MathDB

1995 IMO Shortlist

Part of IMO Shortlist

Subcontests

(8)
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Inequality with n(n-1)/2

Let n n be an integer,n≥3. n \geq 3. Let x1,x2,…,xn x_1, x_2, \ldots, x_n be real numbers such that x_i < x_{i\plus{}1} for 1 \leq i \leq n \minus{} 1. Prove that \frac{n(n\minus{}1)}{2} \sum_{i < j} x_ix_j > \left(\sum^{n\minus{}1}_{i\equal{}1} (n\minus{}i)\cdot x_i \right) \cdot \left(\sum^{n}_{j\equal{}2} (j\minus{}1) \cdot x_j \right)

Tetrahedron and centroid inequality

Let A1A2A3A4 A_1A_2A_3A_4 be a tetrahedron, G G its centroid, and A1′,A2′,A3′, A'_1, A'_2, A'_3, and A4′ A'_4 the points where the circumsphere of A1A2A3A4 A_1A_2A_3A_4 intersects GA1,GA2,GA3, GA_1,GA_2,GA_3, and GA4, GA_4, respectively. Prove that GA1⋅GA2⋅GA3⋅GA⋅4≤GA1′⋅GA2′⋅GA3′⋅GA4′ GA_1 \cdot GA_2 \cdot GA_3 \cdot GA_ \cdot4 \leq GA'_1 \cdot GA'_2 \cdot GA'_3 \cdot GA'_4 and \frac{1}{GA'_1} \plus{} \frac{1}{GA'_2} \plus{} \frac{1}{GA'_3} \plus{} \frac{1}{GA'_4} \leq \frac{1}{GA_1} \plus{} \frac{1}{GA_2} \plus{} \frac{1}{GA_3} \plus{} \frac{1}{GA_4}.

f(m + f(n)) = n + f(m + 95)

Let N \mathbb{N} denote the set of all positive integers. Prove that there exists a unique function f:N↦N f: \mathbb{N} \mapsto \mathbb{N} satisfying f(m \plus{} f(n)) \equal{} n \plus{} f(m \plus{} 95) for all m m and n n in N. \mathbb{N}. What is the value of \sum^{19}_{k \equal{} 1} f(k)?
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3

f(x + 1/x^2) = f(x) + [f(1/x)]^2

Let R \mathbb{R} be the set of real numbers. Does there exist a function f:R↦R f: \mathbb{R} \mapsto \mathbb{R} which simultaneously satisfies the following three conditions?
(a) There is a positive number M M such that ∀x: \forall x: \minus{} M \leq f(x) \leq M. (b) The value of f(1)f(1) is 11. (c) If x≠0, x \neq 0, then f \left(x \plus{} \frac {1}{x^2} \right) \equal{} f(x) \plus{} \left[ f \left(\frac {1}{x} \right) \right]^2

How many people are at the meeting?

At a meeting of 12k 12k people, each person exchanges greetings with exactly 3k\plus{}6 others. For any two people, the number who exchange greetings with both is the same. How many people are at the meeting?

Arithmetic progression of positive integers

For positive integers n, n, the numbers f(n) f(n) are defined inductively as follows: f(1) \equal{} 1, and for every positive integer n, n, f(n\plus{}1) is the greatest integer m m such that there is an arithmetic progression of positive integers a_1 < a_2 < \ldots < a_m \equal{} n for which f(a_1) \equal{} f(a_2) \equal{} \ldots \equal{} f(a_m). Prove that there are positive integers a a and b b such that f(an\plus{}b) \equal{} n\plus{}2 for every positive integer n. n.
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3

Integers satisfying squared and product sum

Let a a and b b be non-negative integers such that ab≥c2, ab \geq c^2, where c c is an integer. Prove that there is a number n n and integers x1,x2,…,xn,y1,y2,…,yn x_1, x_2, \ldots, x_n, y_1, y_2, \ldots, y_n such that \sum^n_{i\equal{}1} x^2_i \equal{} a, \sum^n_{i\equal{}1} y^2_i \equal{} b, \text{ and } \sum^n_{i\equal{}1} x_iy_i \equal{} c.

Unique point X

Let A,B A, B and C C be non-collinear points. Prove that there is a unique point X X in the plane of ABC ABC such that XA^2 \plus{} XB^2 \plus{} AB^2 \equal{} XB^2 \plus{} XC^2 \plus{} BC^2 \equal{} XC^2 \plus{} XA^2 \plus{} CA^2.

