id stringlengths 10 15 | question stringlengths 63 2.3k | solutions stringlengths 20 28.5k |
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IMOLL-1967-51 | A subset \(S\) of the set of integers \(0, \ldots , 99\) is said to have property A if it is impossible to fill a crossword puzzle with 2 rows and 2 columns with numbers in \(S\) (0 is written as 00, 1 as 01, and so on). Determine the maximal number of elements in sets \(S\) with property A. | If there exist two numbers \(\overline{ab},\overline{bc}\in S\) , then one can fill a crossword puzzle as \(\left( \begin{array}{c}a\\ b\\ c \end{array} \right)\) . The converse is obvious. Hence the set \(S\) has property \(A\) if and only if
the set of first digits and the set of second digits of numbers in \(S\) ar... |
IMOLL-1967-52 | In the plane a point \(O\) and a sequence of points \(P_{1}, P_{2}, P_{3}, \ldots\) are given. The distances \(OP_{1}, OP_{2}, OP_{3}, \ldots\) are \(r_{1}, r_{2}, r_{3}, \ldots\) , where \(r_{1} \leq r_{2} \leq r_{3} \leq \ldots\) . Let \(\alpha\) satisfy \(0 < \alpha < 1\) . Suppose that for every \(n\) the distance ... | This problem is not elementary. The solution offered by the proposer was not quite clear and complete (the existence was not proved). |
IMOLL-1967-53 | In making Euclidean constructions in geometry it is permitted to use a straightedge and compass. In the constructions considered in this question, no compasses are permitted, but the straightedge is assumed to have two parallel edges, which can be used for constructing two parallel lines through two given points whose ... | (a) We can construct two lines parallel to the rays of the angle, at equal distances from the rays. The intersection of these two lines lies on the bisector of the angle.
(b) If the length of a segment \(AB\) exceeds the breadth of the ruler, we can construct parallel lines through \(A\) and \(B\) in two different way... |
IMOLL-1967-54 | Is it possible to put 100 (or 200) points on a wooden cube such that by all rotations of the cube the points map into themselves? Justify your answer. | Let \(S\) be the given set of points on the cube. Let \(x, y, z\) denote the numbers of points from \(S\) lying at a vertex, at the midpoint of an edge, at the midpoint of a face of the cube, respectively, and let \(u\) be the number of all other points from \(S\) . Either there are no points from \(S\) at the vertices... |
IMOLL-1967-55 | Find all \(x\) for which for all \(n\)
\[\sin x + \sin 2x + \sin 3x + \dots +\sin nx\leq \frac{\sqrt{3}}{2}.\] | It is enough to find all \(x\) from \((0, 2\pi ]\) such that the given inequality holds for all integers \(n\) .
Suppose \(0 < x < 2\pi /3\) . If \(n\) is the maximum integer for which \(nx \leq 2\pi /3\) , we have \(\pi /3 < nx \leq 2\pi /3\) , and consequently \(\sin nx \geq \sqrt{3} /2\) . Thus \(\sin x + \sin 2x... |
IMOLL-1967-56 | In a group of interpreters each one speaks one or several foreign languages; 24 of them speak Japanese, 24 Malay, 24 Farsi. Prove that it is possible to select a subgroup in which exactly 12 interpreters speak Japanese, exactly 12 speak Malay, and exactly 12 speak Farsi. | We shall prove by induction on \(n\) the following statement: If in some group of interpreters exactly \(n\) persons, \(n \geq 2\) , speak each of the three languages, then it is possible to select a subgroup in which each language is spoken by exactly two persons.
The statement of the problem easily follows from th... |
IMOLL-1967-57 | Consider the sequence \((c_{n})\) :
\[c_{1} = a_{1} + a_{2} + \dots +a_{8},\] \[c_{2} = a_{1}^{2} + a_{2}^{2} + \dots +a_{8}^{2},\] \[\dots\] \[c_{n} = a_{1}^{n} + a_{2}^{n} + \dots +a_{8}^{n},\] \[\dots\]
where \(a_{1},a_{2},\ldots ,a_{8}\) are real numbers, not all equal to zero. Given that among the numbers of... | Obviously \(c_{n} > 0\) for all even \(n\) . Thus \(c_{n} = 0\) is possible only for an odd \(n\) . Let us assume \(a_{1} \leq a_{2} \leq \dots \leq a_{8}\) : in particular, \(a_{1} \leq 0 \leq a_{8}\) .
If \(|a_{1}| < |a_{8}|\) , then there exists \(n_{0}\) such that for every odd \(n > n_{0}\) , \(7|a_{1}|^{n} < a... |
IMOLL-1967-58 | A linear binomial \(l(z) = Az + B\) with complex coefficients \(A\) and \(B\) is given. It is known that the maximal value of \(|l(z)|\) on the segment \(-1\leq x\leq 1\) \((y = 0)\) of the real line in the complex plane \((z = x + iy)\) is equal to \(M\) . Prove that for every \(z\)
\[|l(z)|\leq M\rho ,\]
where ... | The following sequence of equalities and inequalities gives an even stronger estimate than needed.
\[|I(z)| = |Az + B| = \frac{1}{2} |(z + 1)(A + B) + (z - 1)(A - B)|\] \[\qquad = \frac{1}{2} |(z + 1)f(1) + (z - 1)f(-1)|\] \[\qquad \leq \frac{1}{2} |(z + 1|\cdot |f(1)| + |z - 1|\cdot |f(-1)|)\] \[\qquad \leq \frac{1... |
IMOLL-1967-59 | On the circle with center \(O\) and radius 1 the point \(A_{0}\) is fixed and points \(A_{1},A_{2},\ldots ,A_{999},A_{1000}\) are distributed in such a way that \(\angle A_{0}O A_{k} = k\) (in radians). Cut the circle at points \(A_{0},A_{1},\ldots ,A_{1000}\) . How many arcs with different lengths are obtained? | By the arc \(AB\) we shall always mean the positive arc \(AB\) . We denote by \(|AB|\) the length of arc \(AB\) . Let a basic arc be one of the \(n + 1\) arcs into which the circle is partitioned by the points \(A_{0}, A_{1}, \ldots , A_{n}\) , where \(n \in \mathbb{N}\) .
Suppose that \(A_{p}A_{0}\) and \(A_{0}A_{q}\... |
IMOSL-1968-1 | Two ships sail on the sea with constant speeds and fixed directions. It is known that at 9:00 the distance between them was 20 miles; at 9:35, 15 miles; and at 9:55, 13 miles. At what moment were the ships the smallest distance from each other, and what was that distance? | Since the ships are sailing with constant speeds and directions, the second ship is sailing at a constant speed and direction in reference to the first ship. Let \(A\) be the constant position of the first ship in this frame. Let \(B_{1}, B_{2}, B_{3}\) , and \(B\) on line \(b\) defining the trajectory of the ship be p... |
IMOSL-1968-2 | Prove that there exists a unique triangle whose side lengths are consecutive natural numbers and one of whose angles is twice the measure of one of the others. | The sides \(a,b,c\) of a triangle \(A B C\) with \(\angle A B C = 2\angle B A C\) satisfy \(b^{2} = a(a + c)\) (this statement is the lemma in (SL98-7)). Taking into account the remaining condition that \(a,b,c\) are consecutive integers with \(a< b\) , we obtain three cases:
(i) \(a = n,b = n + 1,c = n + 2\) . We g... |
IMOSL-1968-3 | Prove that in any tetrahedron there is a vertex such that the lengths of its sides through that vertex are sides of a triangle. | A triangle cannot be formed out of three lengths if and only if one of them is larger than the sum of the other two. Let us assume this is the case for all triplets of edges out of each vertex in a tetrahedron \(A B C D\) . Let w.l.o.g. \(A B\) be the largest edge of the tetrahedron. Then \(A B\geq A C + A D\) and \(A ... |
IMOSL-1968-4 | Let \(a, b, c\) be real numbers. Prove that the system of equations
\[
\begin{cases}
a x_1^2 + b x_1 + c = x_2, \\
a x_2^2 + b x_2 + c = x_3, \\
\vdots \\
a x_{n-1}^2 + b x_{n-1} + c = x_n, \\
a x_n^2 + b x_n + c = x_1.
\end{cases}
\]
has a unique real solution if and only if \((b - 1)^2 - 4ac = 0\) .
