MathDB

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Part of 2017 District Olympiad

Problems(6)

1+(3n+3)/(a²+b²+c²) perfect square if 3n+1 is perfect square

Source: Romanian District Olympiad 2017, Grade VII, Problem 1

10/9/2018
Let be a natural number n≥3 n\ge 3 with the property that 1+3n 1+3n is a perfect square. Show that there are three natural numbers a,b,c, a,b,c, such that the number 1+3n+3a2+b2+c2 1+\frac{3n+3}{a^2+b^2+c^2} is a perfect square.
number theory
sqrtm-sqrtn =p implies m,n, are squares

Source: Romanian District Olympiad 2017, Grade VIII, Problem 1

10/10/2018
a) Let m,n,p∈Z≥0 m,n,p\in\mathbb{Z}_{\ge 0} such that m>n m>n and m−n=p. \sqrt{m} -\sqrt n=p. Prove that m m and n n are perfect squares.
b) Find the numbers of four digits abcd‾ \overline{abcd} that satisfy the equation: abcd‾−acd‾=bb‾. \sqrt {\overline{abcd} } -\sqrt{\overline{acd}} =\overline{bb} .
number theoryeasy
ratios of segments and ratios of cosines determine concurrence of some lines

Source: Romanian District Olympiad 2017, Grade IX, Problem 1

10/10/2018
Let A1,B1,C1 A_1,B_1,C_1 be the feet of the heights of an acute triangle ABC. ABC. On the segments B1C1,C1A1,A1B1, B_1C_1,C_1A_1,A_1B_1, take the points X,Y, X,Y, respectively, Z, Z, such that {C1XXB1=bcos⁡∠BCAccos⁡∠ABCA1YYC1=ccos⁡∠BACacos⁡∠BCAB1ZZA1=acos⁡∠ABCbcos⁡∠BAC. \left\{\begin{matrix}\frac{C_1X}{XB_1} =\frac{b\cos\angle BCA}{c\cos\angle ABC} \\ \frac{A_1Y}{YC_1} =\frac{c\cos\angle BAC}{a\cos\angle BCA} \\ \frac{B_1Z}{ZA_1} =\frac{a\cos\angle ABC}{b\cos\angle BAC} \end{matrix}\right. . Show that AX,BY,CZ, AX,BY,CZ, are concurrent.
geometrytrigonometryratio
infinitely many x,y such that sqrt (x+sqrt x) =y

Source: Romanian District Olympiad 2017, Grade X, Problem 1

10/10/2018
a) Determine x∈N x\in\mathbb{N} and y∈Q y\in\mathbb{Q} such that x+x=y. \sqrt{x+\sqrt{x}}=y.
b) Show that there are infinitely many pairs (x,y)∈Q2 (x,y)\in\mathbb{Q}^2 such that x+x=y. \sqrt{x+\sqrt{x}} =y .
inequality involving a recurrent sequence

Source: Romanian District Olympiad 2017, Grade XI, Problem 1

10/10/2018
Let (an)n≥1 \left( a_n \right)_{n\ge 1} be a sequence of real numbers such that a1>2 a_1>2 and an+1=a1+2an, a_{n+1} =a_1+\frac{2}{a_n} , for all natural numbers n. n.
a) Show that a2n−1+a2n>4, a_{2n-1} +a_{2n} >4 , for all natural numbers n, n, and lim⁡n→∞an=2. \lim_{n\to\infty} a_n =2. b) Find the biggest real number a a for which the following inequality is true: \sqrt{x^2+a_1^2} +\sqrt{x^2+a_2^2} +\sqrt{x^2+a_3^2} +\cdots +\sqrt{x^2+a_n^2} > n\sqrt{x^2+a^2},  \forall x\in\mathbb{R} , \forall n\in\mathbb{N} .
inequalitiesSequenceslimit
fg>=4id^2 implies int f or int g greater or equal than 1

Source: Romanian District Olympiad 2017, Grade XII, Problem 1

10/10/2018
Let f,g:[0,1]⟶R f,g:[0,1]\longrightarrow{R} be two continuous functions such that f(x)g(x)≥4x2, f(x)g(x)\ge 4x^2, for all x∈[0,1]. x\in [0,1] . Prove that ∣∫01f(x)dx∣≥1 or ∣∫01g(x)dx∣≥1. \left| \int_0^1 f(x)dx \right| \ge 1\text{ or } \left| \int_0^1 g(x)dx \right| \ge 1.
functionIntegralinequalitiescalculus