{x^2 + Ax + B} and {2x^2 + 2x + C} do not intersect

Let Z \mathbb{Z} denote the set of all integers. Prove that for any integers A A and B, B, one can find an integer C C for which M_1 \equal{} \{x^2 \plus{} Ax \plus{} B : x \in \mathbb{Z}\} and M_2 \equal{} {2x^2 \plus{} 2x \plus{} C : x \in \mathbb{Z}} do not intersect.
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3

Base values between 2 and 3 satisfy fractional inequality

Let n n be an integer, n≥3. n \geq 3. Let a1,a2,…,an a_1, a_2, \ldots, a_n be real numbers such that 2≤ai≤3 2 \leq a_i \leq 3 for i \equal{} 1, 2, \ldots, n. If s \equal{} a_1 \plus{} a_2 \plus{} \ldots \plus{} a_n, prove that
\frac{a^2_1 \plus{} a^2_2 \minus{} a^2_3}{a_1 \plus{} a_2 \minus{} a_3} \plus{} \frac{a^2_2 \plus{} a^2_3 \minus{} a^2_4}{a_2 \plus{} a_3 \minus{} a_4} \plus{} \ldots \plus{} \frac{a^2_n \plus{} a^2_1 \minus{} a^2_2}{a_n \plus{} a_1 \minus{} a_2} \leq 2s \minus{} 2n.

E, F, Z, Y are concyclic

The incircle of triangle △ABC \triangle ABC touches the sides BC BC, CA CA, AB AB at D,E,F D, E, F respectively. X X is a point inside triangle of △ABC \triangle ABC such that the incircle of triangle △XBC \triangle XBC touches BC BC at D D, and touches CX CX and XB XB at Y Y and Z Z respectively. Show that E,F,Z,Y E, F, Z, Y are concyclic.

q(x) to be the product of all primes less than p(x)

For an integer x≥1x \geq 1, let p(x)p(x) be the least prime that does not divide xx, and define q(x)q(x) to be the product of all primes less than p(x)p(x). In particular, p(1)=2.p(1) = 2. For xx having p(x)=2p(x) = 2, define q(x)=1q(x) = 1. Consider the sequence x0,x1,x2,…x_0, x_1, x_2, \ldots defined by x0=1x_0 = 1 and xn+1=xnp(xn)q(xn) x_{n+1} = \frac{x_n p(x_n)}{q(x_n)} for n≥0n \geq 0. Find all nn such that xn=1995x_n = 1995.
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2

sqrt([AKON]) + sqrt([CLOM]) &lt;= sqrt([ABCD])

Let ABCD be a convex quadrilateral and O a point inside it. Let the parallels to the lines BC, AB, DA, CD through the point O meet the sides AB, BC, CD, DA of the quadrilateral ABCD at the points E, F, G, H, respectively. Then, prove that \sqrt {\left|AHOE\right|} \plus{} \sqrt {\left|CFOG\right|}\leq\sqrt {\left|ABCD\right|}, where ∣P1P2...Pn∣ \left|P_1P_2...P_n\right| is an abbreviation for the non-directed area of an arbitrary polygon P1P2...Pn P_1P_2...P_n.

One taken from each of any n − 1 of the subsets

Does there exist an integer n>1 n > 1 which satisfies the following condition? The set of positive integers can be partitioned into n n nonempty subsets, such that an arbitrary sum of n \minus{} 1 integers, one taken from each of any n \minus{} 1 of the subsets, lies in the remaining subset.
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2

Orthocentres of triangles ABC and AB’C’

Suppose that ABCD ABCD is a cyclic quadrilateral. Let E \equal{} AC\cap BD and F \equal{} AB\cap CD. Denote by H1 H_{1} and H2 H_{2} the orthocenters of triangles EAD EAD and EBC EBC, respectively. Prove that the points F F, H1 H_{1}, H2 H_{2} are collinear.
Original formulation:
Let ABC ABC be a triangle. A circle passing through B B and C C intersects the sides AB AB and AC AC again at C′ C' and B′, B', respectively. Prove that BB′ BB', CC′CC' and HH′ HH' are concurrent, where H H and H′ H' are the orthocentres of triangles ABC ABC and AB′C′ AB'C' respectively.

SQRT(2p) - SQRT(x) - SQRT(y) non-negative, smallest

Let p p be an odd prime. Determine positive integers x x and y y for which x≤y x \leq y and \sqrt{2p} \minus{} \sqrt{x} \minus{} \sqrt{y} is non-negative and as small as possible.
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