Remark. It... | We will prove the equivalence in the two directions separately:
( \(\Rightarrow\) ) Suppose \(\{x_{1},\ldots ,x_{n}\}\) is the unique solution of the equation. Since \(\{x_{n},x_{1}\) \(x_{2},\ldots ,x_{n - 1}\}\) is also a solution, it follows that \(x_{1} = x_{2} = \dots = x_{n} = x\) and
the system of equations ... |
IMOSL-1968-5 | Let \(h_n\) be the apothem (distance from the center to one of the sides) of a regular \(n\) -gon \((n \geq 3)\) inscribed in a circle of radius \(r\) . Prove the inequality
\[(n + 1)h_{n + 1} - nh_n > r.\]
Also prove that if \(r\) on the right side is replaced with a greater number, the inequality will not remai... | We have \(h_{k} = r\cos (\pi /k)\) for all \(k\in \mathbb{N}\) . Using \(\cos x = 1 - 2\sin^{2}(x / 2)\) and \(\cos x =\) \(2 / (1 + \tan^{2}(x / 2)) - 1\) and \(\tan x > x > \sin x\) for all \(0< x< \pi /2\) , it suffices to prove
\[\quad (n + 1)\left(1 - 2\frac{\pi^{2}}{4(n + 1)^{2}}\right) - n\left(\frac{2}{1 + \... |
IMOSL-1968-6 | If \(a_i (i = 1, 2, \ldots , n)\) are distinct non-zero real numbers, prove that the equation
\[\frac{a_1}{a_1 - x} + \frac{a_2}{a_2 - x} + \dots + \frac{a_n}{a_n - x} = n\]
has at least \(n - 1\) real roots. | We define \(f(x) = \frac{a_{1}}{a_{1} - x} +\frac{a_{2}}{a_{2} - x} +\dots +\frac{a_{n}}{a_{n} - x}\) . Let us assume w.l.o.g. \(a_{1}< a_{2}< \dots < a_{n}\) . We note that for all \(1\leq i< n\) the function \(f\) is continuous in the interval \((a_{i},a_{i + 1})\) and satisfies \(\lim_{x\to a_{i}}f(x) = -\infty\) an... |
IMOSL-1968-7 | Prove that the product of the radii of three circles exscribed to a given triangle does not exceed \(\frac{3\sqrt{3}}{8}\) times the product of the side lengths of the triangle. When does equality hold? | Let \(r_{a},r_{b},r_{c}\) denote the radii of the exscribed circles corresponding to the sides of lengths \(a,b,c\) respectively, and \(R\) \(p\) and \(S\) denote the circumradius, semiperimeter, and area of the given triangle. It is well-known that \(r_{a}(p - a) = r_{b}(p - b) =\) \(r_{c}(p - c) = S = \sqrt{p(p - a)(... |
IMOSL-1968-8 | Given an oriented line \(\Delta\) and a fixed point \(A\) on it, consider all trapezoids \(ABCD\) one of whose bases \(AB\) lies on \(\Delta\) , in the positive direction. Let \(E, F\) be the midpoints of \(AB\) and \(CD\) respectively.
Find the loci of vertices \(B, C, D\) of trapezoids that satisfy the following: ... | Let \(G\) be the point such that \(BCD G\) is a parallelogram and let \(H\) be the midpoint of \(AG\) . Obviously \(H E F D\) is also a parallelogram, and thus \(D H = E F = l\) . If \(A D^{2} + B C^{2} = m^{2}\) is fixed, then from the Stewart theorem we have
\[D H^{2} = \frac{2D A^{2} + 2D G^{2} - A G^{2}}{4} = \fra... |
IMOSL-1968-9 | Let \(ABC\) be an arbitrary triangle and \(M\) a point inside it. Let \(d_a, d_b, d_c\) be the distances from \(M\) to sides \(BC, CA, AB\) ; \(a, b, c\) the lengths of the sides respectively, and \(S\) the area of the triangle \(ABC\) . Prove the inequality
\[a b_{d}d_{b} + b c d_{b}d_{c} + c a d_{c}d_{a}\leq \frac{4... | We note that \(S_{a} = a d_{a} / 2\) , \(S_{b} = b d_{b} / 2\) , and \(S_{c} = c d_{c} / 2\) are the areas of the triangles \(M B C\) , \(M C A\) , and \(M A B\) respectively. The desired inequality now follows from
\[S_{a}S_{b} + S_{b}S_{c} + S_{c}S_{a}\leq \frac{1}{3} (S_{a} + S_{b} + S_{c})^{2} = \frac{S^{2}}{3}.... |
IMOSL-1968-10 | Consider two segments of length \(a, b\) ( \(a > b\) ) and a segment of length \(c = \sqrt{ab}\) .
(a) For what values of \(a / b\) can these segments be sides of a triangle?
(b) For what values of \(a / b\) is this triangle right-angled, obtuse-angled, or acute-angled? | (a) Let us set \(k = a / b > 1\) . Then \(a = k b\) and \(c = \sqrt{k} b\) , and \(a > c > b\) . The segments \(a, b, c\) form a triangle if and only if \(k < \sqrt{k} + 1\) , which holds if and only if \(1 < k < \frac{3 + \sqrt{5}}{2}\) . (b) The triangle is right-angled if and only if \(a^{2} = b^{2} + c^{2} \Leftrig... |
IMOSL-1968-11 | Find all solutions \((x_{1}, x_{2}, \ldots , x_{n})\) of the equation
\[1 + \frac{1}{x_{1}} +\frac{x_{1} + 1}{x_{1}x_{2}} +\frac{(x_{1} + 1)(x_{2} + 1)}{x_{1}x_{2}x_{3}} +\dots +\frac{(x_{1} + 1)\cdot\cdot\cdot(x_{n - 1} + 1)}{x_{1}x_{2}\cdot\cdot\cdot x_{n}} = 0.\] | Introducing \(y_{i} = \frac{1}{x_{i}}\) , we transform our equation to
\[0 = 1 + y_{1} + (1 + y_{1})y_{2} + \dots +(1 + y_{1})\dots (1 + y_{n - 1})y_{n}\] \[= (1 + y_{1})(1 + y_{2})\dots (1 + y_{n}).\]
The solutions are \(n\) - tuples \((y_{1},\ldots ,y_{n})\) with \(y_{i}\neq 0\) for all \(i\) and \(y_{j} = - 1\... |
IMOSL-1968-12 | If \(a\) and \(b\) are arbitrary positive real numbers and \(m\) an integer, prove that
\[\left(1 + \frac{a}{b}\right)^{m} + \left(1 + \frac{b}{a}\right)^{m}\geq 2^{m + 1}.\] | The given inequality is equivalent to \((a + b)^{m} / b^{m} + (a + b)^{m} / a^{m} \geq 2^{m + 1}\) , which can be rewritten as
\[\frac{1}{2}\left(\frac{1}{a^{m}} +\frac{1}{b^{m}}\right)\geq \left(\frac{2}{a + b}\right)^{m}.\]
Since \(f(x) = 1 / x^{m}\) is a convex function for every \(m \in \mathbb{Z}\) , the las... |
IMOSL-1968-13 | Given two congruent triangles \(A_{1}A_{2}A_{3}\) and \(B_{1}B_{2}B_{3}\) \((A_{i}A_{k} = B_{i}B_{k})\) prove that there exists a plane such that the orthogonal projections of these triangles onto it are congruent and equally oriented. | Translating one of the triangles if necessary, we may assume w.l.o.g. that \(B_{1} \equiv A_{1}\) . We also assume that \(B_{2} \neq A_{2}\) and \(B_{3} \neq A_{3}\) , since the result is obvious otherwise.
