id stringlengths 10 15 | question stringlengths 63 2.3k | solutions stringlengths 20 28.5k |
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IMOSL-1978-3 | Let \(n > m \geq 1\) be natural numbers such that the groups of the last three digits in the decimal representation of \(1978^{m}\) , \(1978^{n}\) coincide. Find the ordered pair \((m, n)\) of such \(m, n\) for which \(m + n\) is minimal. | What we need are \(m,n\) for which \(1978^{m}(1978^{n - m} - 1)\) is divisible by \(1000 = 8\cdot 125\) . Since \(1978^{n - m} - 1\) is odd, it follows that \(1978^{m}\) is divisible by 8, so \(m\geq 3\) .
Also, \(1978^{n - m} - 1\) is divisible by 125, i.e., \(1978^{n - m}\equiv 1\) (mod 125). Note that \(1978\equi... |
IMOSL-1978-4 | Let \(T_1\) be a triangle having \(a, b, c\) as lengths of its sides and let \(T_2\) be another triangle having \(u, v, w\) as lengths of its sides. If \(P, Q\) are the areas of the two triangles, prove that
\[16PQ \leq a^2 (-u^2 +v^2 +w^2) + b^2 (u^2 -v^2 +w^2) + c^2 (u^2 +v^2 -w^2).\]
When does equality hold? | Let \(\gamma ,\phi\) be the angles of \(T_{1}\) and \(T_{2}\) opposite to \(c\) and \(w\) respectively. By the cosine theorem, the inequality is transformed into
\[a^{2}(2\nu^{2} - 2u\nu \cos \phi) + b^{2}(2u^{2} - 2u\nu \cos \phi)\] \[+2(a^{2} + b^{2} - 2ab\cos \gamma)u\nu \cos \phi \geq 4ab\nu \sin \gamma \sin \ph... |
IMOSL-1978-5 | For every integer \(d \geq 1\) , let \(M_d\) be the set of all positive integers that cannot be written as a sum of an arithmetic progression with difference \(d\) , having at least two terms and consisting of positive integers. Let \(A = M_1\) , \(B = M_2 \setminus \{2\}\) , \(C = M_3\) . Prove that every \(c \in C\) ... | We first explicitly describe the elements of the sets \(M_{1},M_{2}\) .
\(x\notin M_{1}\) is equivalent to \(x = a + (a + 1) + \dots +(a + n - 1) = n(2a + n - 1) / 2\) for some natural numbers \(n,a,n\geq 2\) . Among \(n\) and \(2a + n - 1\) , one is odd and the other even, and both are greater than 1; so \(x\) has ... |
IMOSL-1978-6 | Let \(\phi :\{1,2,3,\ldots \} \to \{1,2,3,\ldots \}\) be injective. Prove that for all \(n\) ,
\[\sum_{k = 1}^{n}\frac{\phi(k)}{k^{2}}\geq \sum_{k = 1}^{n}\frac{1}{k}.\] | For fixed \(n\) and the set \(\{\phi (1),\ldots ,\phi (n)\}\) , there are finitely many possibilities for a mapping \(\phi\) on \(\{1,\ldots ,n\}\) . Suppose \(\phi\) is the one among these for which \(\sum_{k = 1}^{n}\phi (k) / k^{2}\) is minimal. If \(i< j\) and \(\phi (i) > \phi (j)\) for some \(i,j\in \{1,\ldots ,n... |
IMOSL-1978-7 | We consider three distinct half-lines \(Ox, Oy, Oz\) in a plane. Prove the existence and uniqueness of three points \(A \in Ox, B \in Oy, C \in Oz\) such that the perimeters of the triangles \(OAB, OBC, OCA\) are all equal to a given number \(2p > 0\) . | Let \(x = OA\) , \(y = OB\) , \(z = OC\) , \(\alpha = \angle BOC\) , \(\beta = \angle COA\) , \(\gamma = \angle AOB\) . The conditions yield the equation \(x + y + \sqrt{x^2 + y^2 - 2xy\cos\gamma} = 2p\) , which transforms to \((2p - x - y)^2 = x^2 + y^2 - 2xy\cos\gamma\) , i.e. \((p - x)(p - y) = xy(1 - \cos\gamma)\) ... |
IMOSL-1978-8 | Let \(S\) be the set of all the odd positive integers that are not multiples of 5 and that are less than \(30m, m\) being an arbitrary positive integer. What is the smallest integer \(k\) such that in any subset of \(k\) integers from \(S\) there must be two different integers, one of which divides the other? | Take the subset \(\{a_i\} = \{1,7,11,13,17,19,23,29,\ldots ,30m - 1\}\) of \(S\) containing all the elements of \(S\) that are not multiples of 3. There are \(8m\) such elements. Every element in \(S\) can be uniquely expressed as \(3^i a_i\) for some \(i\) and \(t \geq 0\) . In a subset of \(S\) with \(8m + 1\) elemen... |
IMOSL-1978-9 | Let \(\{f(n)\}\) be a strictly increasing sequence of positive integers: \(0 < f(1) < f(2) < f(3) < \dots\) Of the positive integers not belonging to the sequence, the \(n\) th in order of magnitude is \(f(f(n)) + 1\) . Determine \(f(240)\) . | Since the \(n\) th missing number (gap) is \(f(f(n)) + 1\) and \(f(f(n))\) is a member of the sequence, there are exactly \(n - 1\) gaps less than \(f(f(n))\) . This leads to
\[f(f(n)) = f(n) + n - 1. \quad (1)\]
Since 1 is not a gap, we have \(f(1) = 1\) . The first gap is \(f(f(1)) + 1 = 2\) . Two consecutive i... |
IMOSL-1978-10 | An international society has its members in 6 different countries. The list of members contains 1978 names, numbered \(1,2,\ldots ,1978\) . Prove that there is at least one member whose number is the sum of the numbers of two, not necessarily distinct, of his compatriots. | Assume the opposite. One of the countries, say \(A\) , contains at least 330 members \(a_{1}, a_{2}, \ldots , a_{330}\) of the society ( \(6 \cdot 329 = 1974\) ). Consider the differences \(a_{330} - a_{i}\) , \(i = 1, 2, \ldots , 329\) : the members with these numbers are not in \(A\) , so at least 66 of them, \(a_{33... |
IMOSL-1978-11 | A function \(f:I\to \mathbb{R}\) , defined on an interval \(I\) , is called concave if \(f(\theta x + (1 - \theta)y)\geq \theta f(x) + (1 - \theta)f(y)\) for all \(x,y\in I\) and \(0\leq \theta \leq 1\) . Assume that the functions \(f_{1},\ldots ,f_{n}\) , having all nonnegative values, are concave. Prove that the func... | Set \(F(x) = f_{1}(x)f_{2}(x) \dots f_{n}(x)\) : we must prove concavity of \(F^{1 / n}\) . By the assumption,
\[F(\theta x + (1 - \theta)y)\geq \prod_{i = 1}^{n}[\theta f_{i}(x) + (1 - \theta)f(y)]\] \[\qquad = \sum_{k = 0}^{n}\theta^{k}(1 - \theta)^{n - k}\sum f_{i_{1}}(x)\dots f_{i_{k}}(x)f_{i_{k + 1}}(y)f_{i_{n}... |
IMOSL-1978-12 | In a triangle \(ABC\) we have \(AB = AC\) . A circle is tangent internally to the circumcircle of \(ABC\) and also to the sides \(AB,AC\) , at \(P, Q\) respectively. Prove that the midpoint of \(PQ\) is the center of the incircle of \(ABC\) . | Let \(O\) be the center of the smaller circle, \(T\) its contact point with the circumcircle of \(ABC\) , and \(J\) the midpoint of segment \(BC\) . The figure is symmetric with respect to the line through \(A, O, J, T\) .
A homothety centered at \(A\) taking \(T\) into \(J\) will take the smaller circle into the in... |
IMOSL-1978-13 | Given any point \(P\) in the interior of a sphere with radius \(R\) , three mutually perpendicular segments \(PA, PB, PC\) are drawn terminating on the sphere and having one common vertex in \(P\) . Consider the rectangular parallelepiped of which \(PA, PB, PC\) are coterminal edges. Find the locus of the point \(Q\) t... | Lemma. If \(MNPQ\) is a rectangle and \(O\) any point in space, then \(OM^2 + OP^2 = ON^2 + OQ^2\) .
Proof. Let \(O_{1}\) be the projection of \(O\) onto \(MNPQ\) , and \(m, n, p, q\) denote the distances of \(O_{1}\) from \(MN, NP, PQ, QM\) , respectively. Then \(OM^2 = OQ_1^2 + q^2 + m^2\) , \(ON^2 = OQ_1^2 + m^2 ... |
IMOSL-1978-14 | Prove that it is possible to place \(2n(2n + 1)\) parallelepipedic (rectangular) pieces of soap of dimensions \(1 \times 2 \times (n + 1)\) in a cubic box with edge \(2n + 1\) if and only if \(n\) is even or \(n = 1\) .
