MathDB

2016 Indonesia TST

Part of Indonesia TST

Subcontests

(4)
1
3

Equation with Surd

Determine all real numbers xx which satisfy x=a−a+x x = \sqrt{a - \sqrt{a+x}} where a>0a > 0 is a parameter.

existence of a subset with sum divisible by n

Let kk and nn be positive integers. Determine the smallest integer N≥kN \ge k such that the following holds: If a set of NN integers contains a complete residue modulo kk, then it has a non-empty subset whose sum of elements is divisible by nn.

Coloring a grid with black and white

Let n≥3n \ge 3 be a positive integer. We call a 3×33 \times 3 grid beautiful if the cell located at the center is colored white and all other cells are colored black, or if it is colored black and all other cells are colored white. Determine the minimum value of a+ba+b such that there exist positive integers aa, bb and a coloring of an a×ba \times b grid with black and white, so that it contains n2−nn^2 - n beautiful subgrids.
3
3

A classic extremal combi on sets and memberships

Let {E1,E2,…,Em}\{E_1, E_2, \dots, E_m\} be a collection of sets such that Ei⊆X={1,2,…,100}E_i \subseteq X = \{1, 2, \dots, 100\}, Ei≠XE_i \neq X, i=1,2,…,mi = 1, 2, \dots, m. It is known that every two elements of XX is contained together in exactly one EiE_i for some ii. Determine the minimum value of mm.

Two tangent circles, prove complementary

Circles Ω\Omega and ω\omega are tangent at a point PP (ω\omega lies inside Ω\Omega ). A chord ABAB of Ω\Omega is tangent to ω\omega at C;C; the line PCPC meets Ω\Omega again at Q.Q. Chords QRQR and QSQS of Ω \Omega are tangent to ω.\omega . Let I,X,I,X, and YY be the incenters of the triangles APB,APB, ARB,ARB, and ASB,ASB, respectively. Prove that ∠PXI+∠PYI=90∘.\angle PXI+\angle PYI=90^{\circ }.

summation involving sine squared

Let nn be a positive integer greater than 11. Evaluate the following summation: ∑k=0n−111+8sin⁡2(kπn). \sum_{k=0}^{n-1} \frac{1}{1 + 8 \sin^2 \left( \frac{k \pi}{n} \right)}.
4
2

Loyal subsets and functions F and G

We call a subset BB of natural numbers loyal if there exists natural numbers i≤ji\le j such that B={i,i+1,…,j}B=\{i,i+1,\ldots,j\}. Let QQ be the set of all loyal sets. For every subset A={a1<a2<…<ak}A=\{a_1<a_2<\ldots<a_k\} of {1,2,…,n}\{1,2,\ldots,n\} we set f(A)=max⁡1≤i≤k−1ai+1−aiandg(A)=max⁡B⊆A,B∈Q∣B∣.f(A)=\max_{1\le i \le k-1}{a_{i+1}-a_i}\qquad\text{and}\qquad g(A)=\max_{B\subseteq A, B\in Q} |B|. Furthermore, we define F(n)=∑A⊆{1,2,…,n}f(A)andG(n)=∑A⊆{1,2,…,n}g(A).F(n)=\sum_{A\subseteq \{1,2,\ldots,n\}} f(A)\qquad\text{and}\qquad G(n)=\sum_{A\subseteq \{1,2,\ldots,n\}} g(A). Prove that there exists m∈Nm\in \mathbb N such that for each natural number n>mn>m we have F(n)>G(n)F(n)>G(n). (By ∣A∣|A| we mean the number of elements of AA, and if ∣A∣≤1|A|\le 1, we define f(A)f(A) to be zero).
Proposed by Javad Abedi

a configuration with gergonne point and its cevians

In a non-isosceles triangle ABCABC, let II be its incenter. The incircle of ABCABC touches BCBC, CACA, and ABAB at DD, EE, and FF, respectively. A line passing through DD and perpendicular to ADAD intersects IBIB and ICIC at AbA_b and AcA_c, respectively. Define the points BcB_c, BaB_a, CaC_a, and CbC_b similarly. Let GG be the intersection of the cevians ADAD, BEBE, and CFCF. The points O1O_1 and O2O_2 are the circumcenter of the triangles AbBcCaA_bB_cC_a and AcBaCbA_cB_aC_b, respectively. Prove that IGIG is the perpendicular bisector of O1O2O_1O_2.
2
3

already well-known, but yet strangely difficult

Let a,ba,b be two positive integers, such that ab≠1ab\neq 1. Find all the integer values that f(a,b)f(a,b) can take, where f(a,b)=a2+ab+b2ab−1. f(a,b) = \frac { a^2+ab+b^2} { ab- 1} .

Geometric Ineq on convex n-gons

Given a convex polygon with nn sides and perimeter SS, which has an incircle ω\omega with radius RR. A regular polygon with nn sides, whose vertices lie on ω\omega, has a perimeter ss. Determine whether the following inequality holds: S≥2sRn4n2R2−s2. S \ge \frac{2sRn}{\sqrt{4n^2R^2-s^2}}.

three variables three equations

Determine all triples of real numbers (x,y,z)(x, y, z) which satisfy the following system of equations: {x+y+z=0x3+y3+z3=90x5+y5+z5=2850. \begin{cases} x+y+z=0 \\ x^3+y^3+z^3 = 90 \\ x^5+y^5+z^5=2850. \end{cases}