There exists a plane \(\pi\) through \(A_{1}\) that is parallel to both \(A_{2}B_{2}\) and \(A_{3}B_{3}\) . Le... |
IMOSL-1968-14 | A line in the plane of a triangle \(ABC\) intersects the sides \(AB\) and \(AC\) respectively at points \(X\) and \(Y\) such that \(BX = CY\) . Find the locus of the center of the circumcircle of triangle \(XAY\) . | Let \(O, D, E\) be the circumcenter of \(\triangle ABC\) and the midpoints of \(AB\) and \(AC\) , and given arbitrary \(X \in AB\) and \(Y \in AC\) such that \(BX = CY\) , let \(O_1, D_1, E_1\) be the circumcenter of \(\triangle AXY\) and the midpoints of \(AX\) and \(AY\) , respectively. Since \(AD = AB / 2\) and \(AD... |
IMOSL-1968-15 | Let \([x]\) denote the integer part of \(x\) , i.e., the greatest integer not exceeding \(x\) . If \(n\) is a positive integer, express as a simple function of \(n\) the sum
\[\left[\frac{n + 1}{2}\right] + \left[\frac{n + 2}{4}\right] + \dots +\left[\frac{n + 2^{i}}{2^{i + 1}}\right] + \dots .\] | Set
\[f(n) = \left[\frac{n + 1}{2}\right] + \left[\frac{n + 2}{4}\right] + \dots +\left[\frac{n + 2^i}{2^{i + 1}}\right] + \dots\]
We prove by induction that \(f(n) = n\) . This obviously holds for \(n = 1\) . Let us assume that \(f(n - 1) = n - 1\) . Define
\[g(i,n) = \left[\frac{n + 2^i}{2^{i + 1}}\right] - ... |
IMOSL-1968-16 | A polynomial \(p(x) = a_{0}x^{k} + a_{1}x^{k - 1} + \dots +a_{k}\) with integer coefficients is said to be divisible by an integer \(m\) if \(p(x)\) is divisible by \(m\) for all integers \(x\) . Prove that if \(p(x)\) is divisible by \(m\) , then \(k!a_{0}\) is also divisible by \(m\) . Also prove that if \(a_{0},k,m\... | We shall prove the result by induction on \(k\) . It trivially holds for \(k = 0\) . Assume that the statement is true for some \(k - 1\) , and let \(p(x)\) be a polynomial of degree \(k\) . Let us set \(p_1(x) = p(x + 1) - p(x)\) . Then \(p_1(x)\) is a polynomial of degree \(k - 1\) with leading coefficient \(ka_0\) .... |
IMOSL-1968-17 | Given a point \(O\) and lengths \(x,y,z\) , prove that there exists an equilateral triangle \(ABC\) for which \(OA = x\) , \(OB = y\) , \(OC = z\) , if and only if \(x + y \geq z\) , \(y + z \geq x\) , \(z + x \geq y\) (the points \(O,A,B,C\) are coplanar). | Let there be given an equilateral triangle \(ABC\) and a point \(O\) such that \(OA = x\) , \(OB = y\) , \(OC = z\) . Let \(X\) be the point in the plane such that \(\triangle CXB\) and \(\triangle COA\) are congruent and equally oriented. Then \(BX = x\) and the triangle \(XOC\) is equilateral, which implies \(OX = z\... |
IMOSL-1968-18 | If an acute-angled triangle \(ABC\) is given, construct an equilateral triangle \(A'B'C'\) in space such that lines \(AA', BB', CC'\) pass through a given point. | The required construction is not feasible. In fact, let us consider the special case \(\angle BOC = 135^{\circ}\) , \(\angle AOC = 120^{\circ}\) , \(\angle AOB = 90^{\circ}\) , where \(AA' \cap BB' \cap CC' = \{O\}\) . Denoting \(OA', OB', OC'\) by \(a, b, c\) respectively we obtain the system of equations
\[a^{2} +... |
IMOSL-1968-19 | We are given a fixed point on the circle of radius 1, and going from this point along the circumference in the positive direction on curved distances
\(0,1,2,\ldots\) from it we obtain points with abscissas \(n = 0,1,2,\ldots\) respectively. How many points among them should we take to ensure that some two of them are... | We shall denote by \(d_{n}\) the shortest curved distance from the initial point to the \(n\) th point in the positive direction. The sequence \(d_{n}\) goes as follows: 0, 1, 2, 3, 4, 5, 6, 0.72, 1.72, ..., 5.72, 0.43, 1.43, ..., 5.43, 0.15 = d_{19}. Hence the required number of points is 20. |
IMOSL-1968-20 | Given \(n\) ( \(n \geq 3\) ) points in space such that every three of them form a triangle with one angle greater than or equal to \(120^{\circ}\) , prove that these points can be denoted by \(A_{1}, A_{2}, \ldots , A_{n}\) in such a way that for each \(i, j, k\) , \(1 \leq i < j < k \leq n\) , angle \(A_{i} A_{j} A_{k... | Let us denote the points \(A_{1}, A_{2}, \ldots , A_{n}\) in such a manner that \(A_{1}A_{n}\) is a diameter of the set of given points, and \(A_{1}A_{2} \leq A_{1}A_{3} \leq \dots \leq A_{1}A_{n}\) . Since for each \(1 < i < n\) it holds that \(A_{1}A_{i} < A_{1}A_{n}\) , we have \(\angle A_{i}A_{1}A_{n} < 120^{\circ}... |
IMOSL-1968-21 | Let \(a_{0}, a_{1}, \ldots , a_{k}\) ( \(k \geq 1\) ) be positive integers. Find all positive integers \(y\) such that
\[a_{0} \mid y; (a_{0} + a_{1}) \mid (y + a_{1}); \ldots ; (a_{0} + a_{n}) \mid (y + a_{n}).\] | The given conditions are equivalent to \(y - a_{0}\) being divisible by \(a_{0}, a_{0} + a_{1}, a_{0} + a_{2}, \ldots , a_{0} + a_{n}\) , i.e., to \(y = k[a_{0}, a_{0} + a_{1}, \ldots , a_{0} + a_{n}] + a_{0}, k \in \mathbb{N}_{0}\) . |
IMOSL-1968-22 | Find all positive integers \(x\) for which \(p(x) = x^{2} - 10x - 22\) , where \(p(x)\) denotes the product of the digits of \(x\) . | It can be shown by induction on the number of digits of \(x\) that \(p(x) \leq x\) for all \(x \in \mathbb{N}\) . It follows that \(x^{2} - 10x - 22 \leq x\) , which implies \(x \leq 12\) . Since \(0 < x^{2} - 10x - 22 = (x - 12)(x + 2) + 2\) , one easily obtains \(x \geq 12\) . Now one can directly check that \(x = 12... |
IMOSL-1968-23 | Find all complex numbers \(m\) such that polynomial
\[x^{3} + y^{3} + z^{3} + m x y z\]
can be represented as the product of three linear trinomials. | We may assume w.l.o.g. that in all the factors the coefficient of \(x\) is 1. Suppose that \(x + ay + bz\) is one of the linear factors of \(p(x,y,z) = x^{3} + y^{3} + z^{3} + mxyz\) . Then \(p(x)\) is 0 at every point \((x,y,z)\) with \(z = -ax - by\) . Hence \(x^{3} + y^{3} + (-ax - by)^{3} + mxy(-ax - by) = (1 - a^{... |
IMOSL-1968-24 | Find the number of all \(n\) -digit numbers for which some fixed digit stands only in the \(i\) th ( \(1 < i < n\) ) place and the last \(j\) digits are distinct. \(^{3}\) | If the \(i\) th digit is 0, then the result is
\[\left\{ \begin{array}{l}9^{k - j}\frac{9!}{(10 - j)!}, \mathrm{if} i > k - j, \\ 9^{k - j - 1}\frac{9!}{(9 - j)!}, \mathrm{otherwise} \end{array} \right..\]
If the \(i\) th digit is not 0, then the above results are multiplied by 8. |
IMOSL-1968-25 | Given \(k\) parallel lines and a few points on each of them, find the number of all possible triangles with vertices at these given points. \(^{4}\) | The answer is
\[\sum_{1\leq p< q< r\leq k}n_{p}n_{q}n_{r} + \sum_{1\leq p< q\leq k}\left[n_{p}\binom{n_{q}}{2} + n_{q}\binom{n_{p}}{2}\right].\] |
IMOSL-1968-26 | Let \(a > 0\) be a real number and \(f(x)\) a real function defined on all of \(\mathbb{R}\) , satisfying for all \(x \in \mathbb{R}\) ,
\[f(x + a) = \frac{1}{2} +\sqrt{f(x) - f(x)^{2}}.\]
(a) Prove that the function \(f\) is periodic; i.e., there exists \(b > 0\) such that for all \(x, f(x + b) = f(x)\) .