Remark. It is assumed that the edges of the pieces of soap are parallel to the edges of the box. | We label the cells of the cube by \((a_{1},a_{2},a_{3})\) \(a_{i}\in \{1,2,\ldots ,2n + 1\}\) , in a natural way: for example, as Cartesian coordinates of centers of the cells \(((1,1,1)\) is one corner, etc.). Notice that there should be \((2n + 1)^{3} - 2n(2n + 1)\cdot 2(n + 1) = 2n + 1\) void cells, i.e., those not ... |
IMOSL-1978-15 | Let \(p\) be a prime and \(A = \{a_1, \ldots , a_{p-1}\}\) an arbitrary subset of the set of natural numbers such that none of its elements is divisible by \(p\) . Let us define a mapping \(f\) from \(\mathcal{P}(A)\) (the set of all subsets of \(A\) ) to the set \(P = \{0, 1, \ldots , p-1\}\) in the following way:
(i... | Let \(C_{n} = \{a_{1},\ldots ,a_{n}\}\) \((C_{0} = \emptyset)\) and \(P_{n} = \{f(B) | B\subseteq C_{n}\}\) . We claim that \(P_{n}\) contains at least \(n + 1\) distinct elements. First note that \(P_{0} = \{0\}\) contains one element. Suppose that \(P_{n + 1} = P_{n}\) for some \(n\) . Since \(P_{n + 1}\supseteq \{a_... |
IMOSL-1978-16 | Determine all the triples \((a,b,c)\) of positive real numbers such that the system
\[a x + b y - c z = 0,\] \[a\sqrt{1 - x^{2}} + b\sqrt{1 - y^{2}} - c\sqrt{1 - z^{2}} = 0,\]
is compatible in the set of real numbers, and then find all its real solutions. | Clearly \(|x|\leq 1\) . As \(x\) runs over \([-1,1]\) , the vector \(u = (ax,a\sqrt{1 - x^2})\) runs over all vectors of length \(a\) in the plane having a nonnegative vertical component. Putting \(v = (by,b\sqrt{1 - y^2})\) , \(w = (cz,c\sqrt{1 - z^2})\) , the system becomes \(u + v = w\) , with vectors \(u,v,w\) of l... |
IMOSL-1978-17 | Prove that for any positive integers \(x,y,z\) with \(x y - z^{2} = 1\) one can find nonnegative integers \(a,b,c,d\) such that \(x = a^{2} + b^{2}\) , \(y = c^{2} + d^{2}\) , \(z = a c + b d\) . Set \(z = (2q)!\) to deduce that for any prime number \(p = 4q + 1\) , \(p\) can be represented as the sum of squares of two... | Let \(z_{0}\geq 1\) be a positive integer. Supposing that the statement is true for all triples \((x,y,z)\) with \(z< z_{0}\) , we shall prove that it is true for \(z = z_{0}\) too. If \(z_{0} = 1\) , verification is trivial, while \(x_{0} = y_{0}\) is obviously impossible. So let there be given a triple \((x_{0},y_{0}... |
IMOSL-1979-1 | Prove that in the Euclidean plane every regular polygon having an even number of sides can be dissected into lozenges. (A lozenge is a quadrilateral whose four sides are all of equal length). | We prove more generally, by induction on \(n\) , that any \(2n\) -gon with equal edges and opposite edges parallel to each other can be dissected. For \(n = 2\) the only possible such \(2n\) -gon is a single lozenge, so our theorem holds in this case. We will now show that it holds for general \(n\) . Assume by inducti... |
IMOSL-1979-2 | From a bag containing 5 pairs of socks, each pair a different color, a random sample of 4 single socks is drawn. Any complete pairs in the sample are discarded and replaced by a new pair drawn from the bag. The process continues until the bag is empty or there are 4 socks of different colors held outside the bag. What ... | The only way to arrive at the latter alternative is to draw four different socks in the first drawing or to draw only one pair in the first drawing and then draw two different socks in the last drawing. We will call these probabilities respectively \(p_{1}, p_{2}, p_{3}\) . We calculate them as follows:
\[p_{1} = \f... |
IMOSL-1979-3 | Find all polynomials \(f(x)\) with real coefficients for which
\[f(x)f(2x^{2}) = f(2x^{3} + x).\] | An obvious solution is \(f(x) = 0\) . We now look for nonzero solutions. We note that plugging in \(x = 0\) we get \(f(0)^{2} = f(0)\) ; hence \(f(0) = 0\) or \(f(0) = 1\) . If \(f(0) = 0\) , then \(f\) is of the form \(f(x) = x^{k}g(x)\) , where \(g(0) \neq 0\) . Plugging this formula into \(f(x)f(2x^{2}) = f(2x^{3} +... |
IMOSL-1979-4 | A pentagonal prism \(A_1A_2\ldots A_5B_1B_2\ldots B_5\) is given. The edges, the diagonals of the lateral walls and the internal diagonals of the prism are each colored either red or green in such a way that no triangle whose vertices are vertices of the prism has its three edges of the same color. Prove that all edges... | Let us prove first that the edges \(A_{1}A_{2}, A_{2}A_{3}, \ldots , A_{5}A_{1}\) are of the same color. Assume the contrary, and let w.l.o.g. \(A_{1}A_{2}\) be red and \(A_{2}A_{3}\) be green. Three of the segments \(A_{2}B_{l}\) \((l = 1,2,3,4,5)\) , say \(A_{2}B_{i}, A_{2}B_{j}, A_{2}B_{k}\) , have to be of the same... |
IMOSL-1979-5 | Let \(n \geq 2\) be an integer. Find the maximal cardinality of a set \(M\) of pairs \((j, k)\) of integers, \(1 \leq j < k \leq n\) , with the following property: If \((j, k) \in M\) , then \((k, m) \notin M\) for any \(m\) . | Let \(A = \{x \mid (x,y) \in M\}\) and \(B = \{y \mid (x,y) \in M\) . Then \(A\) and \(B\) are disjoint and hence
\[|M| \leq |A| \cdot |B| \leq \frac{(|A| + |B|)^2}{4} \leq \left[\frac{n^2}{4}\right].\]
These cardinalities can be achieved for \(M = \{(a,b) \mid a = 1,2,\ldots ,[n / 2], b = [n / 2] + 1,\ldots ,n\}... |
IMOSL-1979-6 | Find the real values of \(p\) for which the equation
\[\sqrt{2p + 1 - x^2} +\sqrt{3x + p + 4} = \sqrt{x^2 + 9x + 3p + 9}\]
in \(x\) has exactly two real distinct roots ( \(\sqrt{t}\) means the positive square root of \(t\) ). | Setting \(q = x^{2} + x - p\) , the given equation becomes
\[\sqrt{(x + 1)^{2} - 2q} +\sqrt{(x + 2)^{2} - q} = \sqrt{(2x + 3)^{2} - 3q}. \quad (1)\]
Taking squares of both sides we get \(2\sqrt{((x + 1)^{2} - 2q)((x + 2)^{2} - q)} = 2(x + 1)(x + 2)\) . Taking squares again we get
\[q\left(2q - 2(x + 2)^{2} - (... |
IMOSL-1979-7 | Given that \(1 - \frac{1}{2} +\frac{1}{3} -\frac{1}{4} +\dots -\frac{1}{1318} +\frac{1}{1319} = \frac{p}{q}\) , where \(p\) and \(q\) are natural numbers having no common factor, prove that \(p\) is divisible by 1979. | We denote the sum mentioned above by \(S\) . We have the following equalities:
\[S = 1 - \frac{1}{2} +\frac{1}{3} -\frac{1}{4} +\dots -\frac{1}{1318} +\frac{1}{1319\] \[= 1 + \frac{1}{2} +\dots +\frac{1}{1319} -2\left(\frac{1}{2} +\frac{1}{4} +\dots +\frac{1}{1318}\right)\] \[= 1 + \frac{1}{2} +\dots +\frac{1}{1319} -... |
IMOSL-1979-8 | For all rational \(x\) satisfying \(0 \leq x < 1\) , \(f\) is defined by
\[f(x) = \left\{ \begin{array}{ll}f(2x) / 4, & \mathrm{for} 0 \leq x < 1 / 2, \\ 3 / 4 + f(2x - 1) / 4, & \mathrm{for} 1 / 2 \leq x < 1. \end{array} \right.\]
Given that \(x = 0. b_1 b_2 b_3 \ldots\) is the binary representation of \(x\) , f... | By the definition of \(f\) , it holds that \(f(0.b_{1}b_{2}\dots) = 3b_{1} / 4 + f(0.b_{2}b_{3}\dots) / 4 = 0.b_{1}b_{1} + f(0.b_{2}b_{3}\dots) / 4\) . Continuing this argument we obtain
\[f(0.b_{1}b_{2}b_{3}\dots) = 0.b_{1}b_{1}\dots b_{n}b_{n} + \frac{1}{2^{2n}} f(0.b_{n + 1}b_{n + 2}\dots). \quad (1)\]
The bin... |
IMOSL-1979-9 | Let \(S\) and \(F\) be two opposite vertices of a regular octagon. A counter starts at \(S\) and each second is moved to one of the two neighboring vertices of the octagon. The direction is determined by the toss of a coin. The
process ends when the counter reaches \(F\) . We define \(a_{n}\) to be the number of disti... | Let us number the vertices, starting from \(S\) and moving clockwise. In that case \(S = 1\) and \(F = 5\) . After an odd number of moves to a neighboring point we can be only on an even point, and hence it follows that \(a_{2n - 1} = 0\) for all \(n \in \mathbb{N}\) . Let us define respectively \(z_{n}\) and \(w_{n}\)... |
IMOSL-1979-10 | Show that for any vectors \(a, b\) in Euclidean space,
\[|a\times b|^{3}\leq \frac{3\sqrt{3}}{8} |a|^{2}|b|^{2}|a - b|^{2}.\]
Remark. Here \(\times\) denotes the vector product. | In the cases \(a = \overrightarrow{0}\) , \(b = \overrightarrow{0}\) , and \(a \parallel b\) the inequality is trivial. Otherwise, let us consider a triangle \(ABC\) such that \(\overrightarrow{CB} = a\) and \(\overrightarrow{CA} = b\) . From this point on we shall refer to \(\alpha\) , \(\beta\) , \(\gamma\) as angles... |
IMOSL-1979-11 | Given real numbers \(x_{1}, x_{2}, \ldots , x_{n} (n \geq 2)\) , with \(x_{i} \geq 1 / n (i = 1,2, \ldots , n)\) and with \(x_{1}^{2} + x_{2}^{2} + \dots + x_{n}^{2} = 1\) , find whether the product \(P = x_{1}x_{2}x_{3}\dots x_{n}\) has a greatest and/or least value and if so, give these values. | Let us define \(y_{i} = x_{i}^{2}\) . We thus have \(y_{1} + y_{2} + \dots + y_{n} = 1\) , \(y_{i} \geq 1 / n^{2}\) , and \(P = \sqrt{y_{1}y_{2}\dots y_{n}}\) .
