MathDB

2017 Vietnamese Southern Summer School contest

Part of Vietnamese Southern Summer School

Subcontests

(4)
Problem 4
3

A nice way to color the square board

In a square board of size 1001 x 1001, we color some mm cells in such a way that: i. Of any two cells that share an edge, at least one is colored. ii. Of any 6 consecutive cells in a column or a row, at least 2 consecutive ones are colored. Determine the smallest possible value of mm.

With Geogebra, geometry is easy

Let ABCABC be a triangle. A point PP varies inside BCBC. Let Q,RQ, R be the points on AC,ABAC, AB in that order, such that PQ∥AB,PR∥ACPQ\parallel AB, PR\parallel AC. 1. Prove that, when PP varies, the circumcircle of triangle AQRAQR always passes through a fixed point XX other than AA. 2. Extend AXAX so that it cuts the circumcircle of ABCABC a second time at point KK. Prove that AX=XKAX=XK.

Funny group of students

In a summer school, there are n>4n>4 students. It is known that, among these students, i. If two ones are friends, then they don't have any common friends. ii If two ones are not friends, then they have exactly two common friends. 1. Prove that 8n−78n-7 must be a perfect square. 2. Determine the smallest possible value of nn.
Problem 3
3

Kissing circles

Let ABCABC be a triangle with right angle ACBACB. Denote by FF the projection of CC on ABAB. A circle ω\omega touches FBFB at point PP, touches CFCF at point QQ, and the circumcircle of ABCABC at point RR. Prove that the points A,Q,RA, Q, R all lie on the same line and AP=ACAP=AC.

3-addict valuation

Prove that, for any integer n≥2n\geq 2, there exists an integer xx such that 3n∣x3+20173^n|x^3+2017, but 3n+1∤x3+20173^{n+1}\not | x^3+2017.

A lot of symmetry

Let ω\omega be a circle with center OO and a non-diameter chord BCBC of ω\omega. A point AA varies on ω\omega such that ∠BAC<90∘\angle BAC<90^{\circ}. Let SS be the reflection of OO through BCBC. Let TT be a point on OSOS such that the bisector of ∠BAC\angle BAC also bisects ∠TAS\angle TAS. 1. Prove that TB=TC=TOTB=TC=TO. 2. TB,TCTB, TC cut ω\omega the second times at points E,FE, F, respectively. AE,AFAE, AF cut BCBC at M,NM, N, respectively. Let SMSM intersects the tangent line at CC of ω\omega at XX, SNSN intersects the tangent line at BB of ω\omega at YY. Prove that the bisector of ∠BAC\angle BAC also bisects ∠XAY\angle XAY.
Problem 2
3

Simple cyclic inequality with simple constraint

Let a,b,ca,b,c be the positive real numbers satisfying a2+b2+c2=3a^2+b^2+c^2=3. Prove that: ab(a+c)+bc(b+a)+ca(c+b)≥32.\frac{a}{b(a+c)}+\frac{b}{c(b+a)}+\frac{c}{a(c+b)}\geq \frac{3}{2}.

Functional equations over the reals

Find all functions f:R↦Rf:\mathbb{R}\mapsto \mathbb{R} satisfy: f(x2)+f(xy)=f(x)f(y)+yf(x)+xf(x+y)f(x^2)+f(xy)=f(x)f(y)+yf(x)+xf(x+y) for all real numbers x,yx,y.

Roots of three-degree polynomials

Let P,QP,Q be the polynomials: x3−4x2+39x−46,x3+3x2+4x−3,x^3-4x^2+39x-46, x^3+3x^2+4x-3, respectively. 1. Prove that each of P,QP, Q has an unique real root. Let them be α,β\alpha,\beta, respectively. 2. Prove that {α}>{β}2\{ \alpha\}>\{ \beta\} ^2, where {x}=x−⌊x⌋\{ x\}=x-\lfloor x\rfloor is the fractional part of xx.
Problem 1
3

Asking for address

A and B are friends at a summer school. When B asks A for his address, he answers: "My house is on XYZ street, and my house number is a 3-digit number with distinct digits, and if you permute its digits, you will have other 5 numbers. The interesting thing is that the sum of these 5 numbers is exactly 2017. That's all.". After a while, B can determine A's house number. And you, can you find his house number?

Weird inequality

Let x,y,zx,y,z be the non-negative real numbers satisfying xy+yz+zx≤1xy+yz+zx\leq 1. Prove that: 1−xy−yz−zx≤(6−26)(1−min⁡{x,y,z}).1-xy-yz-zx\leq (6-2\sqrt{6})(1-\min\{x,y,z\}).

Convergent sequence of real numbers

Given a real number aa and a sequence (xn)n=1∞(x_n)_{n=1}^\infty defined by: {x1=1x2=0xn+2=xn2+xn+124+a\left\{\begin{matrix} x_1=1 \\ x_2=0 \\ x_{n+2}=\frac{x_n^2+x_{n+1}^2}{4}+a\end{matrix}\right. for all positive integers nn. 1. For a=0a=0, prove that (xn)(x_n) converges. 2. Determine the largest possible value of aa such that (xn)(x_n) converges.