(b)... | (a) We shall show that the period of \(f\) is \(2a\) . From \((f(x + a) - 1 / 2)^2 = f(x) - f(x)^2\) we obtain
\[(f(x) - f(x)^2) + (f(x + a) - f(x + a)^2) = \frac{1}{4}.\]
Subtracting the above relation for \(x + a\) in place of \(x\) we get \(f(x) - f(x)^2 = f(x + 2a) - f(x + 2a)^2\) , which implies \((f(x) - 1 ... |
IMO-1969-1 | Prove that there exist infinitely many natural numbers \(a\) with the following property: the number \(z = n^{4} + a\) is not prime for any natural number \(n\) . | Set \(a = 4m^{4}\) , where \(m\in \mathbb{N}\) and \(m > 1\) . We then have \(z = n^{4} + 4m^{4} = (n^{2}+\) \(2m^{2})^{2} - (2mn)^{2} = (n^{2} + 2m^{2} + 2mn)(n^{2} + 2m^{2} - 2mn)\) . Since \(n^{2} + 2m^{2} - 2mn =\) \((n - m)^{2} + m^{2}\geq m^{2} > 1\) , it follows that \(z\) must be composite. Thus we have found i... |
IMO-1969-2 | Let \(a_{1},a_{2},\ldots ,a_{n}\) be real constants and
\[y(x) = \cos (a_{1} + x) + \frac{\cos(a_{2} + x)}{2} +\frac{\cos(a_{3} + x)}{2^{2}} +\dots +\frac{\cos(a_{n} + x)}{2^{n - 1}}.\]
If \(x_{1},x_{2}\) are real and \(y(x_{1}) = y(x_{2}) = 0\) , prove that \(x_{1} - x_{2} = m\pi\) for some integer \(m\) . | Using \(\cos (a + x) = \cos a\cos x - \sin a\sin x\) , we obtain \(y(x) = A\sin x + B\cos x\) where \(A = - \sin a_{1} - \sin a_{2} / 2 - \dots - \sin a_{n} / 2^{n - 1}\) and \(B = \cos a_{1} + \cos a_{2} / 2+\) \(\dots +\cos a_{n} / 2^{n - 1}\) . Numbers \(A\) and \(B\) cannot both be equal to 0, for otherwise \(y\) w... |
IMO-1969-3 | Find conditions on the positive real number \(a\) such that there exists a tetrahedron \(k\) of whose edges \((k = 1,2,3,4,5)\) have length \(a\) , and the other \(6 - k\) edges have length 1. | We have several cases:
\(1^{\circ} k = 1\) . W.l.o.g. let \(AB = a\) and the remaining segments have length 1. Let \(M\) be the midpoint of \(CD\) . Then \(AM = BM = \sqrt{3} /2\) ( \(\triangle CDA\) and \(\triangle CDB\) are equilateral) and \(0< AB< AM + BM = \sqrt{3}\) , i.e., \(0< a< \sqrt{3}\) . It is evident t... |
IMO-1969-4 | Let \(AB\) be a diameter of a circle \(\gamma\) . A point \(C\) different from \(A\) and \(B\) is on the circle \(\gamma\) . Let \(D\) be the projection of the point \(C\) onto the line \(AB\) . Consider three other circles \(\gamma_{1}\) , \(\gamma_{2}\) , and \(\gamma_{3}\) with the common tangent \(AB\) : \(\gamma_{... | Let \(O\) be the midpoint of \(AB\) , i.e., the center of \(\gamma\) . Let \(O_{1}\) , \(O_{2}\) , and \(O_{3}\) respectively be the centers of \(\gamma_{1}\) , \(\gamma_{2}\) , and \(\gamma_{3}\) and let \(r_{1}, r_{2}, r_{3}\) respectively be the radii of \(\gamma_{1}\) , \(\gamma_{2}\) and \(\gamma_{3}\) . Let \(C_{... |
IMO-1969-5 | Given \(n\) points in the plane such that no three of them are collinear, prove that one can find at least \(\binom{n-3}{2}\) convex quadrilaterals with their vertices at these points. | We first prove the following lemma.
Lemma. If of five points in a plane no three belong to a single line, then there exist four that are the vertices of a convex quadrilateral.
Proof. If the convex hull of the five points \(A,B,C,D,E\) is a pentagon or a quadrilateral, the statement automatically holds. If the co... |
IMO-1969-6 | Under the conditions \(x_{1},x_{2} > 0,x_{1}y_{1} > z_{1}^{2}\) , and \(x_{2}y_{2} > z_{2}^{2}\) , prove the inequality
\[\frac{8}{(x_{1} + x_{2})(y_{1} + y_{2}) - (z_{1} + z_{2})^{2}}\leq \frac{1}{x_{1}y_{1} - z_{1}^{2}} +\frac{1}{x_{2}y_{2} - z_{2}^{2}}.\] | Define \(u_{1} = \sqrt{x_{1}y_{1}} +z_{1}\) \(u_{2} = \sqrt{x_{2}y_{2}} +z_{2}\) \(v_{1} = \sqrt{x_{1}y_{1}} -z_{1}\) , and \(v_{2} = \sqrt{x_{2}y_{2}} -z_{2}\) By expanding both sides of the equation we can easily verify \((x_{1} + x_{2})(y_{1} + y_{2})-\) \((z_{1} + z_{2})^{2} = (u_{1} + u_{2})(v_{1} + v_{2}) + (\sqr... |
IMOSL-1970-1 | Consider a regular \(2n\) -gon and the \(n\) diagonals of it that pass through its center. Let \(P\) be a point of the inscribed circle and let \(a_1, a_2, \ldots , a_n\) be the angles in which the diagonals mentioned are visible from the point \(P\) . Prove that
\[\sum_{i = 1}^{n}\tan^{2}a_{i} = 2n\frac{\cos^{2}\fr... | Denote respectively by \(R\) and \(r\) the radii of the circumcircle and incircle, by \(A_{1},\ldots ,A_{n},B_{1},\ldots ,B_{n}\) , the vertices of the \(2n\) -gon and by \(O\) its center. Let \(P^{\prime}\) be the point symmetric to \(P\) with respect to \(O\) . Then \(A_{i}P^{\prime}B_{i}P\) is a parallelogram, and a... |
IMOSL-1970-2 | Let \(a\) and \(b\) be the bases of two number systems and let
\[A_{n} = \overline{x_{1}x_{2}\ldots x_{n}}^{(a)},\qquad A_{n + 1} = \overline{x_{0}x_{1}x_{2}\ldots x_{n}}^{(a)},\] \[B_{n} = \overline{x_{1}x_{2}\ldots x_{n}}^{(b)},\qquad B_{n + 1} = \overline{x_{0}x_{1}x_{2}\ldots x_{n}}^{(b)},\]
be numbers in the... | Suppose that \(a > b\) . Consider the polynomial \(P(X) = x_{1}X^{n - 1} + x_{2}X^{n - 2} + \dots +\) \(x_{n - 1}X + x_{n}\) . We have \(A_{n} = P(a)\) \(B_{n} = P(b)\) \(A_{n + 1} = x_{0}a^{n} + P(a)\) , and \(B_{n + 1} =\) \(x_{0}b^{n} + P(b)\) . The inequality \(A_{n} / A_{n + 1}< B_{n} / B_{n + 1}\) becomes \(P(a) ... |
IMOSL-1970-3 | In the tetrahedron SABC the angle BSC is a right angle, and the projection of the vertex \(S\) to the plane \(ABC\) is the intersection of the altitudes of the triangle \(ABC\) . Let \(z\) be the radius of the inscribed circle of the triangle \(ABC\) . Prove that \(SA^2 + SB^2 + SC^2 \geq 18z^2\) . | We shall use the following lemma
Lemma. If an altitude of a tetrahedron passes through the orthocenter of the opposite side, then each of the other altitudes possesses the same property.