The upper bound is obtained immediately from the AM-GM inequality: \(P \leq 1 / n^{n / 2}\) , where equality holds when \(x_{i} = \sqrt{y_{i}} = 1 / \sqrt{n... |
IMOSL-1979-12 | Let \(R\) be a set of exactly 6 elements. A set \(F\) of subsets of \(R\) is called an \(S\) -family over \(R\) if and only if it satisfies the following three conditions:
(i) For no two sets \(X,Y\) in \(F\) is \(X\subseteq Y\) (ii) For any three sets \(X,Y,Z\) in \(F\) \(X\cup Y\cup Z\neq R\) (iii) \(\textstyle \b... | The first criterion ensures that all sets in an \(S\) -family are distinct. Since the number of different families of subsets is finite, \(h\) has to exist. In fact, we will show that \(h = 11\) . First of all, if there exists \(X \in F\) such that \(|X| \geq 5\) , then by (3) there exists \(Y \in F\) such that \(X \cu... |
IMOSL-1979-13 | Show that \(\frac{20}{60} < \sin 20^{\circ} < \frac{21}{60}\) . | From elementary trigonometry we have \(\sin 3t = 3\sin t - 4\sin^{3}t\) . Hence, if we denote \(y = \sin 20^{\circ}\) , we have \(\sqrt{3} /2 = \sin 60^{\circ} = 3y - 4y^{3}\) . Obviously \(0< y< 1 / 2 = \sin 30^{\circ}\) . The function \(f(x) = 3x - 4x^{3}\) is strictly increasing on \([0,1 / 2)\) because \(f^{\prime}... |
IMOSL-1979-14 | Find all bases of logarithms in which a real positive number can be equal to its logarithm or prove that none exist. | Let us assume that \(a\in \mathbb{R}\backslash \{1\}\) is such that there exist \(a\) and \(x\) such that \(x = \log_{a}x\) or equivalently \(f(x):= \ln x / x = \ln a\) . Then \(a\) is a value of the function \(f(x)\) for \(x\in \mathbb{R}^{+}\backslash \{1\}\) , and the converse also holds.
First we observe that \(... |
IMOSL-1979-15 | The nonnegative real numbers \(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, a\) satisfy the following relations:
\[\sum_{i = 1}^{5}i x_{i} = a,\qquad \sum_{i = 1}^{5}i^{3}x_{i} = a^{2},\qquad \sum_{i = 1}^{5}i^{5}x_{i} = a^{3}.\]
What are the possible values of \(a\) ? | We note that
\[\sum_{i = 1}^{5}i(a - i^{2})^{2}x_{i} = a^{2}\sum_{i = 1}^{5}ix_{i} - 2a\sum_{i = 1}^{5}i^{3}x_{i} + \sum_{i = 1}^{5}i^{5}x_{i} = a^{2}\cdot a - 2a\cdot a^{2} + a^{3} = 0.\]
Since the terms in the sum on the left are all nonnegative, it follows that all the terms have to be 0. Thus, either \(x_{i} ... |
IMOSL-1979-16 | Let \(K\) denote the set \(\{a, b, c, d, e\}\) . \(F\) is a collection of 16 different subsets of \(K\) , and it is known that any three members of \(F\) have at least one element in common. Show that all 16 members of \(F\) have exactly one element in common. | Obviously, no two elements of \(F\) can be complements of each other. If one of the sets has one element, then the conclusion is trivial. If there exist two different 2-element sets, then they must contain a common element, which in turn must then be contained in all other sets. Thus we can assume that there exists at ... |
IMOSL-1979-17 | Inside an equilateral triangle \(ABC\) one constructs points \(P\) , \(Q\) and \(R\) such that
\[\angle QAB = \angle PBA = 15^{\circ},\] \[\angle RBC = \angle QCB = 20^{\circ},\] \[\angle PCA = \angle RAC = 25^{\circ}.\]
Determine the angles of triangle \(PQR\) . | Let \(K\) , \(L\) , and \(M\) be intersections of \(CQ\) and \(BR\) , \(AR\) and \(CP\) , and \(AQ\) and \(BP\) , respectively. Let \(\angle X\) denote the angle of the hexagon \(KQMPLR\) at the vertex \(X\) , where \(X\) is one of the six points. By an elementary calculation of angles we get
\[\angle K = 140^{\circ... |
IMOSL-1979-18 | Let \(m\) positive integers \(a_1, \ldots , a_m\) be given. Prove that there exist fewer than \(2^m\) positive integers \(b_1, \ldots , b_n\) such that all sums of distinct \(b_k\)'s are distinct and all \(a_i\) \((i \leq m)\) occur among them. | Let us write all \(a_{i}\) in binary representation. For \(S \subseteq \{1, 2, \ldots , m\}\) let us define \(b(S)\) as the number in whose binary representation ones appear in exactly the slots where ones appear in all \(a_{i}\) where \(i \subseteq S\) and don't appear in any other \(a_{i}\) . Some \(b(S)\) , includin... |
IMOSL-1979-19 | Consider the sequences \((a_n)\) , \((b_n)\) defined by
\[a_1 = 3,\qquad b_1 = 100,\qquad a_{n + 1} = 3^{a_n},\qquad b_{n + 1} = 100^{b_n}.\]
Find the smallest integer \(m\) for which \(b_m > a_{100}\) . | Let us define \(i_{j}\) for two positive integers \(i\) and \(j\) in the following way: \(i_{1} = i\) and \(i_{j + 1} = i^{j}\) for all positive integers \(j\) . Thus we must find the smallest \(m\) such that \(100_{m} > 3100\) . Since \(100_{1} = 100 > 27 = 32\) , we inductively have \(100_{j} = 10^{100_{j - 1}} > 3^{... |
IMOSL-1979-20 | Given the integer \(n > 1\) and the real number \(a > 0\) determine the maximum of \(\sum_{i = 1}^{n - 1} x_i x_{i + 1}\) taken over all nonnegative numbers \(x_i\) with sum \(a\) . | Let \(x_{k} = \max \{x_{1}, x_{2}, \ldots , x_{n}\}\) . Then \(x_{i}x_{i + 1} \leq x_{i}x_{k}\) for \(i = 1, 2, \ldots , k - 1\) and \(x_{i}x_{i + 1} \leq x_{k}x_{i + 1}\) for \(i = k, \ldots , n - 1\) . Summing up these inequalities for \(i = 1, 2, \ldots , n - 1\) we obtain
\[\sum_{i = 1}^{n - 1} \leq x_{k}(x_{1} ... |
IMOSL-1979-21 | Let \(N\) be the number of integral solutions of the equation
\[x^{2} - y^{2} = z^{3} - t^{3}\]
satisfying the condition \(0 \leq x, y, z, t \leq 10^6\) , and let \(M\) be the number of integral solutions of the equation
\[x^{2} - y^{2} = z^{3} - t^{3} + 1\]
satisfying the condition \(0 \leq x, y, z, t \leq... | Denote \(m = 10^{6}\) and let \(f(n)\) be the number of different ways \(n \in \mathbb{N}\) can be expressed as \(x^{2} + y^{3}\) with \(x, y \in \{0, 1, \ldots , m\}\) . Clearly \(f(n) = 0\) for \(n < 0\) or \(n > m^{2} + m^{3}\) . The first equation can be written as \(x^{2} + t^{3} = y^{2} + z^{3} = n\) , whereas th... |
IMOSL-1979-22 | There are two circles in the plane. Let a point \(A\) be one of the points of intersection of these circles. Two points begin moving simultaneously with constant speeds from the point \(A\) , each point along its own circle. The two points return to the point \(A\) at the same time. Prove that there is a point \(P\) in... | Let the centers of the two circles be denoted by \(O\) and \(O_{1}\) and their respective
\(O Q M(t) = Q N(t)\) for all \(t\) . We note that \(O Q = O_{1}A = r_{1}\) , \(O_{1}Q = O A = r\) and \(\angle Q O A = \angle Q O_{1}A = \phi\) . Since the two points return to \(A\) at the same time, it follows that \(\angle ... |
IMOSL-1979-23 | Find all natural numbers \(n\) for which \(2^8 + 2^{11} + 2^n\) is a perfect square. | It is easily verified that no solutions exist for \(n \leq 8\) . Let us now assume that \(n > 8\) . We note that \(2^{8} + 2^{11} + 2^{n} = 2^{8} \cdot (9 + 2^{n - 8})\) . Hence \(9 + 2^{n - 8}\) must also be a square, say \(9 + 2^{n - 8} = x^{2}\) , \(x \in \mathbb{N}\) , i.e., \(2^{n - 8} = x^{2} - 9 = (x - 3)(x + 3)... |
IMOSL-1979-24 | A circle \(O\) with center \(O\) on base \(BC\) of an isosceles triangle \(ABC\) is tangent to the equal sides \(AB, AC\) . If point \(P\) on \(AB\) and point \(Q\) on \(AC\) are selected such that \(PB \times CQ = (BC / 2)^2\) , prove that line segment \(PQ\) is tangent to circle \(O\) , and prove the converse. | Clearly \(O\) is the midpoint of \(BC\) . Let \(M\) and \(N\) be the points of tangency of the circle with \(AB\) and \(AC\) , respectively, and let \(\angle BAC = 2\phi\) . Then \(\angle BOM = \angle CON = \phi\) .
