MathDB

2

Part of MathLinks Contest 1st

Problems(7)

0142 polynomial 1st edition Round 4 p2

Source:

5/9/2021
Let ff be a polynomial with real coefficients such that for each positive integer n the equation f(x)=nf(x) = n has at least one rational solution. Find ff.
algebrapolynomial1st edition
0112 inequalities 1st edition Round 1 p2

Source:

5/9/2021
Prove that for all positive integers a,b,ca, b, c the following inequality holds: a+ba+c+b+cb+a+c+ac+b≤ab+bc+ca\frac{a + b}{a + c}+\frac{b + c}{b + a}+\frac{c + a}{c + b} \le \frac{a}{b}+\frac{b}{c}+\frac{c}{a}
inequalities1st edition
0132 polynomial 1st edition Round 3 p2

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5/9/2021
Let a be a non-zero integer, and n≥3n \ge 3 another integer. Prove that the following polynomial is irreducible in the ring of integer polynomials (i.e. it cannot be written as a product of two non-constant integer polynomials): f(x)=xn+axn−1+axn−2+...+ax−1f(x) = x^n + ax^{n-1} + ax^{n-2} +... + ax -1
algebrapolynomial1st edition
0122 geometry 1st edition Round 2 p2

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5/9/2021
In a triangle △ABC\vartriangle ABC, ∠B=2∠C\angle B = 2\angle C. Let PP and QQ be points on the perpendicular bisector of segment BCBC such that rays APAP and AQAQ trisect ∠A\angle A. Prove that PQPQ is smaller than ABAB if and only if ∠B\angle B is obtuse.
geometry1st edition
0152 complex numbers inequalities 1st edition Round 5 p2

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5/9/2021
Let mm be the greatest number such that for any set of complex numbers having the sum of all modulus of all the elements 11, there exists a subset having the modulus of the sum of the elements in the subset greater than mm. Prove that 14≤m≤12.\frac14 \le m \le \frac12.
(Optional Task for 3p) Find a smaller value for the RHS.
complex numbersinequalities1st editionalgebra
0162 geometry 1st edition Round 6 p2

Source:

5/9/2021
Given is a triangle ABCABC and on its sides the triangles ABM,BCNABM, BCN and CAPCAP are build such that ∠AMB=150o\angle AMB = 150^o, AM=MBAM = MB, ∠CAP=∠CBN=30o\angle CAP = \angle CBN = 30^o, ∠ACP=∠BCN=45o\angle ACP = \angle BCN = 45^o. Prove that the triangle MNPMNP is an equilateral triangle.
geometry1st edition
0172 geo inequalities 1st edition Round 7 p2

Source:

5/9/2021
Consider the circles ω\omega, ω1\omega_1, ω2\omega_2, where ω1\omega_1, ω2\omega_2 pass through the center OO of ω\omega. The circle ω\omega cuts ω1\omega_1 at A,EA, E and ω2\omega_2 at C,DC, D. The circles ω1\omega_1 and ω2\omega_2 intersect at OO and MM. If ADD cuts CECE at BB and if MN⊥BOMN \perp BO, (N∈BON \in BO) prove that 2MN2≤BM⋅MO2MN^2 \le BM \cdot MO.
geometry1st edition