Proof. Denote the tetrahedron by \(SABC\) and let \(a = BC\) , \(b = CA\) , \(c = AB\) , \(m = SA\) , \(n = SB\) , \(p = SC\) .... |
IMOSL-1970-4 | For what natural numbers \(n\) can the product of some of the numbers \(n, n + 1, n + 2, n + 3, n + 4, n + 5\) be equal to the product of the remaining ones? | Suppose that \(n\) is such a natural number. If a prime number \(p\) divides any of the numbers \(n, n + 1, \ldots , n + 5\) , then it must divide another one of them, so the only possibilities are \(p = 2, 3, 5\) . Moreover, \(n + 1, n + 2, n + 3, n + 4\) have no prime divisors other than 2 and 3 (if some prime number... |
IMOSL-1970-5 | Let \(M\) be an interior point of the tetrahedron \(ABCD\) . Prove that
\[\overrightarrow{MA}\mathrm{vol}(MBCD) + \overrightarrow{MB}\mathrm{vol}(MACD)\] \[+\overrightarrow{MC}\mathrm{vol}(MABD) + \overrightarrow{MD}\mathrm{vol}(MABC) = 0\]
(vol(PQRS) denotes the volume of the tetrahedron \(PQRS\) ). | Denote respectively by \(A_1, B_1, C_1\) and \(D_1\) the points of intersection of the lines \(AM, BM, CM\) , and \(DM\) with the opposite sides of the tetrahedron. Since \(\text{vol}(MBCD) = \text{vol}(ABCD) \overline{MA_1} / \overline{AA_1}\) , the relation we have to prove is equivalent to
\[\overrightarrow{MA} \... |
IMOSL-1970-6 | In the triangle \(ABC\) let \(B'\) and \(C'\) be the midpoints of the sides \(AC\) and \(AB\) respectively and \(H\) the foot of the altitude passing through the vertex \(A\) . Prove that the circumcircles of the triangles \(AB'C', BC'H\) , and \(B'CH\) have a common point \(I\) and that the line \(HI\) passes through ... | Let \(F\) be the midpoint of \(B'C'\) , \(A'\) the midpoint of \(BC\) , and \(I\) the intersection point of the line \(HF\) and the circle circumscribed about \(\triangle BHC'\) . Denote by \(M\) the intersection point of the line \(AA'\) with the circumscribed circle about the triangle \(ABC\) . Triangles \(HB'C'\) an... |
IMOSL-1970-7 | For which digits \(a\) do exist integers \(n \geq 4\) such that each digit of \(\frac{n(n+1)}{2}\) equals \(a\) ? | For \(a = 5\) one can take \(n = 10\) , while for \(a = 6\) one takes \(n = 11\) . Now assume \(a \notin \{5,6\}\) .
If there exists an integer \(n\) such that each digit of \(n(n + 1) / 2\) is equal to \(a\) , then there is an integer \(k\) such that \(n(n + 1) / 2 = (10^{k} - 1)a / 9\) . After multiplying both sid... |
IMOSL-1970-8 | Given a point \(M\) on the side \(AB\) of the triangle \(ABC\) , let \(r_1\) and \(r_2\) be the radii of the inscribed circles of the triangles \(ACM\) and \(BCM\) respectively and let \(\rho_1\) and \(\rho_2\) be the radii of the excircles of the triangles \(ACM\) and \(BCM\) at the sides \(AM\) and \(BM\) respectivel... | Let \(A C = b,B C = a,A M = x,B M = y,C M = l\) . Denote by \(I_{1}\) the incenter and by \(S_{1}\) the center of the excircle of \(\Delta A M C\) . Suppose that \(P_{1}\) and \(Q_{1}\) are feet of perpendiculars from \(I_{1}\) and \(S_{1}\) , respectively, to the line \(A C\) . Then \(\triangle I_{1}C P_{1}\sim \trian... |
IMOSL-1970-9 | Let \(u_1, u_2, \ldots , u_n, v_1, v_2, \ldots , v_n\) be real numbers. Prove that
\[1 + \sum_{i = 1}^{n}(u_{i} + v_{i})^{2}\leq \frac{4}{3}\left(1 + \sum_{i = 1}^{n}u_{i}^{2}\right)\left(1 + \sum_{i = 1}^{n}v_{i}^{2}\right).\]
In what case does equality hold? | Let us set \(a = \sqrt{\sum_{i = 1}^{n}u_{i}^{2}}\) and \(b = \sqrt{\sum_{i = 1}^{n}v_{i}^{2}}\) . By Minkowski's inequality (for \(p = 2\) ) we have \(\sum_{i = 1}^{n}(u_{i} + v_{i})^{2} \leq (a + b)^{2}\) . Hence the LHS of the desired inequality
is not greater than \(1 + (a + b)^2\) , while the RHS is equal to \(4(... |
IMOSL-1970-10 | Let \(1 = a_0 \leq a_1 \leq a_2 \leq \dots \leq a_n \leq \dots\) be a sequence of real numbers. Consider the sequence \(b_1, b_2, \ldots\) defined by:
\[b_n = \sum_{k = 1}^{n}\left(1 - \frac{a_{k - 1}}{a_k}\right)\frac{1}{\sqrt{a_k}}.\]
Prove that:
(a) For all natural numbers \(n\) , \(0 \leq b_n < 2\) .
(b) ... | (a) Since \(a_{n - 1} < a_n\) , we have
\[\left(1 - \frac{a_{k - 1}}{a_k}\right)\frac{1}{\sqrt{a_k}} = \frac{a_k - a_{k - 1}}{a_k^{3 / 2}}\] \[\leq \frac{2(\sqrt{a_k} - \sqrt{a_{k - 1}})\sqrt{a_k}}{a_k\sqrt{a_{k - 1}}} = 2\left(\frac{1}{\sqrt{a_{k - 1}}} -\frac{1}{\sqrt{a_k}}\right).\]
Summing up all these inequa... |
IMOSL-1970-11 | Let \(P, Q, R\) be polynomials and let \(S(x) = P(x^3) + xQ(x^3) + x^2R(x^3)\) be a polynomial of degree \(n\) whose roots \(x_1, \ldots , x_n\) are distinct. Construct with the aid of the polynomials \(P, Q, R\) a polynomial \(T\) of degree \(n\) that has the roots \(x_1^3, x_2^3, \ldots , x_n^3\) . | Let \(S(x) = (x - x_1)(x - x_2) \dots (x - x_n)\) . We have \(x^3 - x_i^3 = (x - x_i)(\omega x - x_i)(\omega^2 x - x_i)\) , where \(\omega\) is a primitive third root of 1. Multiplying these equalities for \(i = 1, \ldots , n\) we obtain
\[T(x^{3}) = (x^{3} - x_{1}^{3})(x^{3} - x_{2}^{3})\cdot \cdot \cdot (x^{3} - x_{... |
IMOSL-1970-12 | We are given 100 points in the plane, no three of which are on the same line. Consider all triangles that have all vertices chosen from the 100 given points. Prove that at most \(70\%\) of these triangles are acute angled. | Lemma. Five points are given in the plane such that no three of them are collinear. Then there are at least three triangles with vertices at these points that are not acute-angled.
Proof. We consider three cases, according to whether the convex hull of these points is a triangle, quadrilateral, or pentagon.
(i) L... |
IMOSL-1971-1 | Consider a sequence of polynomials \(P_{0}(x), P_{1}(x), P_{2}(x), \ldots , P_{n}(x), \ldots\) , where \(P_{0}(x) = 2\) , \(P_{1}(x) = x\) and for every \(n \geq 1\) the following equality holds: \(P_{n+1}(x) + P_{n-1}(x) = xP_{n}(x)\) . Prove that there exist three real numbers \(a, b, c\) such that for all \(n \geq 1... | Assuming that \(a,b,c\) in (1) exist, let us find what their values should be. Since \(P_{2}(x) = x^{2} - 2\) , equation (1) for \(n = 1\) becomes \((x^{2} - 4)^{2} = [a(x^{2} - 2) + bx + 2c]^{2}\) . Therefore, there are two possibilities for \((a,b,c)\) : \((1,0, - 1)\) and \((- 1,0,1)\) . In both cases we must prove ... |
IMOSL-1971-2 | Prove that for every natural number \(m\geq 1\) there exists a finite set \(S_{m}\) of points in the plane satisfying the following condition: If \(A\) is any point in \(S_{m}\) , then there are exactly \(m\) points in \(S_{m}\) whose distance to \(A\) equals 1. | We will construct such a set \(S_{m}\) of \(2^{m}\) points.
