- In the plane the points with integer coordinates are the vertices of unit squares. The squares are coloured alternately black and white (as on a chessboard). For any pair of positive integers $ m$ and $ n$, consider a right-angled triangle whose vertices have integer coordinates and whose legs, of lengths $ m$ and $ n$, lie along edges of the squares. Let $ S_1$ be the total area of the black part of the triangle and $ S_2$ be the total area of the white part. Let $ f(m,n) = | S_1 - S_2 |$.
a) Calculate $ f(m,n)$ for all positive integers $ m$ and $ n$ which are either both even or both odd.
b) Prove that $ f(m,n) \leq \frac 12 \max \{m,n \}$ for all $ m$ and $ n$.
c) Show that there is no constant $ C\in\mathbb{R}$ such that $ f(m,n) < C$ for all $ m$ and $ n$. - Let $ R_1,R_2, \ldots$ be the family of finite sequences of positive integers defined by the following rules: $ R_1 = (1),$ and if $ R_{n - 1} = (x_1, \ldots, x_s),$ then \[ R_n = (1, 2, \ldots, x_1, 1, 2, \ldots, x_2, \ldots, 1, 2, \ldots, x_s, n).\] For example, $ R_2 = (1, 2),$ $ R_3 = (1, 1, 2, 3),$ $ R_4 = (1, 1, 1, 2, 1, 2, 3, 4).$ Prove that if $ n > 1,$ then the $ k$th term from the left in $ R_n$ is equal to 1 if and only if the $ k$th term from the right in $ R_n$ is different from 1.
- For each finite set $ U$ of nonzero vectors in the plane we define $ l(U)$ to be the length of the vector that is the sum of all vectors in $ U.$ Given a finite set $ V$ of nonzero vectors in the plane, a subset $ B$ of $ V$ is said to be maximal if $ l(B)$ is greater than or equal to $ l(A)$ for each nonempty subset $ A$ of $ V.$
sisting of $ n \geq 1$ vectors the number of maximal subsets is less than or equal to $ 2n.$ - An $ n \times n$ matrix whose entries come from the set $ S = \{1, 2, \ldots , 2n - 1\}$ is called a silver matrix if, for each $ i = 1, 2, \ldots , n$, the $ i$-th row and the $ i$-th column together contain all elements of $ S$. Show that
a) There is no silver matrix for $ n = 1997$;
b) Silver matrices exist for infinitely many values of $ n$. - Let $ ABCD$ be a regular tetrahedron and $ M,N$ distinct points in the planes $ ABC$ and $ ADC$ respectively. Show that the segments $ MN,BN,MD$ are the sides of a triangle.
- a) Let $ n$ be a positive integer. Prove that there exist distinct positive integers $ x, y, z$ such that \[ x^{n-1} + y^n = z^{n+1}.\] b) Let $ a, b, c$ be positive integers such that $ a$ and $ b$ are relatively prime and $ c$ is relatively prime either to $ a$ or to $ b.$ Prove that there exist infinitely many triples $ (x, y, z)$ of distinct positive integers $ x, y, z$ such that \[ x^a + y^b = z^c.\]
- The lengths of the sides of a convex hexagon $ ABCDEF$ satisfy $ AB = BC$, $ CD = DE$, $ EF = FA$. Prove that \[ \frac {BC}{BE} + \frac {DE}{DA} + \frac {FA}{FC} \geq \frac {3}{2}. \]
- It is known that $ \angle BAC$ is the smallest angle in the triangle $ ABC$. The points $ B$ and $ C$ divide the circumcircle of the triangle into two arcs. Let $ U$ be an interior point of the arc between $ B$ and $ C$ which does not contain $ A$. The perpendicular bisectors of $ AB$ and $ AC$ meet the line $ AU$ at $ V$ and $ W$, respectively. The lines $ BV$ and $ CW$ meet at $ T$. Show that $ AU = TB + TC$.
- Let $ A_1A_2A_3$ be a non-isosceles triangle with incenter $ I.$ Let $ C_i,$ $ i = 1, 2, 3,$ be the smaller circle through $ I$ tangent to $ A_iA_{i+1}$ and $ A_iA_{i+2}$ (the addition of indices being mod 3). Let $ B_i, i = 1, 2, 3,$ be the second point of intersection of $ C_{i+1}$ and $ C_{i+2}.$ Prove that the circumcentres of the triangles $ A_1 B_1I,A_2B_2I,A_3B_3I$ are collinear.
- Find all positive integers $ k$ for which the following statement is true: If $ F(x)$ is a polynomial with integer coefficients satisfying the condition $ 0 \leq F(c) \leq k$ for each $ c\in \{0,1,\ldots,k + 1\}$, then $$ F(0) = F(1) = \ldots = F(k + 1).$$
- Let $ P(x)$ be a polynomial with real coefficients such that $ P(x) > 0$ for all $ x \geq 0.$ Prove that there exists a positive integer n such that $ (1 + x)^n \cdot P(x)$ is a polynomial with nonnegative coefficients.
- Let $ p$ be a prime number and $ f$ an integer polynomial of degree $ d$ such that $ f(0) = 0,f(1) = 1$ and $ f(n)$ is congruent to $ 0$ or $ 1$ modulo $ p$ for every integer $ n$. Prove that $ d\geq p - 1$.
