MathDB

3

Part of 2008 Tournament Of Towns

Problems(8)

TT2008 Junior O-Level - P3

Source:

9/4/2010
Acute triangle A1A2A3A_1A_2A_3 is inscribed in a circle of radius 22. Prove that one can choose points B1,B2,B3B_1, B_2, B_3 on the arcs A1A2,A2A3,A3A1A_1A_2, A_2A_3, A_3A_1 respectively, such that the numerical value of the area of the hexagon A1B1A2B2A3B3A_1B_1A_2B_2A_3B_3 is equal to the numerical value of the perimeter of the triangle A1A2A3.A_1A_2A_3.
geometryperimetergeometry proposed
TT2008 Junior A-Level - P3

Source:

9/4/2010
In his triangle ABCABC Serge made some measurements and informed Ilias about the lengths of median ADAD and side ACAC. Based on these data Ilias proved the assertion: angle CABCAB is obtuse, while angle DABDAB is acute. Determine a ratio AD/ACAD/AC and prove Ilias' assertion (for any triangle with such a ratio).
ratioinequalitiesgeometry proposedgeometry
TT2008 Senior O-Level - P3

Source:

9/4/2010
A 3030-gon A1A2⋯A30A_1A_2\cdots A_{30} is inscribed in a circle of radius 22. Prove that one can choose a point BkB_k on the arc AkAk+1A_kA_{k+1} for 1≤k≤291 \leq k \leq 29 and a point B30B_{30} on the arc A30A1A_{30}A_1, such that the numerical value of the area of the 6060-gon A1B1A2B2…A30B30A_1B_1A_2B_2 \dots A_{30}B_{30} is equal to the numerical value of the perimeter of the original 3030-gon.
geometryperimeterperpendicular bisectorgeometry unsolved
TT2008 Senior A-Level - P3

Source:

9/4/2010
There are NN piles each consisting of a single nut. Two players in turns play the following game. At each move, a player combines two piles that contain coprime numbers of nuts into a new pile. A player who can not make a move, loses. For every N>2N > 2 determine which of the players, the first or the second, has a winning strategy.
symmetrycombinatorics unsolvedcombinatorics
2008 ToT Spring Junior O P3 ten cards with no a, 10 with b ,10 ten with c

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3/7/2020
There are ten cards with the number aa on each, ten with bb and ten with cc, where a,ba, b and cc are distinct real numbers. For every five cards, it is possible to add another five cards so that the sum of the numbers on these ten cards is 00. Prove that one of a,ba, b and cc is 00.
combinatorics
2008 ToT Spring Junior A P3 game on a 1x(N + 2)

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3/7/2020
Alice and Brian are playing a game on a 1×(N+2)1\times (N + 2) board. To start the game, Alice places a checker on any of the NN interior squares. In each move, Brian chooses a positive integer nn. Alice must move the checker to the nn-th square on the left or the right of its current position. If the checker moves off the board, Alice wins. If it lands on either of the end squares, Brian wins. If it lands on another interior square, the game proceeds to the next move. For which values of NN does Brian have a strategy which allows him to win the game in a finite number of moves?
combinatoricstable
2008 ToT Spring Senior O P3 equal segments, circumcircle related, right triangle

Source:

2/26/2020
In triangle ABC,∠A=90oABC, \angle A = 90^o. MM is the midpoint of BCBC and HH is the foot of the altitude from AA to BCBC. The line passing through MM and perpendicular to ACAC meets the circumcircle of triangle AMCAMC again at PP. If BPBP intersects AHAH at KK, prove that AK=KHAK = KH.
geometrycircumcircleequal segmentsright triangle
2008 ToT Spring Senior A P3 sum x_i^2/n > sum y_i^2/(n-1)

Source:

3/7/2020
A polynomial xn+a1xn−1+a2xn−2+...+an−2x2+an−1x+anx^n + a_1x^{n-1} + a_2x^{n-2} +... + a_{n-2}x^2 + a_{n-1}x + a_n has nn distinct real roots x1,x2,...,xnx_1, x_2,...,x_n, where n>1n > 1. The polynomial nxn−1+(n−1)a1xn−2+(n−2)a2xn−3+...+2an−2x+an−1nx^{n-1}+ (n - 1)a_1x^{n-2} + (n - 2)a_2x^{n-3} + ...+ 2a_{n-2}x + a_{n-1} has roots y1,y2,...,yn1y_1, y_2,..., y_{n_1}. Prove that x12+x22+...+xn2n>y12+y22+...+yn−12n−1\frac{x^2_1+ x^2_2+ ...+ x^2_n}{n}>\frac{y^2_1 + y^2_2 + ...+ y^2_{n-1}}{n - 1}
algebrapolynomialinequalitiesSumroots