Take vectors \(u_{1},\ldots ,u_{m}\) in a given plane such that \(|u_{i}| = 1 / 2\) and \(0\neq |c_{1}u_{1}+\) \(c_{2}u_{2} + \dots +c_{n}u_{n}|\neq 1 / 2\) for any choice of numbers \(c_{i}\) equal to 0 or \(\pm 1\) (where two or more of the numbers \(c_{i... |
IMOSL-1971-3 | Knowing that the system
\[x + y + z = 3,\] \[x^{3} + y^{3} + z^{3} = 15,\] \[x^{4} + y^{4} + z^{4} = 35,\]
has a real solution \(x,y,z\) for which \(x^{2} + y^{2} + z^{2}< 10\) , find the value of \(x^{5} + y^{5} + z^{5}\) for that solution. | Let \(x,y,z\) be a solution of the given system with \(x^{2} + y^{2} + z^{2} = \alpha < 10\) . Then
\[xy + yz + zx = \frac{(x + y + z)^{2} - (x^{2} + y^{2} + z^{2})}{2} = \frac{9 - \alpha}{2}.\]
Furthermore, \(3xyz = x^{3} + y^{3} + z^{3} - (x + y + z)(x^{2} + y^{2} + z^{2} - xy - yz - zx)\) , which gives us \(xy... |
IMOSL-1971-4 | We are given two mutually tangent circles in the plane, with radii \(r_{1},r_{2}\) . A line intersects these circles in four points, determining three segments of equal length. Find this length as a function of \(r_{1}\) and \(r_{2}\) and the condition for the solvability of the problem. | In the coordinate system in which the \(x\) -axis passes through the centers of the circles and the \(y\) -axis is their common tangent, the circles have equations
\[x^{2} + y^{2} + 2r_{1}x = 0,\quad x^{2} + y^{2} - 2r_{2}x = 0.\]
Let \(p\) be the desired line with equation \(y = ax + b\) . The abscissas of point... |
IMOSL-1971-5 | Let \(a,b,c,d,e\) be real numbers. Prove that the expression
\[(a - b)(a - c)(a - d)(a - e) + (b - a)(b - c)(b - d)(b - e)\] \[+(c - a)(c - b)(c - d)(c - e) + (d - a)(d - b)(d - c)(d - e)\] \[+(e - a)(e - b)(e - c)(e - d).\]
is nonnegative. | Without loss of generality, we may assume that \(a \geq b \geq c \geq d \geq e\) . Then \(a - b = -(b - a) \geq 0\) , \(a - c \geq b - c \geq 0\) , \(a - d \geq b - d \geq 0\) and \(a - e \geq b - e \geq 0\) , and hence
\[(a - b)(a - c)(a - d)(a - e) + (b - a)(b - c)(b - d)(b - e) \geq 0.\]
Analogously, \((d - a)... |
IMOSL-1971-6 | Let \(n\geq 2\) be a natural number. Find a way to assign natural numbers to the vertices of a regular \(2^{n}\) -gon such that the following conditions are satisfied:
(i) only digits 1 and 2 are used; (ii) each number consists of exactly \(n\) digits; (iii) different numbers are assigned to different vertices; (iv)... | The proof goes by induction on \(n\) . For \(n = 2\) , the following labeling satisfies the conditions (i)-(iv): \(C_{1} = 11\) , \(C_{2} = 12\) , \(C_{3} = 22\) , \(C_{4} = 21\) .
Suppose that \(n > 2\) , and that the numeration \(C_{1}, C_{2}, \ldots , C_{2^{n - 1}}\) of a regular \(2^{n - 1}\) - gon, in cyclical ... |
IMOSL-1971-7 | Given a tetrahedron \(ABCD\) all of whose faces are acute-angled triangles, set
\[\sigma = \angle D A B + \angle B C D - \angle A B C - \angle C D A.\]
Consider all closed broken lines \(XYZTX\) whose vertices \(X,Y,Z,T\) lie in the interior of segments \(AB,BC,CD,DA\) respectively. Prove that:
(a) if \(\sigma... | (a) Suppose that \(X, Y, Z\) are fixed on segments \(AB, BC, CD\) . It is proven in a standard way that if \(\angle ATX \neq \angle ZTD\) , then \(ZT + TX\) can be reduced. It follows that if there exists a broken line \(XYZTX\) of minimal length, then the following conditions hold:
\[\angle DAB = \pi -\angle ATX - ... |
IMOSL-1971-8 | Determine whether there exist distinct real numbers \(a,b,c,t\) for which:
(i) the equation \(ax^{2} + btx + c = 0\) has two distinct real roots \(x_{1},x_{2}\) (ii) the equation \(bx^{2} + ctx + a = 0\) has two distinct real roots \(x_{2},x_{3}\) (iii) the equation \(cx^{2} + atx + b = 0\) has two distinct real roo... | Suppose that \(a, b, c, t\) satisfy all the conditions. Then \(abc \neq 0\) and
\[x_{1}x_{2} = \frac{c}{a},\qquad x_{2}x_{3} = \frac{a}{b},\qquad x_{3}x_{1} = \frac{b}{c}.\]
Multiplying these equations, we obtain \(x_{1}^{2}x_{2}^{2}x_{3}^{2} = 1\) , and hence \(x_{1}x_{2}x_{3} = \epsilon = \pm 1\) . From the abo... |
IMOSL-1971-9 | Let \(T_{k} = k - 1\) for \(k = 1,2,3,4\) and
\[T_{2k - 1} = T_{2k - 2} + 2^{k - 2},\quad T_{2k} = T_{2k - 5} + 2^{k}\qquad (k\geq 3).\]
Show that for all \(k\) ,
\[1 + T_{2n - 1} = \left[\frac{12}{7} 2^{n - 1}\right]\quad \mathrm{and}\quad 1 + T_{2n} = \left[\frac{17}{7} 2^{n - 1}\right],\]
where \([x]\) den... | We use induction. Since \(T_{1} = 0\) , \(T_{2} = 1\) , \(T_{3} = 2\) , \(T_{4} = 3\) , \(T_{5} = 5\) , \(T_{6} = 8\) , the statement is true for \(n = 1, 2, 3\) . Suppose that both formulas from the problem hold for some \(n \geq 3\) . Then
\[T_{2n + 1} = 1 + T_{2n} + 2^{n - 1} = \left[\frac{17}{7} 2^{n - 1} + 2^{n... |
IMOSL-1971-10 | Prove that the sequence \(2^{n} - 3\) \((n > 1)\) contains a subsequence of numbers relatively prime in pairs. | We use induction. Suppose that every two of the numbers \(a_{1} = 2^{n_{1}} - 3, a_{2} = 2^{n_{2}} - 3, \ldots, a_{k} = 2^{n_{k}} - 3\) , where \(2 = n_{1} < n_{2} < \dots < n_{k}\) , are coprime. Then one can construct \(a_{k+1} = 2^{n_{k+1}} - 3\) in the following way:
Set \(s = a_{1}a_{2}\ldots a_{k}\) . Among the ... |
IMOSL-1971-11 | The matrix
\[
\begin{pmatrix}
a_{11} & \cdots & a_{1n} \\
\vdots & \ddots & \vdots \\
a_{n1} & \cdots & a_{nn}
\end{pmatrix}
\]
satisfies the inequality \(\textstyle \sum_{j = 1}^{n}|a_{j1}x_{1} + \dots +a_{jn}x_{n}|\leq M\) for each choice of numbers \(x_{i}\) equal to \(\pm 1\) .Show that
\[|a_{11} + a_{22} + ... | We use induction. The statement for \(n = 1\) is trivial. Suppose that it holds for \(n = k\) and consider \(n = k + 1\) . From the given condition, we have
\[{ \begin{array}{l}{{\sum_{j=1}^{k}|a_{j,1}x_{1}+\cdots+a_{j,k}x_{k}+a_{j,k+1}|}\] \[{+\left|a_{k+1,1}x_{1}+\cdots+a_{k+1,k}x_{k}+a_{k+1,k+1}\right|\leq M,}\] ... |
IMOSL-1971-12 | Two congruent equilateral triangles \(ABC\) and \(A'B'C'\) in the plane are given. Show that the midpoints of the segments \(AA',BB',CC'\) either are collinear or form an equilateral triangle. | Let us start with the case \(A = A'\) . If the triangles \(ABC\) and \(A'B'C'\) are oppositely oriented, then they are symmetric with respect to some axis, and the statement is true. Suppose that they are equally oriented. There is a rotation around \(A\) by \(60^{\circ}\) that maps \(ABB'\) onto \(ACC'\) . This rotati... |
IMOSL-1971-13 | Consider the \(n\times n\) array of nonnegative integers
\[
\begin{pmatrix}
a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{n1} & a_{n2} & \cdots & a_{nn}
\end{pmatrix},
\]
with the following property: If an element \(a_{ij}\) is zero, then the sum o... | Let \(p\) be the least of all the sums of elements in one row or column. If \(p \geq n / 2\) , then the sum of all elements of the array is \(s \geq np \geq n^2 /2\) . Now suppose that \(p < n / 2\) . Without loss of generality, one can assume that the sum of elements in the first row is \(p\) , and that exactly the fi... |
IMOSL-1971-14 | A broken line \(A_1A_2\ldots A_n\) is drawn in a \(50\times 50\) square, so that the distance from any point of the square to the broken line is less than 1. Prove that its total length is greater than 1248. | Denote by \(V\) the figure made by a circle of radius 1 whose center moves along the broken line. From the condition of the problem, \(V\) contains the whole \(50\times 50\) square, and thus the area \(S(V)\) of \(V\) is not less than 2500.