Let us assume that \(PQ\) touches the circle in \(X\) . If we set \(\angle POM = \angle POX = x\) and... |
IMOSL-1979-25 | Given a point \(P\) in a given plane \(\pi\) and also a given point \(Q\) not in \(\pi\) , show how to determine a point \(R\) in \(\pi\) such that \(\frac{QP + PR}{QR}\) is a maximum. | Let us first look for such a point \(R\) on a ray \(l\) in \(\pi\) going through \(P\) . Let \(\angle QPR = 2\theta\) . Consider a point \(Q'\) on the extension of \(l\) beyond \(P\) such that \(Q'P = QP\) . Then we have
\[\frac{QP + PR}{QR} = \frac{RQ'}{QR} = \frac{\sin\angle Q'QR}{\sin\angle QQ'R}.\]
Since \(\a... |
IMOSL-1979-26 | Prove that the functional equations
\[f(x + y) = f(x) + f(y),\] \[\mathrm{and}\quad f(x + y + xy) = f(x) + f(y) + f(xy)\quad (x,y\in \mathbb{R})\]
are equivalent. | Let us assume that \(f(x + y) = f(x) + f(y)\) for all reals. In this case we trivially apply the equation to get \(f(x + y + xy) = f(x + y) + f(xy) = f(x) + f(y) + f(xy)\) . Hence the equivalence is proved in the first direction.
Now let us assume that \(f(x + y + xy) = f(x) + f(y) + f(xy)\) for all reals. Plugging ... |
IMOSL-1981-1 | (a) For which values of \(n > 2\) is there a set of \(n\) consecutive positive integers such that the largest number in the set is a divisor of the least common multiple of the remaining \(n - 1\) numbers?
(b) For which values of \(n > 2\) is there a unique set having the stated property? | Assume that the set \(\{a - n + 1,a - n + 2,\ldots ,a\}\) of \(n\) consecutive numbers satisfies the condition \(a\mid \operatorname {lcm}[a - n + 1,\ldots ,a - 1]\) . Let \(a = p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}}\ldots p_{r}^{\alpha_{r}}\) be the canonic representation of \(a\) , where \(p_{1}< p_{2}< \dots < p_{r}\)... |
IMOSL-1981-2 | A sphere \(S\) is tangent to the edges \(AB, BC, CD, DA\) of a tetrahedron \(ABCD\) at the points \(E, F, G, H\) respectively. The points \(E, F, G, H\) are the vertices of a square. Prove that if the sphere is tangent to the edge \(AC\), then it is also tangent to the edge \(BD\). | Lemma. Let \(E,F,G,H,I\) , and \(K\) be points on edges \(AB,BC,CD,DA,AC\) , and \(BD\) of a tetrahedron. Then there is a sphere that touches the edges at these points if and only if
\[A E = A H = A I,\quad B E = B F = B K,\] \[C F = C G = C I,\quad D G = D H = D K.\]
\((*)\)
Proof. The "only if" side of the e... |
IMOSL-1981-3 | Find the minimum value of
\[\max (a + b + c, b + c + d, c + d + e, d + e + f, e + f + g)\]
subject to the constraints
\[a,b,c,d,e,f,g\geq 0, \qquad (ii) a + b + c + d + e + f + g = 1.\] | Denote \(\max (a + b + c, b + c + d, c + d + e, d + e + f, e + f + g)\) by \(p\) . We have
\[(a + b + c) + (c + d + e) + (e + f + g) = 1 + c + e \leq 3p,\]
which implies that \(p \geq 1 / 3\) . However, \(p = 1 / 3\) is achieved by taking \((a, b, c, d, e, f, g) = (1 / 3, 0, 0, 1 / 3, 0, 0, 1 / 3)\) . Therefore the... |
IMOSL-1981-4 | Let \(\{f_{n}\}\) be the Fibonacci sequence \(\{1,1,2,3,5,\ldots \}\) .
(a) Find all pairs \((a,b)\) of real numbers such that for each \(n\) , \(af_{n} + bf_{n + 1}\) is a member of the sequence.
(b) Find all pairs \((u,v)\) of positive real numbers such that for each \(n\) , \(uf_{n}^{2} + vf_{n + 1}^{2}\) is a m... | We shall use the known formula for the Fibonacci sequence
\[f_{n} = \frac{1}{\sqrt{5}} (\alpha^{n} - (-1)^{n}\alpha^{-n}),\qquad \mathrm{where} \alpha = \frac{1 + \sqrt{5}}{2}. \quad (1)\]
(a) Suppose that \(a f_{n} + b f_{n + 1} = f_{k_{n}}\) for all \(n\) , where \(k_{n} > 0\) is an integer depending on \(n\) .... |
IMOSL-1981-5 | A cube is assembled with 27 white cubes. The larger cube is then painted black on the outside and disassembled. A blind man reassembles it. What is the probability that the cube is now completely black on the outside? Give an approximation of the size of your answer. | There are four types of small cubes upon disassembling:
(1) 8 cubes with three faces, painted black, at one corner;
(2) 12 cubes with two black faces, both at one edge;
(3) 6 cubes with one black face;
(4) 1 completely white cube.
All cubes of type (1) must go to corners, and be placed in a correct way (... |
IMOSL-1981-6 | Let \(P(z)\) and \(Q(z)\) be complex-variable polynomials, with degree not less than 1. Let
\[P_{k} = \{z\in \mathbb{C}\mid P(z) = k\} ,\qquad Q_{k} = \{z\in \mathbb{C}\mid Q(z) = k\} .\]
Let also \(P_{0} = Q_{0}\) and \(P_{1} = Q_{1}\) . Prove that \(P(z) \equiv Q(z)\) . | Assume w.l.o.g. that \(n = \deg P \geq \deg Q\) , and let \(P_0 = \{z_1, z_2, \ldots , z_k\}\) , \(P_1 = \{z_{k+1}, z_{k+2}, \ldots , z_{k+m}\}\) . The polynomials \(P\) and \(Q\) match at \(k + m\) points \(z_1, z_2, \ldots , z_{k+m}\) ; hence if we prove that \(k + m > n\) , the result will follow. By the assumption,... |
IMOSL-1981-7 | Assume that \(f(x,y)\) is defined for all positive integers \(x\) and \(y\) , and that the following equations are satisfied:
\[f(0,y) = y + 1,\]
\[f(x + 1,0) = f(x,1),\]
\[f(x + 1,y + 1) = f(x,f(x + 1,y)).\]
Determine \(f(2,2),f(3,3)\) and \(f(4,4)\) . | We immediately find that \(f(1,0) = f(0,1) = 2\) . Then \(f(1,y + 1) = f(0,f(1,y)) = f(1,y) + 1\) ; hence \(f(1,y) = y + 2\) for \(y \geq 0\) . Next we find that \(f(2,0) = f(1,1) = 3\) and \(f(2,y + 1) = f(1,f(2,y)) = f(2,y) + 2\) , from which \(f(2,y) = 2y + 3\) . Particularly, \(f(2,2) = 7\) . Further, \(f(3,0) = f(... |
IMOSL-1981-8 | Let \(f(n,r)\) be the arithmetic mean of the minima of all \(r\) -subsets of the set \(\{1,2,\ldots ,n\}\) . Prove that \(f(n,r) = \frac{n + 1}{r + 1}\) . | Since the number \(k\) , \(k = 1, 2, \ldots , n - r + 1\) , is the minimum in exactly \(\binom{n-k}{r-1}\) \(r\) -element subsets of \(\{1, 2, \ldots , n\}\) , it follows that
\[f(n, r) = \frac{1}{\binom{n}{r}} \sum_{k=1}^{n-r+1} k \binom{n-k}{r-1}.\]
Using the equality \(\binom{r+j}{j} = \sum_{i=0}^{j} \binom{r+... |
IMOSL-1981-9 | A sequence \((a_{n})\) is defined by means of the recursion
\[a_{1} = 1,\quad a_{n + 1} = \frac{1 + 4a_{n} + \sqrt{1 + 24a_{n}}}{16}.\]
Find an explicit formula for \(a_{n}\) . | If we put \(1 + 24a_{n} = b_{n}^{2}\) , the given recurrent relation becomes
\[\frac{2}{3} b_{n + 1}^{2} = \frac{3}{2} +\frac{b_{n}^{2}}{6} +b_{n} = \frac{2}{3}\left(\frac{3}{2} +\frac{b_{n}}{2}\right)^{2},\qquad \mathrm{i.e.,}\qquad b_{n + 1} = \frac{3 + b_{n}}{2}, \quad (1)\]
where \(b_{1} = 5\) . To solve this... |
IMOSL-1981-10 | Determine the smallest natural number \(n\) having the following property: For every integer \(p\) , \(p \geq n\) , it is possible to subdivide (partition) a given square into \(p\) squares (not necessarily equal). | It is easy to see that partitioning into \(p = 2k\) squares is possible for \(k \geq 2\) (Fig. 1). Furthermore, whenever it is possible to partition the square into \(p\) squares, there is a partition of the square into \(p + 3\) squares: namely, in the partition into \(p\) squares, divide one of them into four new squ... |
IMOSL-1981-11 | On a semicircle with unit radius four consecutive chords \(AB, BC, CD, DE\) with lengths \(a, b, c, d\) , respectively, are given. Prove that
\[a^{2} + b^{2} + c^{2} + d^{2} + abc + bcd < 4.\] | Let us denote the center of the semicircle by \(O\) , and \(\angle AOB = 2\alpha\) , \(\angle BOC = 2\beta\) , \(AC = m\) , \(CE = n\) .
