MathDB

2004 Unirea

Part of Unirea

Subcontests

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Problem 7 night 3

Let a,b,ca,b,c be real numbers. Show that a3+b3+c3=a+b+c3\sqrt[3]{a} + \sqrt[3]{b} +\sqrt[3]{c} = \sqrt[3]{a+b+c} if and only if a3+b3+c3=(a+b+c)3 a^3 + b^3 + c^3 = (a + b + c)^3

trigonometric equation |\sin 3x+\cos (7\pi /2 -5x)|=2

Solve in the real numbers the equation ∣sin⁡3x+cos⁡(7π/2−5x)∣=2. |\sin 3x+\cos (7\pi /2 -5x)|=2.

Geometry with beautiful analytic solution

Let be a point P P on the diagonal BD BD (excluding its endpoints) of a quadrilateral ABCD, ABCD, and Q Q be a point in the interior of ABD. ABD. The projections of P P on AB,AD AB,AD are P1,P2, P_1,P_2, respectively, and the projections of Q Q on AB,AD AB,AD are Q1,Q2, Q_1,Q_2, respectively, and verify the equations AQ1=14AB AQ_1=\frac{1}{4}AB and AQ2=14AD. AQ_2=\frac{1}{4}AD. Show that the point Q Q is not in the interior of AP1P2. AP_1P_2.
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Combinatorics identity

Prove that there exist 2004 2004 pairwise distinct numbers n1,n2,…,n2004, n_1,n_2,\ldots ,n_{2004} , all greater than 1, 1, satisfying: (n12)+(n22)+⋯+(n20032)=(n20042). \binom{n_1}{2} +\binom{n_2}{2} +\cdots +\binom{n_{2003}}{2} =\binom{n_{2004}}{2} .

Two related problems involving properties of Euler's number

a) Prove that for any natural numbers n, n, the inequality e2−1/n>∏k=1n(1+1/k2) e^{2-1/n} >\prod_{k=1}^n (1+1/k^2) holds.
b) Prove that the sequence (an)n≥1 \left( a_n \right)_{n\ge 1} with a1=1 a_1=1 and defined by the recursive relation an+1=2n2∑k=1nkak a_{n+1}=\frac{2}{n^2}\sum_{k=1}^n ka_k is nondecreasing. Is it convergent?

Some primitive

Hello, I've been trying to solve this for a while now, but no success! I mean, I have an expression for this but not a closed one. I derived something in terms of Tchebychev Polynomials : cos(nx) = P_n(cos(x)). Any help is appreciated. Compute the following primitive: ∫xsin⁡(2004x)tan⁡x dx \int \frac{x\sin\left(2004 x\right)}{\tan x}\ dx