.

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

Alice Smith - 4 months ago

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SAT1000 - P873

If there exists tRt \in \mathbb R, so that the following system of nn equations all holds:

{t=1t2=2t3=3tn=n\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 nn.

Alice Smith - 4 months ago

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SAT1000 - P866

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

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

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

C. x+yx+yC.\ \lfloor x+y \rfloor \leq \lfloor x \rfloor + \lfloor y \rfloor

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

Alice Smith - 4 months ago

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SAT1000 - P277

Without using calculator, Find 4cos50°tan40°4 \cos 50 \degree - \tan 40 \degree.

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

Alice Smith - 4 months ago

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SAT1000 - P276

If tanα=2tanπ5\tan \alpha=2 \tan \dfrac{\pi}{5}, find cos(α3π10)sin(απ5)\dfrac{\cos(\alpha-\dfrac{3\pi}{10})}{\sin(\alpha-\dfrac{\pi}{5})}.

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

Alice Smith - 4 months ago

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SAT1000 - P275

Without using calculator, Find cos10°tan20°+3sin10°tan70°2cos40°\dfrac{\cos 10 \degree}{\tan 20 \degree}+\sqrt{3} \sin 10 \degree \tan 70 \degree - 2 \cos 40 \degree.

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

Alice Smith - 4 months ago

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SAT1000 - P333

Given that f(x)=cos2x,g(x)=sinxf(x)=\cos 2x, g(x)=\sin x.

Find all possible real number aa and positive integer nn, so that f(x)+ag(x)=0f(x)+a g(x)=0 has exactly 20132013 roots on the interval (0,nπ)(0,n \pi).

How to submit:

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

  • Then sort the solutions by aa from smallest to largest, if aa is the same, then sort by nn from smallest to largest.

  • Let the sorted solutions be: (a1,n1),(a2,n2),(a3,n3),,(aN,nN)(a_1,n_1), (a_2,n_2), (a_3,n_3), \cdots ,(a_N,n_N), then M=k=1Nk(ak+nk)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)(-1,2), (-1,1), (1,3), (0,4)

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

Then N=4N=4, M=k=14k(ak+nk)=1×(1+1)+2×(1+2)+3×(0+4)+4×(1+3)=30M=\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 M+N\lfloor M+N \rfloor.

Alice Smith - 4 months ago

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SAT1000 - P329

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

For 0t10 \leq t \leq 1, what is the best graph for f(t)f(t)?

Alice Smith - 4 months ago

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SAT1000 - P153

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

  • f(0)=f(1)=0f(0)=f(1)=0.

  • x,y[0,1] (xy),f(x)f(y)<12xy\forall x,y \in [0,1]\ (x \neq y), |f(x)-f(y)| < \dfrac{1}{2} |x-y|.

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

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

Alice Smith - 4 months ago

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SAT1000 - P461

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

Let Tn=17SnS2nan+1 (nN+)T_n=\dfrac{17 S_n - S_{2n}}{a_{n+1}}\ (n \in \mathbb N^+). If TmT_{m} is the maximum term of sequence {Tn}\{T_n\}, find mm.

Alice Smith - 4 months ago

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SAT1000 - P500

{an}\{a_n\} is a sequence such that a1=m (mN+)a_1=m\ (m \in \mathbb N^+), an+1={an2, an0(mod2)3an+1, an1(mod2)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 a6=1a_6=1, find the sum of all possible value(s) for mm.

Alice Smith - 4 months ago

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SAT1000 - P496

Given that an=n2(cos2nπ3sin2nπ3) (nN+)a_n=n^2(\cos^2 \dfrac{n\pi}{3}-\sin^2 \dfrac{n\pi}{3})\ (n \in \mathbb N^+), let Sn=k=1nakS_n=\displaystyle \sum_{k=1}^{n} a_k.

bn=S3nn4n (nN+)b_n=\dfrac{S_{3n}}{n \cdot 4^n}\ (n \in \mathbb N^+), Tn=k=1nbkT_n=\displaystyle \sum_{k=1}^{n} b_k.

T10=pqT_{10} = \dfrac{p}{q}, where p,qp,q are positive coprime integers.

Submit pq+2(S100+S201+S302)\lfloor p-q+2(S_{100}+S_{201}+S_{302}) \rfloor