We claim that \(a^{2} + b^{2} + n^{2} + abn = 4\) . Indeed, since \(a = 2\sin \alpha\) , \(b = 2\sin \beta\) , \(n = 2\cos (\alpha +\beta)\) , we have
\[a^{2} + b^{2} + n^{2} + ... |
IMOSL-1981-12 | Determine the maximum value of \(m^{2} + n^{2}\) where \(m\) and \(n\) are integers satisfying
\[m,n\in \{1,2,\ldots ,100\} \quad \mathrm{and}\quad (n^{2} - mn - m^{2})^{2} = 1.\] | We will solve the contest problem (in which \(m, n \in \{1, 2, \ldots , 1981\}\) ). For \(m = 1\) , \(n\) can be either 1 or 2. If \(m > 1\) , then \(n(n - m) = m^{2} \pm 1 > 0\) ; hence \(n - m > 0\) . Set \(p = n - m\) . Since \(m^{2} - mp - p^{2} = m^{2} - p(m + p) = -(n^{2} - nm - m^{2})\) , we see that \((m, n)\) ... |
IMOSL-1981-13 | Let \(P\) be a polynomial of degree \(n\) satisfying
\[P(k) = \binom{n+1}{k}^{-1}\quad \mathrm{for} k = 0,1,\ldots ,n.\]
Determine \(P(n + 1)\) . | Lemma. For any polynomial \(P\) of degree at most \(n\) ,
\[\sum_{i = 0}^{n + 1}(-1)^{i}\binom{n + 1}{i} P(i) = 0. \quad (1)\]
Proof. We shall use induction on \(n\) . For \(n = 0\) it is trivial. Assume that it is true for \(n = k\) and suppose that \(P(x)\) is a polynomial of degree \(k + 1\) . Then \(P(x) - P(... |
IMOSL-1981-14 | Prove that a convex pentagon (a five-sided polygon) \(ABCDE\) with equal sides and for which the interior angles satisfy the condition \(\angle A \geq \angle B \geq \angle C \geq \angle D \geq \angle E\) is a regular pentagon. | We need the following lemma.
Lemma. If a convex quadrilateral \(PQR\) satisfies \(PS = QR\) and \(\angle SPQ \geq \angle RQP\) , then \(\angle QRS \geq \angle PSR\) .
Proof. If the lines \(PS\) and \(QR\) are parallel, then this quadrilateral is a parallelogram, and the statement is trivial. Otherwise, let \(X\) be... |
IMOSL-1981-15 | Find the point \(P\) inside the triangle \(ABC\) for which
\[\frac{BC}{PD} + \frac{CA}{PE} + \frac{AB}{PF}\]
is minimal, where \(PD, PE, PF\) are the perpendiculars from \(P\) to \(BC, CA, AB\) respectively. | Set \(BC = a\) , \(CA = b\) , \(AB = c\) , and denote the area of \(\triangle ABC\) by \(P\) , and \(a / PD + b / PE + c / PF\) by \(S\) . Since \(a \cdot PD + b \cdot PE + c \cdot PF = 2P\) , by the Cauchy-Schwarz inequality we have
\[2PS = (a\cdot PD + b\cdot PE + c\cdot PF)\left(\frac{a}{PD} +\frac{b}{PE} +\frac{... |
IMOSL-1981-16 | A sequence of real numbers \(u_1, u_2, u_3, \ldots\) is determined by \(u_1\) and the following recurrence relation for \(n \geq 1\) :
\[4u_{n + 1} = \sqrt[3]{64u_n + 15}.\]
Describe, with proof, the behavior of \(u_n\) as \(n \to \infty\) . | The sequence \(\{u_{n}\}\) is bounded, whatever \(u_{1}\) is. Indeed, assume the opposite, and let \(u_{m}\) be the first member of the sequence such that \(|u_{m}| > \max \{2, |u_{1}|\}\) . Then \(|u_{m - 1}| = |u_{m}^{3} - 15 / 64| > |u_{m}|\) , which is impossible.
Next, let us see for what values of \(u_{m}\) , ... |
IMOSL-1981-17 | Three equal circles touch the sides of a triangle and have one common point \(O\) . Show that the center of the circle inscribed in and of the circle circumscribed about the triangle \(ABC\) and the point \(O\) are collinear. | Let us denote by \(S_{A},S_{B},S_{C}\) the centers of the given circles, where \(S_{A}\) lies on the bisector of \(\angle A\) , etc. Then \(S_{A}S_{B}\parallel AB\) , \(S_{B}S_{C}\parallel BC\) , \(S_{C}S_{A}\parallel CA\) , so that the inner bisectors of the angles of triangle \(ABC\) are also inner bisectors of the a... |
IMOSL-1981-18 | Several equal spherical planets are given in outer space. On the surface of each planet there is a set of points that is invisible from any of the remaining planets. Prove that the sum of the areas of all these sets is equal to the area of the surface of one planet. | Let \(C\) be the convex hull of the set of the planets: its border consists of parts of planes, parts of cylinders, and parts of the surfaces of some planets. These parts of planets consist exactly of all the invisible points; any point on a planet that is inside \(C\) is visible. Thus it remains to show that the areas... |
IMOSL-1981-19 | A finite set of unit circles is given in a plane such that the area of their union \(U\) is \(S\) . Prove that there exists a subset of mutually disjoint circles such that the area of their union is greater than \(\frac{2S}{9}\) . | Consider the partition of plane \(\pi\) into regular hexagons, each having inradius 2. Fix one of these hexagons, denoted by \(\gamma\) . For any other hexagon \(x\) in the partition, there exists a unique translation \(\tau_{x}\) taking it onto \(\gamma\) . Define the mapping \(\phi :\pi \to \gamma\) as follows: If \(... |
IMOSL-1982-1 | The function \(f(n)\) is defined for all positive integers \(n\) and takes on nonnegative integer values. Also, for all \(m, n\) ,
\[f(m + n) - f(m) - f(n) = 0 \mathrm{or} 1;\]
\[f(2) = 0, \quad f(3) > 0, \quad \text{and} \quad f(9999) = 3333.\]
Determine \(f(1982)\) . | From \(f(1) + f(1)\leq f(2) = 0\) we obtain \(f(1) = 0\) . Since \(0< f(3)\leq f(1)+\) \(f(2) + 1\) , it follows that \(f(3) = 1\) . Note that if \(f(3n)\geq n\) , then \(f(3n + 3)\geq\) \(f(3n) + f(3)\geq n + 1\) . Hence by induction \(f(3n)\geq n\) holds for all \(n\in \mathbb{N}\) . Moreover, if the inequality is st... |
IMOSL-1982-2 | Let \(K\) be a convex polygon in the plane and suppose that \(K\) is positioned in the coordinate system in such a way that
\[\mathrm{area}(K\cap Q_{i}) = \frac{1}{4}\mathrm{area}K(i = 1,2,3,4,),\]
where the \(Q_{i}\) denote the quadrants of the plane. Prove that if \(K\) contains no nonzero lattice point, then t... | Since \(K\) does not contain a lattice point other than \(O(0,0)\) , it is bounded by four lines \(u,v,w,x\) that pass through the points \(U(1,0)\) , \(V(0,1)\) , \(W(-1,0)\) , \(X(0, - 1)\) respectively. Let \(PQRS\) be the quadrilateral formed by these lines, where \(U\in SP\) , \(V\in PQ\) , \(W\in QR\) , \(X\in RS... |
IMOSL-1982-3 | Consider the infinite sequences \(\{x_{n}\}\) of positive real numbers with the following properties:
\[x_{0} = 1 \quad \text{and for all} \quad i \geq 0, \quad x_{i + 1} \leq x_{i}.\]
(a) Prove that for every such sequence there is an \(n \geq 1\) such that \(\frac{x_0^2}{x_1} + \frac{x_1^2}{x_2} + \dots + \frac{x... | (a) By the Cauchy-Schwarz inequality we have \(\left(x_{0}^{2} / x_{1} + \dots +x_{n - 1}^{2} / x_{n}\right)\cdot \left(x_{1} + \dots +x_{n}\right)\geq \left(x_{0} + \dots +x_{n - 1}\right)^{2}\) . Let us set \(X_{n - 1} = x_{1} + x_{2} + \dots +x_{n - 1}\) . Using \(x_{0} = 1\) , the last inequality can be rewritten a... |