- In town $ A,$ there are $ n$ girls and $ n$ boys, and each girl knows each boy. In town $ B,$ there are $ n$ girls $ g_1, g_2, \ldots, g_n$ and $ 2n - 1$ boys $ b_1, b_2, \ldots, b_{2n-1}.$ The girl $ g_i,$ $ i = 1, 2, \ldots, n,$ knows the boys $ b_1, b_2, \ldots, b_{2i-1},$ and no others. For all $ r = 1, 2, \ldots, n,$ denote by $ A(r),B(r)$ the number of different ways in which $ r$ girls from town $ A,$ respectively town $ B,$ can dance with $ r$ boys from their own town, forming $ r$ pairs, each girl with a boy she knows. Prove that $ A(r) = B(r)$ for each $ r = 1, 2, \ldots, n.$
- Let $ b, m, n$ be positive integers such that $ b > 1$ and $ m \neq n.$ Prove that if $ b^m - 1$ and $ b^n - 1$ have the same prime divisors, then $ b + 1$ is a power of 2.
- An infinite arithmetic progression whose terms are positive integers contains the square of an integer and the cube of an integer. Show that it contains the sixth power of an integer.
- In an acute-angled triangle $ ABC,$ let $ AD,BE$ be altitudes and $ AP,BQ$ internal bisectors. Denote by $ I$ and $ O$ the incenter and the circumcentre of the triangle, respectively. Prove that the points $ D, E,$ and $ I$ are collinear if and only if the points $ P, Q,$ and $ O$ are collinear.
- Find all pairs $ (a,b)$ of positive integers that satisfy the equation: $ a^{b^2} = b^a$.
- The altitudes through the vertices $ A,B,C$ of an acute-angled triangle $ ABC$ meet the opposite sides at $ D,E, F,$ respectively. The line through $ D$ parallel to $ EF$ meets the lines $ AC$ and $ AB$ at $ Q$ and $ R,$ respectively. The line $ EF$ meets $ BC$ at $ P.$ Prove that the circumcircle of the triangle $ PQR$ passes through the midpoint of $ BC.$
- Let $ a_1\geq \cdots \geq a_n \geq a_{n + 1} = 0$ be real numbers. Show that \[ \sqrt {\sum_{k = 1}^n a_k} \leq \sum_{k = 1}^n \sqrt k (\sqrt {a_k} - \sqrt {a_{k + 1}}). \]
- Let $ ABC$ be a triangle. $ D$ is a point on the side $ (BC)$. The line $ AD$ meets the circumcircle again at $ X$. $ P$ is the foot of the perpendicular from $ X$ to $ AB$, and $ Q$ is the foot of the perpendicular from $ X$ to $ AC$. Show that the line $ PQ$ is a tangent to the circle on diameter $ XD$ if and only if $ AB = AC$.
- Let $ x_1$, $ x_2$, $ \ldots$, $ x_n$ be real numbers satisfying the conditions \[ \left\{\begin{array}{cccc} |x_1 + x_2 + \cdots + x_n | & = & 1 & \ \\ |x_i| & \leq & \displaystyle \frac {n + 1}{2} & \ \textrm{ for }i = 1, 2, \ldots , n. \end{array} \right. \] Show that there exists a permutation $ y_1$, $ y_2$, $ \ldots$, $ y_n$ of $ x_1$, $ x_2$, $ \ldots$, $ x_n$ such that \[ | y_1 + 2 y_2 + \cdots + n y_n | \leq \frac {n + 1}{2}. \]
- Does there exist functions $ f,g: \mathbb{R}\to\mathbb{R}$ such that $ f(g(x)) = x^2$ and $ g(f(x)) = x^k$ for all real numbers $ x$
a) if $ k = 3$?
b) if $ k = 4$? - Let $ ABCD$ be a convex quadrilateral. The diagonals $ AC$ and $ BD$ intersect at $ K$. Show that $ ABCD$ is cyclic if and only if $$ AK \sin A + CK \sin C = BK \sin B + DK \sin D.$$
- For each positive integer $ n$, let $ f(n)$ denote the number of ways of representing $ n$ as a sum of powers of 2 with nonnegative integer exponents. Representations which differ only in the ordering of their summands are considered to be the same. For instance, $ f(4) = 4$, because the number 4 can be represented in the following four ways: 4; 2+2; 2+1+1; 1+1+1+1. Prove that, for any integer $ n \geq 3$ we have $$ 2^{\frac {n^2}{4}} < f(2^n) < 2^{\frac {n^2}2}.$$
- Let $ X,Y,Z$ be the midpoints of the small arcs $ BC,CA,AB$ respectively (arcs of the circumcircle of $ ABC$). $ M$ is an arbitrary point on $ BC$, and the parallels through $ M$ to the internal bisectors of $ \angle B,\angle C$ cut the external bisectors of $ \angle C,\angle B$ in $ N,P$ respectively. Show that $ XM,YN,ZP$ concur.
- For every integer $ n \geq 2$ determine the minimum value that the sum $ \sum^n_{i=0} a_i$ can take for nonnegative numbers $ a_0, a_1, \ldots, a_n$ satisfying the condition $ a_0 = 1,$ $ a_i \leq a_{i+1} + a_{i+2}$ for $ i = 0, \ldots, n - 2.$
[Shortlists] International Mathematical Olympiad 1997
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