Reach for the Summit problem set - Mathematics

Welcome to the road to Olympics! By this problem set, you will lead your way to the summit of mathematics, just like climbing mountain in Celeste.

To match the Celeste scheme, each section will be seperated into A,B,C sides, each of which has lots of problems, having the similar format. The difficulty of the problems may be randomly shuffled, though.

I'll use this problem set for my games or programs later on, and I will upgrade it frequently.

Problem Format: For example, if the problem is on Stage 1, A side and 5th position, then the name of the problem will be:

Reach for the Summit - M-S1-A5

Table of Contents:

Stage 1:

  • Sets and PIE

  • Concepts, Properties, Graphs of Functions

  • Derivatives and Extrema

Stage 2:

  • Trigonometric Functions

  • Trigonometric Identities

Stage 3:

  • Sequences

Stage 4:

  • Complex Numbers

Stage 5:

  • Euclidean Geometry

Stage 6:

  • Lines

  • Quadratic Curves

Stage 7:

  • Solid Geometry

  • Vectors

Stage 8:

  • Inequalities

  • Geometric Inequalities

Stage 9:

  • Functional Equations

  • Polynomials

Stage 10:

  • Combinatorics

  • Combinatorics and Counting

  • Combinatoric Identities

  • Probability

Stage 11:

  • Divisibility

  • Modular Arithmetic

Stage 12:

  • Number Bases

  • Floor and Ceiling Functions

  • Diophantine Equations

Stage 13:

  • Integer Points

Stage 14:

  • Coloring Problems

Stage 15:

  • Existence Problems

  • Combinatoric Geometry

Stage 16:

  • Graph Theory

Note by Alice Smith
3 weeks, 4 days ago

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Reach for the Summit - M-S1-A1

Let set A={xx=a2+b2,a,bZ}A=\{x|x=a^2+b^2, a,b \in \mathbb Z\}.

If x1,x2Ax_1,x_2 \in A, is it always true that x1x2Ax_1 \cdot x_2 \in A?

Alice Smith - 3 weeks, 4 days ago

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Reach for the Summit - M-S1-A2

Given that a,bR, a>0a,b \in \mathbb R,\ a>0, if 4a3ab=16, log2a=a+1b4^a-3a^b=16,\ \log_{2}a=\dfrac{a+1}{b}, find the sum of all possible value(s) of aba^b.

Alice Smith - 3 weeks, 4 days ago

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Reach for the Summit - M-S1-A3

Find the monotonic increasing interval for function y=log0.5(x2+4x+4)y=\log_{0.5}(x^2+4x+4).

Alice Smith - 3 weeks, 4 days ago

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Reach for the Summit - M-S1-A4

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

Alice Smith - 3 weeks, 4 days ago

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Reach for the Summit - M-S2-A1

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

Alice Smith - 3 weeks, 4 days ago

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Reach for the Summit - M-S2-A2

Without using the calculator, find the value of:

(sec50°+tan10°)(cos2π7+cos4π7+cos6π7)(tan6°tan42°tan66°tan78°)(\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 AA denote the value. Submit 10000A\lfloor 10000A \rfloor.

Alice Smith - 3 weeks, 4 days ago

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Reach for the Summit - M-S3-A1

Given a positive infinite sequence {an}\{a_n\}, Sn=k=1nakS_n = \displaystyle \sum_{k=1}^{n} a_k.

If nN+\forall n \in \mathbb N^+, the arithmetic mean of ana_n and 22 is equal to the geometric mean of SnS_n and 22, then find a2020a_{2020}.

Alice Smith - 2 weeks, 2 days ago

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Reach for the Summit - M-S3-A2

In sequence {an}\{a_n\}, a1=1,a2=3,a3=6a_1=1,a_2=3,a_3=6, for n4, an=3an1an22an3n \geq 4,\ a_n=3a_{n-1}-a_{n-2}-2a_{n-3}.

Then n4\forall n \geq 4, an>λ×2n2 (λN+)a_n>\lambda \times 2^{n-2}\ (\lambda \in \mathbb N^+), then find the maximum value of λ\lambda.

Alice Smith - 2 weeks, 2 days ago

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Reach for the Summit - M-S4-A1

If zz is one of the 7th7^{th} roots of unity, z1z \neq 1, find (z+z2+z4)2(|z+z^2+z^4|)^2.

Alice Smith - 2 weeks, 2 days ago

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Reach for the Summit - M-S4-A2

If three complex roots of the equation: x3+px+1=0x^3+px+1=0 form an equilateral triangle on the complex plane, then find the area of that triangle.

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

Alice Smith - 2 weeks, 2 days ago

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Reach for the Summit - M-S4-A3

Find the value of:

(k=01010(1)k(20202k))2+(k=01009(1)k(20202k+1))2(mod998244353)\displaystyle (\sum_{k=0}^{1010} (-1)^k {2020 \choose 2k} )^2 + (\sum_{k=0}^{1009} (-1)^k {2020 \choose 2k+1} )^2 \pmod {998244353}

Alice Smith - 2 weeks, 2 days ago

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Reach for the Summit - M-S5-A1

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

If the circumcenter of AMN\triangle AMN is D(x1,y1)D(x_1,y_1), the circumcenter of ABC\triangle ABC is E(x2,y2)E(x_2,y_2),

Find 1000(x1+2y1+3x2+4y2)\lfloor 1000(x_1+2y_1+3x_2+4y_2)\rfloor.

Hint: Find the relationship of ADAD and AEAE.

Alice Smith - 1 week, 4 days ago

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Reach for the Summit - M-S5-A2

Find the smallest and largest inscribed equilateral triangle in an 1×11 \times 1 square.

If the smallest triangle has area SminS_{min}, the largest has area SmaxS_{max}, submit 106(SmaxSmin)\lfloor 10^6 (S_{max}-S_{min})\rfloor.

Alice Smith - 1 week, 4 days ago

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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?

Alice Smith - 1 week, 4 days ago

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Reach for the Summit - M-S6-A1

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

How to submit:

  • First, find the number of all possible solutions (x,y)(x,y). Let NN denote the number of solutions.
  • Then sort the solutions by xx from smallest to largest, if xx is the same, then sort by yy from smallest to largest.
  • Let the sorted solutions be: (x1,y1),(x2,y2),(x3,y3),,(xN,yN)(x_1,y_1), (x_2,y_2), (x_3,y_3), \cdots ,(x_N,y_N), then M=k=1Nk(xk+yk)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)(-1,2), (-1,1), (1,3), (0,4), 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 and M=k=14k(xk+yk)=1×(1+1)+2×(1+2)+3×(0+4)+4×(1+3)=30\\ 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 M+N\lfloor M+N \rfloor.

Alice Smith - 1 week, 4 days ago

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Reach for the Summit - M-S6-A2

As shown above, line ll is tangent to curve C:x2+y22x2y+1=0C:x^2+y^2-2x-2y+1=0 and it intersects with y-axis at point AA, x-axis at point BB, OO is the origin, OA>2,OB>2|OA|>2, |OB|>2. Then find the minimum area for AOB\triangle AOB.

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

Alice Smith - 1 week, 4 days ago

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Reach for the Summit - M-S7-A1

If point A,B,C,DA,B,C,D are in the 3D3D space, ABC=BCD=CDA=DAB=90°\angle ABC = \angle BCD = \angle CDA = \angle DAB = 90 \degree, is it always true that A,B,C,DA,B,C,D are on the same plane?

Alice Smith - 1 week, 4 days ago

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