IMOSL-1982-4 | Determine all real values of the parameter \(a\) for which the equation
\[16x^{4} - ax^{3} + (2a + 17)x^{2} - ax + 16 = 0\]
has exactly four distinct real roots that form a geometric progression. | Suppose that \(a\) satisfies the requirements of the problem and that \(x\) , \(qx\) , \(q^{2}x\) , \(q^{3}x\) are the roots of the given equation. Then \(x \neq 0\) and we may assume that \(|q| > 1\) , so that \(|x| < |qx| < |q^{2}x| < |q^{3}x|\) . Since the equation is symmetric, \(1 / x\) is also a root and therefor... |
IMOSL-1982-5 | Let \(A_{1}A_{2}A_{3}A_{4}A_{5}A_{6}\) be a regular hexagon. Each of its diagonals \(A_{i - 1}A_{i + 1}\) is divided into the same ratio \(\frac{\lambda}{1 - \lambda}\) , where \(0 < \lambda < 1\) , by a point \(B_{i}\) in such a way that \(A_{i}\) , \(B_{i}\) , and \(B_{i + 2}\) are collinear \((i \equiv 1, \ldots , 6... | Notice that \(\triangle A_{5}B_{4}A_{4} \cong \triangle A_{3}B_{2}A_{2}\) . We know that \(\angle A_{5}A_{3}A_{2} = 90^{\circ}\) and that \(\angle A_{2}B_{4}A_{4}\) is equal to the sum of the angles \(\angle A_{2}B_{4}A_{3}\) and \(\angle A_{3}B_{4}A_{4}\) . Clearly, \(\angle A_{2}B_{4}A_{3} = 90^{\circ} - \angle B_{2}... |
IMOSL-1982-6 | Let \(S\) be a square with sides of length 100 and let \(L\) be a path within \(S\) that does not meet itself and that is composed of linear segments \(A_{0}A_{1}, A_{1}A_{2}, \ldots , A_{n - 1}A_{n}\) with \(A_{0} \neq A_{n}\) . Suppose that for every point \(P\) of the boundary of \(S\) there is a point of \(L\) at a... | Denote by \(d(U, V)\) the distance between points or sets of points \(U\) and \(V\) . For \(P, Q \in L\) we shall denote by \(L_{PQ}\) the part of \(L\) between points \(P\) and \(Q\) and by \(l_{PQ}\) the length of this part. Let us denote by \(S_{i}\) \((i = 1, 2, 3, 4)\) the vertices of \(S\) and by \(T_{i}\) points... |
IMOSL-1982-7 | Let \(p(x)\) be a cubic polynomial with integer coefficients with leading coefficient 1 and with one of its roots equal to the product of the other two. Show that \(2p(-1)\) is a multiple of \(p(1) + p(-1) - 2(1 + p(0))\) . | Let \(a, b, ab\) be the roots of the cubic polynomial \(p(x) = (x - a)(x - b)(x - ab)\) . Observe that
\[2p(-1) = -2(1 + a)(1 + b)(1 + ab);\] \[p(1) + p(-1) - 2(1 + p(0)) = -2(1 + a)(1 + b).\]
The statement of the problem is trivial if both the expressions are equal to zero. Otherwise, the quotient \(\frac{2p(- 1... |
IMOSL-1982-8 | A convex, closed figure lies inside a given circle. The figure is seen from every point of the circumference at a right angle (that is, the two rays drawn from the point and supporting the convex figure are perpendicular). Prove that the center of the circle is a center of symmetry of the figure. | Let \(\mathcal{F}\) be the given figure. Consider any chord \(AB\) of the circumcircle \(\gamma\) that supports \(\mathcal{F}\) . The other supporting lines to \(\mathcal{F}\) from \(A\) and \(B\) intersect \(\gamma\) again at \(D\) and \(C\) respectively so that \(\angle DAB = \angle ABC = 90^{\circ}\) . Then \(ABCD\)... |
IMOSL-1982-9 | Let \(ABC\) be a triangle, and let \(P\) be a point inside it such that \(\angle PAC = \angle PBC\) . The perpendiculars from \(P\) to \(BC\) and \(CA\) meet these lines at \(L\) and \(M\) , respectively, and \(D\) is the midpoint of \(AB\) . Prove that \(DL = DM\) . | Let \(X\) and \(Y\) be the midpoints of the segments \(AP\) and \(BP\) . Then \(DYPX\) is a
parallelogram. Since \(X\) and \(Y\) are the circumcenters of the triangles \(APM\) and \(BPL\) , we conclude that \(XM = XP = DY\) and \(YL = YP = DX\) . Furthermore, we have \(\angle DXM = \angle DXP + \angle PXM = \angle D... |
IMOSL-1982-10 | A box contains \(p\) white balls and \(q\) black balls. Beside the box there is a pile of black balls. Two balls are taken out of the box. If they have the same color, a black ball from the pile is put into the box. If they have different colors, the white ball is put back into the box. This procedure is repeated until... | If the two balls taken from the box are both white, then the number of white balls decreases by two; otherwise, it remains unchanged. Hence the parity of the number of white balls does not change during the procedure. Therefore if \(p\) is even, the last ball cannot be white; the probability is 0. If \(p\) is odd, the ... |
IMOSL-1982-11 | (a) Find the rearrangement \(\{a_{1}, \ldots , a_{n}\}\) of \(\{1, 2, \ldots , n\}\) that maximizes
\[a_{1}a_{2} + a_{2}a_{3} + \dots +a_{n}a_{1} = Q.\]
(b) Find the rearrangement that minimizes \(Q\) . | (a) Suppose \(\{a_1, a_2, \ldots , a_n\}\) is the arrangement that yields the maximal value \(Q_{\max}\) of \(Q\) . Note that the value of \(Q\) for the rearrangement \(\{a_1, \ldots , a_{i-1}, a_j, a_{j-1}, \ldots , a_i, a_{j+1}, \ldots , a_n\}\) equals \(Q_{\max} - (a_i - a_j)(a_{i-1} - a_{j+1})\) , where \(1 < i < j... |
IMOSL-1982-12 | Four distinct circles \(C, C_{1}, C_{2}, C_{3}\) and a line \(L\) are given in the plane such that \(C\) and \(L\) are disjoint and each of the circles \(C_{1}, C_{2}, C_{3}\) touches the other
two, as well as \(C\) and \(L\) . Assuming the radius of \(C\) to be 1, determine the distance between its center and \(L\) . | Let \(y\) be the line perpendicular to \(L\) passing through the center of \(C\) . It can be shown by a continuity argument that there exists a point \(Y \in y\) such that an inversion \(\Psi\) centered at \(Y\) maps \(C\) and \(L\) onto two concentric circles \(\widehat{C}\) and \(\widehat{L}\) . Let \(\widehat{X}\) d... |
IMOSL-1982-13 | A scalene triangle \(A_{1}A_{2}A_{3}\) is given with sides \(a_{1},a_{2},a_{3}\) ( \(a_{i}\) is the side opposite to \(A_{i}\) ). For all \(i = 1,2,3\) , \(M_{i}\) is the midpoint of side \(a_{i}\) , \(T_{i}\) is the point where the incircle touches side \(a_{i}\) , and the reflection of \(T_{i}\) in the interior bisec... | The points \(S_{1}\) , \(S_{2}\) , \(S_{3}\) clearly lie on the inscribed circle. Let \(\widehat{X}\widehat{Y}\) denote the oriented arc \(XY\) . The arcs \(\widehat{T_2S_1}\) and \(\widehat{T_1T_3}\) are equal, since they are symmetric with respect to the bisector of \(\angle A_{1}\) . Similarly, \(\widehat{T_3T_2} = ... |
IMOSL-1982-14 | Let \(ABCD\) be a convex plane quadrilateral and let \(A_{1}\) denote the circumcenter of \(\triangle BCD\) . Define \(B_{1},C_{1},D_{1}\) in a corresponding way.
