MathDB

1996 IMO Shortlist

Part of IMO Shortlist

Subcontests

(9)
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Determine maximum and minimum value of a(n) over n <= 1996

Let the sequence a(n), n \equal{} 1,2,3, \ldots be generated as follows with a(1) \equal{} 0, and for n>1: n > 1: a(n) \equal{} a\left( \left \lfloor \frac{n}{2} \right \rfloor \right) \plus{} (\minus{}1)^{\frac{n(n\plus{}1)}{2}}. 1.) Determine the maximum and minimum value of a(n) a(n) over n≤1996 n \leq 1996 and find all n≤1996 n \leq 1996 for which these extreme values are attained. 2.) How many terms a(n),n≤1996, a(n), n \leq 1996, are equal to 0?

Polygon F and d^2 - h^2 >= p^2 /4

In the plane, consider a point X X and a polygon F \mathcal{F} (which is not necessarily convex). Let p p denote the perimeter of F \mathcal{F}, let d d be the sum of the distances from the point X X to the vertices of F \mathcal{F}, and let h h be the sum of the distances from the point X X to the sidelines of F \mathcal{F}. Prove that d^2 \minus{} h^2\geq\frac {p^2}{4}.
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k = f(x)*(x+1)^n + g(x)*(x^n + 1)

Let n n be an even positive integer. Prove that there exists a positive inter k k such that k \equal{} f(x) \cdot (x\plus{}1)^n \plus{} g(x) \cdot (x^n \plus{} 1) for some polynomials f(x),g(x) f(x), g(x) having integer coefficients. If k0 k_0 denotes the least such k, k, determine k0 k_0 as a function of n, n, i.e. show that k_0 \equal{} 2^q where q q is the odd integer determined by n \equal{} q \cdot 2^r, r \in \mathbb{N}. Note: This is variant A6' of the three variants given for this problem.

First rectangle can be placed within the second one

Let the sides of two rectangles be {a,b} \{a,b\} and {c,d}, \{c,d\}, respectively, with a<c≤d<b a < c \leq d < b and ab<cd. ab < cd. Prove that the first rectangle can be placed within the second one if and only if \left(b^2 \minus{} a^2\right)^2 \leq \left(bc \minus{} ad \right)^2 \plus{} \left(bd \minus{} ac \right)^2.

Finite number of coins placed on an infinite row of squares

A finite number of coins are placed on an infinite row of squares. A sequence of moves is performed as follows: at each stage a square containing more than one coin is chosen. Two coins are taken from this square; one of them is placed on the square immediately to the left while the other is placed on the square immediately to the right of the chosen square. The sequence terminates if at some point there is at most one coin on each square. Given some initial configuration, show that any legal sequence of moves will terminate after the same number of steps and with the same final configuration.
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(x - a_1)(x - a_2) ... (x - a_n) &lt;= x^n - a^n_1

Let a1≥a2≥…≥an a_1 \geq a_2 \geq \ldots \geq a_n be real numbers such that for all integers k>0, k > 0,
a^k_1 \plus{} a^k_2 \plus{} \ldots \plus{} a^k_n \geq 0.
Let p \equal{}\max\{|a_1|, \ldots, |a_n|\}. Prove that p \equal{} a_1 and that
(x \minus{} a_1) \cdot (x \minus{} a_2) \cdots (x \minus{} a_n) \leq x^n \minus{} a^n_1 for all x>a1. x > a_1.

color the squares

A square (n \minus{} 1) \times (n \minus{} 1) is divided into (n \minus{} 1)^2 unit squares in the usual manner. Each of the n2 n^2 vertices of these squares is to be coloured red or blue. Find the number of different colourings such that each unit square has exactly two red vertices. (Two colouring schemse are regarded as different if at least one vertex is coloured differently in the two schemes.)
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f(x + 13/42) + f(x) = f(x + 1/6) + f(x + 1/7)

Let f f be a function from the set of real numbers R \mathbb{R} into itself such for all x∈R, x \in \mathbb{R}, we have ∣f(x)∣≤1 |f(x)| \leq 1 and f \left( x \plus{} \frac{13}{42} \right) \plus{} f(x) \equal{} f \left( x \plus{} \frac{1}{6} \right) \plus{} f \left( x \plus{} \frac{1}{7} \right). Prove that f f is a periodic function (that is, there exists a non-zero real number c c such f(x\plus{}c) \equal{} f(x) for all x∈R x \in \mathbb{R}).

Geometric Inequality

Let ABCABC be an acute triangle with circumcenter OO and circumradius RR. AOAO meets the circumcircle of BOCBOC at A′A', BOBO meets the circumcircle of COACOA at B′B' and COCO meets the circumcircle of AOBAOB at C′C'. Prove that OA′⋅OB′⋅OC′≥8R3.OA'\cdot OB'\cdot OC'\geq 8R^{3}. Sorry if this has been posted before since this is a very classical problem, but I failed to find it with the search-function.

two injective surjective function

let V V be a finitive set and g g and f f be two injective surjective functions from V VtoV V.let T T and S S be two sets such that they are defined as following" S \equal{} \{w \in V: f(f(w)) \equal{} g(g(w))\} T \equal{} \{w \in V: f(g(w)) \equal{} g(f(w))\} we know that S \cup T \equal{} V, prove: for each w∈V:f(w)∈S w \in V : f(w) \in S if and only if g(w)∈S g(w) \in S
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Inequality 15

Suppose that a,b,c>0a, b, c > 0 such that abc=1abc = 1. Prove that abab+a5+b5+bcbc+b5+c5+caca+c5+a5≤1. \frac{ab}{ab + a^5 + b^5} + \frac{bc}{bc + b^5 + c^5} + \frac{ca}{ca + c^5 + a^5} \leq 1.

Four integers are marked on a circle

Four integers are marked on a circle. On each step we simultaneously replace each number by the difference between this number and next number on the circle, moving in a clockwise direction; that is, the numbers a,b,c,d a,b,c,d are replaced by a\minus{}b,b\minus{}c,c\minus{}d,d\minus{}a. Is it possible after 1996 such to have numbers a,b,c,d a,b,c,d such the numbers |bc\minus{}ad|, |ac \minus{} bd|, |ab \minus{} cd| are primes?

Triangle, orthocenter, parallels - prove that EX || AP

Let ABC ABC be a triangle, and H H its orthocenter. Let P P be a point on the circumcircle of triangle ABC ABC (distinct from the vertices A A, B B, C C), and let E E be the foot of the altitude of triangle ABC ABC from the vertex B B. Let the parallel to the line BP BP through the point A A meet the parallel to the line AP AP through the point B B at a point Q Q. Let the parallel to the line CP CP through the point A A meet the parallel to the line AP AP through the point C C at a point R R. The lines HR HR and AQ AQ intersect at some point X X. Prove that the lines EX EX and AP AP are parallel.