Let \(L\) be the length of the broken line. We shall show that \(S(V)\leq 2L... |
IMOSL-1971-15 | Natural numbers from 1 to 99 (not necessarily distinct) are written on 99 cards. It is given that the sum of the numbers on any subset of cards (including the set of all cards) is not divisible by 100. Show that all the cards contain the same number. | Assume the opposite. Then one can numerate the cards 1 to 99, with a number \(n_{i}\) written on the card \(i\) , so that \(n_{98}\neq n_{99}\) . Denote by \(x_{i}\) the remainder of \(n_{1} + n_{2}+\) \(\dots +n_{i}\) upon division by 100, for \(i = 1,2,\ldots ,99\) . All \(x_{i}\) must be distinct: Indeed, if \(x_{i}... |
IMOSL-1971-16 | Given a convex polyhedron \(P_{1}\) with 9 vertices \(A_{1},\ldots ,A_{9}\) , let us denote by \(P_{2},P_{3},\ldots ,P_{9}\) the images of \(P_{1}\) under the translations mapping the vertex \(A_{1}\) to \(A_{2},A_{3},\ldots ,A_{9}\) respectively. Prove that among the polyhedra \(P_{1},\ldots ,P_{9}\) at least two have... | Denote by \(P^{\prime}\) the polyhedron defined as the image of \(P\) under the homothety with center at \(A_{1}\) and coefficient of similarity 2. It is easy to see that all \(P_{i}\) , \(i = 1,\ldots ,9\) , are contained in \(P^{\prime}\) (indeed, if \(M\in P_{k}\) , then
\[\frac{1}{2}\overline{A_{1}M} = \frac{1}{... |
IMOSL-1971-17 | Prove the inequality
\[\frac{a_{1} + a_{3}}{a_{1} + a_{2}} +\frac{a_{2} + a_{4}}{a_{2} + a_{3}} +\frac{a_{3} + a_{1}}{a_{3} + a_{4}} +\frac{a_{4} + a_{2}}{a_{4} + a_{1}}\geq 4,\]
where \(a_{i} > 0\) , \(i = 1,2,3,4\) . | We use the following obvious consequences of \((a + b)^{2}\geq 4ab\) :
\[\frac{1}{(a_{1} + a_{2})(a_{3} + a_{4})}\geq \frac{4}{(a_{1} + a_{2} + a_{3} + a_{4})^{2}},\]
\[\frac{1}{(a_{1} + a_{4})(a_{2} + a_{3})}\geq \frac{4}{(a_{1} + a_{2} + a_{3} + a_{4})^{2}}.\]
Now we have
\[\frac{a_{1} + a_{3}}{a_{1} + a_{2... |
IMOSL-1972-1 | Let \(f\) and \(\phi\) be real functions defined on the set \(\mathbb{R}\) satisfying the functional equation
\[f(x + y) + f(x - y) = 2\phi (y)f(x), \quad (1)\]
for arbitrary real \(x, y\) (give examples of such functions). Prove that if \(f(x)\) is not identically 0 and \(|f(x)| \leq 1\) for all \(x\) , then \(|... | Suppose that \(f(x_{0})\neq 0\) and for a given \(y\) define the sequence \(x_{k}\) by the formula
\[x_{k + 1} = \left\{ \begin{array}{ll}x_{k} + y, & \mathrm{if}|f(x_{k} + y)|\geq |f(x_{k} - y)|;\\ x_{k} - y, & \mathrm{otherwise}. \end{array} \right.\]
It follows from (1) that \(|f(x_{k + 1})|\geq |\phi (y)||f(x... |
IMOSL-1972-2 | We are given \(3n\) points \(A_{1}, A_{2}, \ldots , A_{3n}\) in the plane, no three of them collinear. Prove that one can construct \(n\) disjoint triangles with vertices at the points \(A_{i}\) . | We use induction. For \(n = 1\) the assertion is obvious. Assume that it is true for a positive integer \(n\) . Let \(A_{1},A_{2},\ldots ,A_{3n + 3}\) be given \(3n + 3\) points, and let w.l.o.g. \(A_{1}A_{2}\ldots A_{m}\) be their convex hull.
Among all the points \(A_{i}\) distinct from \(A_{1},A_{2}\) , we choose... |
IMOSL-1972-3 | Let \(x_{1}, x_{2}, \ldots , x_{n}\) be real numbers satisfying \(x_{1} + x_{2} + \dots + x_{n} = 0\) . Let \(m\) be the least and \(M\) the greatest among them. Prove that
\[x_{1}^{2} + x_{2}^{2} + \dots +x_{n}^{2}\leq -nmM.\] | We have for each \(k = 1,2,\ldots ,n\) that \(m\leq x_{k}\leq M\) , which gives \((M - x_{k})(m - x_{k})\leq\) 0. It follows directly that
\[0\geq \sum_{k = 1}^{n}(M - x_{k})(m - x_{k}) = n m M - (m + M)\sum_{k = 1}^{n}x_{k} + \sum_{k = 1}^{n}x_{k}^{2}.\]
But \(\textstyle \sum_{k = 1}^{n}x_{k} = 0\) , implying th... |
IMOSL-1972-4 | Let \(n_{1}, n_{2}\) be positive integers. Consider in a plane \(E\) two disjoint sets of points \(M_{1}\) and \(M_{2}\) consisting of \(2n_{1}\) and \(2n_{2}\) points, respectively, and such that no three points of the union \(M_{1} \cup M_{2}\) are collinear. Prove that there exists a straight line \(g\) with the fol... | Choose in \(E\) a half-line \(s\) beginning at a point \(O\) . For every \(\alpha\) in the interval \([0,180^{\circ}]\) , denote by \(s(\alpha)\) the line obtained by rotation of \(s\) about \(O\) by \(\alpha\) , and by \(g(\alpha)\) the oriented line containing \(s(\alpha)\) on which \(s(\alpha)\) defines the positive... |
IMOSL-1972-5 | Prove the following assertion: The four altitudes of a tetrahedron \(ABCD\) intersect in a point if and only if
\[AB^{2} + CD^{2} = BC^{2} + AD^{2} = CA^{2} + BD^{2}.\] | Lemma. If \(X, Y, Z, T\) are points in space, then the lines \(XZ\) and \(YT\) are perpendicular if and only if \(XY^{2} + ZT^{2} = YZ^{2} + TX^{2}\) .
Proof. Consider the plane \(\pi\) through \(XZ\) parallel to \(YT\) . If \(Y', T'\) are the feet of the perpendiculars to \(\pi\) from \(Y, T\) respectively, then
... |
IMOSL-1972-6 | Show that for any \(n \neq 0\) (mod 10) there exists a multiple of \(n\) not containing the digit 0 in its decimal expansion. | Let \(n = 2^{\alpha}5^{\beta}m\) , where \(\alpha = 0\) or \(\beta = 0\) . These two cases are analogous, and we treat only \(\alpha = 0\) , \(n = 5^{\beta}m\) . The case \(m = 1\) is settled by the following lemma.
Lemma. For any integer \(\beta \geq 1\) there exists a multiple \(M_{\beta}\) of \(5^{\beta}\) with \... |
IMOSL-1972-7 | (a) A plane \(\pi\) passes through the vertex \(O\) of the regular tetrahedron \(OPQR\) . We define \(p, q, r\) to be the signed distances of \(P, Q, R\) from \(\pi\) measured along a directed normal to \(\pi\) . Prove that
\[p^{2} + q^{2} + r^{2} + (q - r)^{2} + (r - p)^{2} + (p - q)^{2} = 2a^{2},\]
where \(a\) ... | (a) Consider the circumscribing cube \(OQ_{1}PR_{1}O_{1}QP_{1}R\) (that is, the cube in which the edges of the tetrahedron are small diagonals), of side \(b = a\sqrt{2} /2\) . The left-hand side is the sum of squares of the projections of the edges of the tetrahedron onto a perpendicular \(l\) to \(\pi\) . On the other... |
IMOSL-1972-8 | Let \(m\) and \(n\) be nonnegative integers. Prove that \(m!n!(m + n)!\) divides \((2m)!(2n)!\) . | Let \(f(m,n) = \frac{(2m)!(2n)!}{m!n!(m+n)!}\) . Then it is directly shown that
\[f(m,n) = 4f(m,n - 1) - f(m + 1,n - 1),\]
and thus \(n\) may be successively reduced until one obtains \(f(m,n) = \sum_{r}c_{r}f(r,0)\) . Now \(f(r,0)\) is a simple binomial coefficient, and the \(c_{r}\) 's are integers.
