MathDB

1

Part of 2004 China Team Selection Test

Problems(8)

Find the maximum value of OM + ON - MN

Source: China Team Selection Test 2004, Day 1, Problem 1

10/13/2005
Let ∠XOY=π2\angle XOY = \frac{\pi}{2}; PP is a point inside ∠XOY\angle XOY and we have OP=1;∠XOP=π6.OP = 1; \angle XOP = \frac{\pi}{6}. A line passes PP intersects the Rays OXOX and OYOY at MM and NN. Find the maximum value of OM+ON−MN.OM + ON - MN.
LaTeXgeometry unsolvedgeometry
Prove an Integer

Source: China TST 2004 Quiz

2/1/2009
Let m1 m_1, m2 m_2, ⋯ \cdots, mr m_r (may not distinct) and n1 n_1, n2 n_2 ⋯ \cdots, ns n_s (may not distinct) be two groups of positive integers such that for any positive integer d d larger than 1 1, the numbers of which can be divided by d d in group m1 m_1, m2 m_2, ⋯ \cdots, mr m_r (including repeated numbers) are no less than that in group n1 n_1, n2 n_2 ⋯ \cdots, ns n_s (including repeated numbers). Prove that m1⋅m2⋯mrn1⋅n2⋯ns \displaystyle \frac{m_1 \cdot m_2 \cdots m_r}{n_1 \cdot n_2 \cdots n_s} is integer.
number theoryprime factorizationnumber theory unsolved
Cyclic Points

Source: China TST 2004 Quiz

2/1/2009
Using AB AB and AC AC as diameters, two semi-circles are constructed respectively outside the acute triangle ABC ABC. AH⊥BC AH \perp BC at H H, D D is any point on side BC BC (D D is not coinside with B B or C C ), through D D, construct DE∥AC DE \parallel AC and DF∥AB DF \parallel AB with E E and F F on the two semi-circles respectively. Show that D D, E E, F F and H H are concyclic.
geometry unsolvedgeometry
China TST 2004 concurrent sides

Source: China Team Selection Test 2004, Day 2, Problem 1

10/14/2005
Points D,E,FD,E,F are on the sides BC,CABC, CA and ABAB, respectively which satisfy EF∣∣BCEF || BC, D1D_1 is a point on BC,BC, Make D1E1∣∣DE,D1F1∣∣DFD_1E_1 || D_E, D_1F_1 || DF which intersect ACAC and ABAB at E1E_1 and F1F_1, respectively. Make △PBC∼△DEF\bigtriangleup PBC \sim \bigtriangleup DEF such that PP and AA are on the same side of BC.BC. Prove that E,E1F1,PD1E, E_1F_1, PD_1 are concurrent. [Edit by Darij: See my post #4 below for a possible correction of this problem. However, I am not sure that it is in fact the problem given at the TST... Does anyone have a reliable translation?]
geometrygeometric transformationLaTeXgeometry solved
Area Inequality

Source: China TST 2004 Quiz

2/1/2009
Find the largest value of the real number λ \lambda, such that as long as point P P lies in the acute triangle ABC ABC satisfying \angle{PAB}\equal{}\angle{PBC}\equal{}\angle{PCA}, and rays AP AP, BP BP, CP CP intersect the circumcircle of triangles PBC PBC, PCA PCA, PAB PAB at points A1 A_1, B1 B_1, C1 C_1 respectively, then S_{A_1BC}\plus{} S_{B_1CA}\plus{} S_{C_1AB} \geq \lambda S_{ABC}.
geometryinequalitiescircumcirclegeometry unsolved
System of Equations

Source: China TST 2004 Quiz

2/1/2009
Given integer n n larger than 5 5, solve the system of equations (assuming xi≥0x_i \geq 0, for i=1,2,…n i=1,2, \dots n): {x1+22x2+32x3+⋯+n2xn=n+2,x1+22x2+32x3+⋯+n2xn=2n+2,x1+22x2+32x3+⋯+n2xn=n2+n+4,x1+23x2+33x3+⋯+n3xn=n3+n+8. \begin{cases} \displaystyle x_1+ \phantom{2^2} x_2+ \phantom{3^2} x_3 + \cdots + \phantom{n^2} x_n &= n+2, \\ x_1 + 2\phantom{^2}x_2 + 3\phantom{^2}x_3 + \cdots + n\phantom{^2}x_n &= 2n+2, \\ x_1 + 2^2x_2 + 3^2 x_3 + \cdots + n^2x_n &= n^2 + n +4, \\ x_1+ 2^3x_2 + 3^3x_3+ \cdots + n^3x_n &= n^3 + n + 8. \end{cases}
algebrasystem of equationsalgebra unsolved
Functional Equation

Source: China TST 2004 Quiz

2/1/2009
Given non-zero reals a a, b b, find all functions f:R⟼R f: \mathbb{R} \longmapsto \mathbb{R}, such that for every x,y∈R x, y \in \mathbb{R}, y≠0 y \neq 0, f(2x) \equal{} af(x) \plus{} bx and \displaystyle f(x)f(y) \equal{} f(xy) \plus{} f \left( \frac {x}{y} \right).
functionquadraticsalgebra unsolvedalgebra
Sequence

Source: China TST 2004 Quiz

2/1/2009
Given sequence {cn} \{ c_n \} satisfying the conditions that c_0\equal{}1, c_1\equal{}0, c_2\equal{}2005, and c_{n\plus{}2}\equal{}\minus{}3c_n \minus{} 4c_{n\minus{}1} \plus{}2008, ( n\equal{}1,2,3, \cdots). Let {an} \{ a_n \} be another sequence such that a_n\equal{}5(c_{n\plus{}1} \minus{} c_n) \cdot (502 \minus{} c_{n\minus{}1} \minus{} c_{n\minus{}2}) \plus{} 4^n \times 2004 \times 501, ( n\equal{}2,3, \cdots). Is an a_n a perfect square for every n>2 n > 2?
algebra unsolvedalgebra