(a) Prove that either all of \(A_{1},B_{1},C_{1},D_{1}\) coincide in one point, or they are all distinct. Assuming the latter case, show that \(A_{1},C_{1... | (a) If any two of \(A_1, B_1, C_1, D_1\) coincide, say \(A_1 \equiv B_1\) , then \(ABCD\) is inscribed in a circle centered at \(A_1\) and hence all \(A_1, B_1, C_1, D_1\) coincide. Assume now the opposite, and let w.l.o.g. \(\angle DAB + \angle DCB < 180^\circ\) . Then \(A\) is outside the circumcircle of \(\triangle ... |
IMOSL-1982-15 | Show that
\[\frac{1 - s^{a}}{1 - s} \leq (1 + s)^{a - 1}\]
holds for every \(1 \neq s > 0\) real and \(0 < a \leq 1\) rational. | Let \(a = k / n\) , where \(n,k\in \mathbb{N}\) \(n\geq k\) . Putting \(t^{n} = s\) , the given inequality becomes \(\frac{1 - t^{k}}{1 - t^{n}}\leq (1 + t^{n})^{k / n - 1}\) , or equivalently
\[(1 + t + \dots +t^{k - 1})^{n}(1 + t^{n})^{n - k}\leq (1 + t + \dots +t^{n - 1})^{n}.\]
This is clearly true for \(k = ... |
IMOSL-1982-16 | Prove that if \(n\) is a positive integer such that the equation \(x^{3} - 3xy^{2} + y^{3} = n\) has a solution in integers \((x,y)\) , then it has at least three such solutions. Show that the equation has no solution in integers when \(n = 2891\) . | It is easy to verify that whenever \((x,y)\) is a solution of the equation \(x^{3} - 3x y^{2}+\) \(y^{3} = n\) , so are the pairs \((y - x, - x)\) and \((- y,x - y)\) . No two of these three solutions are equal unless \(x = y = n = 0\)
Observe that \(2891\equiv 2\) (mod 9). Since \(x^{3},y^{3}\equiv 0,\pm 1\) (mod 9... |
IMOSL-1982-17 | The right triangles \(ABC\) and \(AB_{1}C_{1}\) are similar and have opposite orientation. The right angles are at \(C\) and \(C_{1}\) , and we also have \(\angle CAB = \angle C_{1}AB_{1}\) . Let \(M\) be the point of intersection of the lines \(BC_{1}\) and \(B_{1}C\) . Prove that if the lines \(AM\) and \(CC_{1}\) ex... | Let \(A\) be the origin of the Cartesian plane. Suppose that \(BC:AC = k\) and that \((a,b)\) and \((a_{1},b_{1})\) are coordinates of the points \(C\) and \(C_{1}\) , respectively. Then the coordinates of the point \(B\) are \((a,b) + k(-b,a) = (a - kb,b + ka)\) , while the coordinates of \(B_{1}\) are \((a_{1},b_{1})... |
IMOSL-1982-18 | Let \(O\) be a point of three-dimensional space and let \(l_{1},l_{2},l_{3}\) be mutually perpendicular straight lines passing through \(O\) . Let \(S\) denote the sphere with center \(O\) and radius \(R\) , and for every point \(M\) of \(S\) , let \(S_{M}\) denote the sphere with center \(M\) and radius \(R\) . We den... | Set the coordinate system with the axes \(x,y,z\) along the lines \(l_{1},l_{2},l_{3}\) respectively. The coordinates \((a,b,c)\) of \(M\) satisfy \(a^{2} + b^{2} + c^{2} = R^{2}\) , and so \(S_{M}\) is given by the equation \((x - a)^{2} + (y - b)^{2} + (z - c)^{2} = R^{2}\) . Hence the coordinates of \(P_{1}\) are \(... |
IMOSL-1982-19 | Let \(M\) be the set of real numbers of the form \(\frac{m + n}{\sqrt{m^{2} + n^{2}}}\) , where \(m\) and \(n\) are positive integers. Prove that for every pair \(x \in M\) , \(y \in M\) with \(x < y\) , there exists an element \(z \in M\) such that \(x < z < y\) . | Let us set \(x = m / n\) . Since \(f(x) = (m + n) / \sqrt{m^2 + n^2} = (x + 1) / \sqrt{1 + x^2}\) is a continuous function of \(x\) , \(f(x)\) takes all values between any two values of \(f\) ; moreover, the corresponding \(x\) can be rational. This completes the proof.
Remark. Since \(f\) is increasing for \(x \geq... |
IMOSL-1982-20 | Let \(ABCD\) be a convex quadrilateral and draw regular triangles \(ABM\) , \(CDP\) , \(BCN\) , \(ADQ\) , the first two outward and the other two inward. Prove that \(MN = AC\) . What can be said about the quadrilateral \(MNPQ\) ? | Since \(MN\) is the image of \(AC\) under rotation about \(B\) for \(60^{\circ}\) , we have \(MN = AC\) . Similarly, \(PQ\) is the image of \(AC\) under rotation about \(D\) through \(60^{\circ}\) , from which it follows that \(PQ \parallel MN\) . Hence either \(M, N, P, Q\) are collinear or \(MNPQ\) is a parallelogram... |
IMOSL-1983-1 | The localities \(P_{1}, P_{2}, \ldots , P_{1983}\) are served by ten international airlines \(A_{1}, A_{2}, \ldots , A_{10}\) . It is noticed that there is direct service (without stops) between any two of these localities and that all airline schedules offer round-trip flights. Prove that at least one of the airlines ... | Suppose that there are \(n\) airlines \(A_{1},\ldots ,A_{n}\) and \(N > 2^{n}\) cities. We shall prove that there is a round trip by at least one \(A_{i}\) containing an odd number of stops.
For \(n = 1\) the statement is trivial, since one airline serves at least 3 cities and hence \(P_{1}P_{2}P_{3}P_{1}\) is a rou... |
IMOSL-1983-2 | Let \(n\) be a positive integer. Let \(\sigma (n)\) be the sum of the natural divisors \(d\) of \(n\) (including 1 and \(n\) ). We say that an integer \(m \geq 1\) is superabundant (P. Erdős, 1944) if \(\forall k \in \{1, 2, \ldots , m - 1\}\) , \(\frac{\sigma(m)}{m} > \frac{\sigma(k)}{k}\) . Prove that there exists an... | By definition, \(\sigma (n) = \sum_{d|n}d = \sum_{d|n}n / d = n\sum_{d|n}1 / d\) , hence \(\sigma (n) / n = \sum_{d|n}1 / d\) . In particular, \(\sigma (n!) / n! = \sum_{d|n}!1 / d\geq \sum_{k = 1}^{n}1 / k\) . It follows that the sequence \(\sigma (n) / n\) is unbounded, and consequently there exist an infinite number... |
IMOSL-1983-3 | We say that a set \(E\) of points of the Euclidian plane is "Pythagorean" if for any partition of \(E\) into two sets \(A\) and \(B\) , at least one of the sets contains the vertices of a right-angled triangle. Decide whether the following sets are Pythagorean:
(a) a circle;
(b) an equilateral triangle (that is, th... | (a) A circle is not Pythagorean. Indeed, consider the partition into two semicircles each closed at one and open at the other end.
(b) An equilateral triangle, call it \(PQR\) , is Pythagorean. Let \(P^{\prime}\) , \(Q^{\prime}\) , and \(R^{\prime}\) be the points on \(QR\) , \(RP\) , and \(PQ\) such that \(PR^{\pri... |
IMOSL-1983-4 | On the sides of the triangle \(ABC\) , three similar isosceles triangles \(ABP\) \((AP = PB)\) , \(AQC\) \((AQ = QC)\) , and \(BRC\) \((BR = RC)\) are constructed. The first two are constructed externally to the triangle \(ABC\) , but the third is placed in the same half-plane determined by the line \(BC\) as the trian... | The rotational homothety centered at \(C\) that sends \(B\) to \(R\) also sends \(A\) to \(Q\) ; hence the triangles \(ABC\) and \(QRC\) are similar. For the same reason, \(\triangle ABC\) and \(\triangle PBR\) are similar. Moreover, \(BR = CR\) ; hence \(\triangle CRQ \cong \triangle RBP\) . Thus \(PR = QC = AQ\) and ... |
IMOSL-1983-5 | Consider the set of all strictly decreasing sequences of \(n\) natural numbers having the property that in each sequence no term divides any other term of the sequence. Let \(A = (a_j)\) and \(B = (b_j)\) be any two such sequences. We say that \(A\) precedes \(B\) if for some \(k\) , \(a_k < b_k\) and \(a_i = b_i\) for... | Each natural number \(p\) can be written uniquely in the form \(p = 2^{q}(2r - 1)\) . We call \(2r - 1\) the odd part of \(p\) . Let \(A_{n} = (a_{1},a_{2},\ldots ,a_{n})\) be the first sequence. Clearly the terms of \(A_{n}\) must have different odd parts, so those parts must be at
least \(1,3,\ldots ,2n - 1\) . Bein... |
IMOSL-1983-6 | Suppose that \(\{x_1, x_2, \ldots , x_n\}\) are positive integers for which \(x_1 + x_2 + \dots + x_n = 2(n + 1)\) . Show that there exists an integer \(r\) with \(0 \leq r \leq n - 1\) for which the following \(n - 1\) inequalities hold:
\[x_{r + 1} + \dots +x_{r + i}\leq 2i + 1\qquad \forall i,1\leq i\leq n - r;\]... | The existence of \(r\) : Let \(S = \{x_{1} + x_{2} + \dots +x_{i} - 2i\mid i = 1,2,\dots ,n\}\) . Let \(\mathrm{max}S\) be attained for the first time at \(r^{\prime}\) .
If \(r^{\prime} = n\) , then \(x_{1} + x_{2} + \dots +x_{i} - 2i< 2\) for \(1\leq i\leq n - 1\) , so one can take \(r = r^{\prime}\) . Suppose tha... |
IMOSL-1983-7 | Let \(a\) be a positive integer and let \(\{a_n\}\) be defined by \(a_0 = 0\) and
\[a_{n + 1} = (a_n + 1)a + (a + 1)a_n + 2\sqrt{a(a + 1)a_n(a_n + 1)}\qquad (n = 1,2\ldots).\]
Show that for each positive integer \(n\) , \(a_n\) is a positive integer. | Clearly, each \(a_{n}\) is positive and \(\sqrt{a_{n + 1}} = \sqrt{a_{n}}\sqrt{a + 1} +\sqrt{a_{n} + 1}\sqrt{a}\) . Notice that \(\sqrt{a_{n + 1} + 1} = \sqrt{a + 1}\sqrt{a_{n} + 1} +\sqrt{a}\sqrt{a_{n}}\) . Therefore
\[(\sqrt{a + 1} -\sqrt{a})(\sqrt{a_{n} + 1} -\sqrt{a_{n}})\] \[\quad = (\sqrt{a + 1}\sqrt{a_{n} + 1... |
IMOSL-1983-8 | In a test, \(3n\) students participate, who are located in three rows of \(n\) students in each. The students leave the test room one by one. If \(N_1(t), N_2(t), N_3(t)\) denote the numbers of students in the first, second, and third row respectively at time \(t\) , find the probability that for each \(t\) during the ... | Situations in which the condition of the statement is fulfilled are the following:
\(S_{1}\) .. \(N_{1}(t) = N_{2}(t) = N_{3}(t)\) \(S_{2}\) .. \(N_{i}(t) = N_{j}(t) = h,N_{k}(t) = h + 1\) , where \((i,j,k)\) is a permutation of the set \(\{1,2,3\}\) . In this case the first student to leave must be from row \(k\) .... |
IMOSL-1983-9 | If \(a, b\) , and \(c\) are sides of a triangle, prove that
\[a^{2}b(a - b) + b^{2}c(b - c) + c^{2}a(c - a)\geq 0.\]
Determine when there is equality. | For any triangle of sides \(a,b,c\) there exist 3 nonnegative numbers \(x,y,z\) such that \(a = y + z\) \(b = z + x\) \(c = x + y\) (these numbers correspond to the division of the sides of a triangle by the point of contact of the incircle). The inequality becomes
\[(y + z)^{2}(z + x)(y - x) + (z + x)^{2}(x + y)(z ... |
IMOSL-1983-10 | Let \(p\) and \(q\) be integers. Show that there exists an interval \(I\) of length \(1 / q\) and a polynomial \(P\) with integral coefficients such that
\[\left|P(x) - \frac{p}{q}\right|< \frac{1}{q^2}\]
for all \(x\in I\) | Choose \(P(x) = \frac{p}{q}\left((qx - 1)^{2n + 1} + 1\right)\) , \(I = [1 / 2q,3 / 2q]\) . Then all the coefficients of \(P\) are integers, and
\[\left|P(x) - \frac{p}{q}\right| = \left|\frac{p}{q} (qx - 1)^{2n + 1}\right| \leq \left|\frac{p}{q}\right|\frac{1}{2^{2n + 1}},\]
for \(x\in I\) . The desired inequali... |
IMOSL-1983-11 | (FIN 2') Let \(f:[0,1]\to \mathbb{R}\) be continuous and satisfy:
\[\begin{array}{r l r} & {} & {b f(2x) = f(x),\qquad 0\leq x\leq 1 / 2;}\\ & {} & {f(x) = b + (1 - b)f(2x - 1),\qquad 1 / 2\leq x\leq 1,} \end{array} \quad (11)\]
where \(b = \frac{1 + c}{2 + c}\) , \(c > 0\) . Show that \(0< f(x) - x< c\) for ever... | First suppose that the binary representation of \(x\) is finite: \(x = 0, a_1 a_2 \ldots a_n = \sum_{j = 1}^{n} a_j 2^{-j}, a_i \in \{0, 1\}\) . We shall prove by induction on \(n\) that
\[f(x) = \sum_{j = 1}^{n} b_0 \ldots b_{j - 1} a_j, \quad \text{where} b_k = \left\{ \begin{array}{ll} -b & \text{if} a_k = 0, \\ ... |
IMOSL-1983-12 | Find all functions \(f\) defined on the positive real numbers and taking positive real values that satisfy the following conditions:
(i) \(f(xf(y)) = yf(x)\) for all positive real \(x,y\)
(ii) \(f(x)\to 0\) as \(x\to +\infty\) | Putting \(y = x\) in (i) we see that there exist positive real numbers \(z\) such that \(f(z) = z\) (this is true for every \(z = x f(x)\) ). Let \(a\) be any of them. Then \(f(a^2) = f(af(a)) = af(a) = a^2\) , and by induction, \(f(a^n) = a^n\) . If \(a > 1\) , then \(a^n \to +\infty\) as \(n \to \infty\) , and we hav... |
IMOSL-1983-13 | Let \(E\) be the set of \(1983^{3}\) points of the space \(\mathbb{R}^{3}\) all three of whose coordinates are integers between 0 and 1982 (including 0 and 1982). A coloring of \(E\) is a map from \(E\) to the set {red, blue}. How many colorings of \(E\) are there satisfying the following property: The number of red ve... | Given any coloring of the \(3 \times 1983 - 2\) points of the axes, we prove that there is a unique coloring of \(E\) having the given property and extending this coloring.
The first thing to notice is that given any rectangle \(R_{1}\) parallel to a coordinate plane and whose edges are parallel to the axes, there is ... |
IMOSL-1983-14 | Prove or disprove: From the interval \([1,\ldots ,30000]\) one can select a set of 1000 integers containing no arithmetic triple (three consecutive numbers of an arithmetic progression). | Let \(T_{n}\) be the set of all nonnegative integers whose ternary representations consist of at most \(n\) digits and do not contain a digit 2. The cardinality of \(T_{n}\) is \(2^{n}\) , and the greatest integer in \(T_{n}\) is \(11 \ldots 1 = 3^{0} + 3^{1} + \dots + 3^{n - 1} = (3^{n} - 1) / 2\) . We claim that ther... |
IMOSL-1983-15 | Decide whether there exists a set \(M\) of natural numbers satisfying the following conditions:
(i) For any natural number \(m > 1\) there are \(a,b\in M\) such that \(a + b = m\) (ii) If \(a,b,c,d\in M,a,b,c,d > 10\) and \(a + b = c + d\) , then \(a = c\) or \(a = d\) | There is no such set. Suppose that \(M\) satisfies the conditions (i) and (ii) and let \(q_{n} = |\{a \in M: a \leq n\}|\) . Consider the differences \(b - a\) , where \(a, b \in M\) and \(10 < a < b \leq k\) . They are all positive and less than \(k\) , and (ii) implies that they are \(\binom{q_{k} - q_{10}}{2}\) diff... |
IMOSL-1983-16 | Let \(F(n)\) be the set of polynomials \(P(x) = a_{0} + a_{1}x + \dots +a_{n}x^{n}\) , with \(a_{0},a_{1},\ldots ,a_{n}\in \mathbb{R}\) and \(0\leq a_{0} = a_{n}\leq a_{1} = a_{n - 1}\leq \dots \leq a_{[n / 2]} = a_{[(n + 1) / 2]}\) . Prove that if \(f\in F(m)\) and \(g\in F(n)\) , then \(f g\in F(m + n)\) . | Set \(h_{n,i}(x) = x^{i} + \dots + x^{n - i}\) , \(2i \leq n\) . The set \(F(n)\) is the set of linear combinations with nonnegative coefficients of the \(h_{n,i}\) 's. This is a convex cone. Hence, it suffices to prove that \(h_{n,i}h_{m,j} \in F(m + n)\) . Indeed, setting \(p = n - 2i\) and \(q = m - 2j\) and assumin... |
IMOSL-1983-17 | Let \(P_{1},P_{2},\ldots ,P_{n}\) be distinct points of the plane, \(n\geq 2\) . Prove that
\[\max_{1\leq i< j\leq n}P_{i}P_{j} > \frac{\sqrt{3}}{2} (\sqrt{n} -1)\min_{1\leq i< j\leq n}P_{i}P_{j}.\] | Set \(a = \min P_{i}P_{j}\) , \(b = \max P_{i}P_{j}\) . We use the following lemma.
Lemma. There exists a disk of radius less than or equal to \(b / \sqrt{3}\) containing all the \(P_{i}\) 's.
Assuming that this is proved, the disks with center \(P_{i}\) and radius \(a / 2\) are disjoint and included in a disk of r... |
IMOSL-1983-18 | Let \(a,b,c\) be positive integers satisfying \((a,b) = (b,c) = (c,a) = 1\) . Show that \(2abc - ab - bc - ca\) is the largest integer not representable as
\[xbc + yca + zab\]
with nonnegative integers \(x,y,z\) . | Let \((x_{0}, y_{0}, z_{0})\) be one solution of \(b cx + cay + abz = n\) (not necessarily nonnegative). By subtracting \(b cx_{0} + cay_{0} + abz_{0} = n\) we get
\[bc(x - x_0) + ca(y - y_0) + ab(z - z_0) = 0.\]
Since \((a, b) = (a, c) = 1\), we must have \(a|x - x_0\) or \(x - x_0 = as\). Substituting this in the ... |
IMOSL-1983-19 | Let \((F_{n})_{n \geq 1}\) be the Fibonacci sequence \(F_{1} = F_{2} = 1\) , \(F_{n + 2} = F_{n + 1} + F_{n}\) \((n \geq 1)\) , and \(P(x)\) the polynomial of degree 990 satisfying
\[P(k) = F_{k},\quad \mathrm{for} k = 992,\ldots ,1982.\]
Prove that \(P(1983) = F_{1983} - 1\) . | For all \(n\), there exists a unique polynomial \(P_{n}\) of degree \(n\) such that \(P_{n}(k) = F_{k}\) for \(n + 2 \le k \le 2n + 2\) and \(P_{n}(2n + 3) = F_{2n+3} - 1\). For \(n = 0\), we have \(F_{1} =\)
\(F_{2} = 1\) , \(F_{3} = 2\) , \(P_{0} = 1\) . Now suppose that \(P_{n - 1}\) has been constructed and let \(... |
IMOSL-1983-20 | Solve the system of equations
\[x_{1}|x_{1}| = x_{2}|x_{2}| + (x_{1} - a)|x_{1} - a|,\] \[x_{2}|x_{2}| = x_{3}|x_{3}| + (x_{2} - a)|x_{2} - a|,\] \[\qquad \ldots\] \[x_{n}|x_{n}| = x_{1}|x_{1}| + (x_{n} - a)|x_{n} - a|,\]
in the set of real numbers, where \(a > 0\) . | If \((x_{1},x_{2},\ldots ,x_{n})\) satisfies the system with parameter \(a\) , then \((-x_{1}, - x_{2},\ldots , - x_{n})\) satisfies the system with parameter \(-a\) . Hence it is sufficient to consider only \(a\geq 0\) .
Let \((x_{1},\ldots ,x_{n})\) be a solution. Suppose \(x_{1}\leq a\) , \(x_{2}\leq a,\ldots ,x_... |
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