Second s... |
IMOSL-1972-9 | Find all solutions in positive real numbers \(x_{i}\) \((i = 1,2,3,4,5)\) of the following system of inequalities:
\[(x_{1}^{2} - x_{3}x_{5})(x_{2}^{2} - x_{3}x_{5})\leq 0,\] \[(x_{2}^{2} - x_{4}x_{1})(x_{3}^{2} - x_{4}x_{1})\leq 0,\] \[(x_{3}^{2} - x_{5}x_{2})(x_{4}^{2} - x_{5}x_{2})\leq 0,\] \[(x_{4}^{2} - x_{1}x_... | Clearly \(x_{1} = x_{2} = x_{3} = x_{4} = x_{5}\) is a solution. We shall show that this describes all solutions.
Suppose that not all \(x_{i}\) are equal. Then among \(x_{3},x_{5},x_{2},x_{4},x_{1}\) two consecutive are distinct: Assume w.l.o.g. that \(x_{3}\neq x_{5}\) . Moreover, since \((1 / x_{1},\ldots ,1 / x_... |
IMOSL-1972-10 | Prove that for each \(n \geq 4\) every cyclic quadrilateral can be decomposed into \(n\) cyclic quadrilaterals. | Consider first a triangle. It can be decomposed into \(k = 3\) cyclic quadrilaterals by perpendiculars from some interior point of it to the sides; also, it can be decomposed into a cyclic quadrilateral and a triangle, and it follows by induction that this decomposition is possible for every \(k\) . Since every triangl... |
IMOSL-1972-11 | Consider a sequence of circles \(K_{1}, K_{2}, K_{3}, K_{4}, \ldots\) of radii \(r_{1}, r_{2}, r_{3}, r_{4}, \ldots\) , respectively, situated inside a triangle \(ABC\) . The circle \(K_{1}\) is tangent to \(AB\) and \(AC\) ; \(K_{2}\) is tangent to \(K_{1}, BA\) , and \(BC\) ; \(K_{3}\) is tangent to \(K_{2}, CA\) , a... | Let \(\angle A = 2x\) , \(\angle B = 2y\) , \(\angle C = 2z\) .
(a) Denote by \(M_{i}\) the center of \(K_{i}\) , \(i = 1,2,\ldots\) . If \(N_{1},N_{2}\) are the projections of \(M_{1},M_{2}\) onto \(AB\) , we have \(AN_{1} = r_{1}\cot x\) , \(N_{2}B = r_{2}\cot y\) , and \(N_{1}N_{2} = \sqrt{(r_{1} + r_{2})^{2} - (... |
IMOSL-1972-12 | A set of 10 positive integers is given such that the decimal expansion of each of them has two digits. Prove that there are two disjoint subsets of the set with equal sums of their elements. | First we observe that it is not essential to require the subsets to be disjoint (if they aren't, one simply excludes their intersection). There are \(2^{10} - 1 = 1023\) different subsets and at most 990 different sums. By the pigeonhole principle there are two different subsets with equal sums. |
IMOSL-1973-1 | Let a tetrahedron \(ABCD\) be inscribed in a sphere \(S\) . Find the locus of points \(P\) inside the sphere \(S\) for which the equality
\[\frac{AP}{PA_1} +\frac{BP}{PB_1} +\frac{CP}{PC_1} +\frac{DP}{PD_1} = 4\]
holds, where \(A_{1},B_{1},C_{1}\) , and \(D_{1}\) are the intersection points of \(S\) with the line... | The condition of the point \(P\) can be written in the form \(\frac{AP^2}{AP\cdot PA_1} +\frac{BP^2}{BP\cdot PB_1} +\) \(\frac{CP^2}{CP\cdot PC_1} +\frac{DP^2}{DP\cdot PD_1} = 4\) . All the four denominators are equal to \(R^2 - OP^2\) , i.e., to the power of \(P\) with respect to \(S\) . Thus the condition becomes
... |
IMOSL-1973-2 | Given a circle \(K\) , find the locus of vertices \(A\) of parallelograms \(ABCD\) with diagonals \(AC \leq BD\) , such that \(BD\) is inside \(K\) . | Let \(D'\) be the reflection of \(D\) across \(A\) . Since \(BCAD'\) is then a parallelogram, the condition \(BD \geq AC\) is equivalent to \(BD \geq BD'\) , which is in turn equivalent to \(\angle BAD \geq \angle BAD'\) , i.e. to \(\angle BAD \geq 90^{\circ}\) . Thus the needed locus is actually the locus of points \(... |
IMOSL-1973-3 | Prove that the sum of an odd number of unit vectors passing through the same point \(O\) and lying in the same half-plane whose border passes through \(O\) has length greater than or equal to 1. | We use induction on odd numbers \(n\) . For \(n = 1\) there is nothing to prove. Suppose that the result holds for \(n - 2\) vectors, and let us be given vectors \(v_1, v_2, \ldots , v_n\) arranged clockwise. Set \(v' = v_2 + v_3 + \dots + v_{n-1}\) , \(u = v_1 + v_n\) , and \(v = v_1 + v_2 + \dots + v_n = v' + u\) . B... |
IMOSL-1973-4 | Let \(P\) be a set of 7 different prime numbers and \(C\) a set of 28 different composite numbers each of which is a product of two (not necessarily different) numbers from \(P\) . The set \(C\) is divided into 7 disjoint four-element subsets such that each of the numbers in one set has a common prime divisor with at l... | Each of the subsets must be of the form \(\{a^2, ab, ac, ad\}\) or \(\{a^2, ab, ac, bc\}\) . It is now easy to count up the partitions. The result is 26460. |
IMOSL-1973-5 | A circle of radius 1 is located in a right-angled trihedron and touches all its faces. Find the locus of centers of such circles. | Let \(O\) be the vertex of the trihedron, \(Z\) the center of a circle \(k\) inscribed in the trihedron, and \(A, B, C\) points in which the plane of the circle meets the edges of the trihedron. We claim that the distance \(OZ\) is constant.
Set \(OA = x\) , \(OB = y\) , \(OC = z\) , \(BC = a\) , \(CA = b\) , \(AB =... |
IMOSL-1973-6 | Does there exist a finite set \(M\) of points in space, not all in the same plane, such that for each two points \(A, B \in M\) there exist two other points \(C, D \in M\) such that lines \(AB\) and \(CD\) are parallel? | Yes. Take for \(\mathcal{M}\) the set of vertices of a cube \(ABCDEFGH\) and two points \(I, J\) symmetric to the center \(O\) of the cube with respect to the laterals \(ABCD\) and \(EFGH\) .
Remark. We prove a stronger result: Given an arbitrary finite set of points \(\mathcal{S}\) , then there is a finite set \(\m... |
IMOSL-1973-7 | Given a tetrahedron \(ABCD\) , let \(x = AB \cdot CD\) , \(y = AC \cdot BD\) , and \(z = AD \cdot BC\) . Prove that there exists a triangle with edges \(x, y, z\) . | The result follows immediately from Ptolemy's inequality. |
IMOSL-1973-8 | Prove that there are exactly \(\binom{k}{k/2}\) arrays \(a_1, a_2, \ldots, a_{k+1}\) of nonnegative integers such that \(a_1 = 0\) and \(|a_i - a_{i+1}| = 1\) for \(i = 1, 2, \ldots, k\) . | Let \(f_{n}\) be the required total number, and let \(f_{n}(k)\) denote the number of sequences \(a_{1}, \ldots , a_{n}\) of nonnegative integers such that \(a_{1} = 0\) , \(a_{n} = k\) , and \(|a_{i} - a_{i + 1}| = 1\) for \(i = 1, \ldots , n - 1\) . In particular, \(f_{1}(0) = 1\) and \(f_{n}(k) = 0\) if \(k < 0\) or... |
IMOSL-1973-9 | Let \(Ox, Oy, Oz\) be three rays, and \(G\) a point inside the trihedron \(Oxyz\) . Consider all planes passing through \(G\) and cutting \(Ox, Oy, Oz\) at points \(A, B, C\) , respectively. How is the plane to be placed in order to yield a tetrahedron \(OABC\) with minimal perimeter? | Let \(a,b,c\) be vectors going along \(O x,O y,O z\) , respectively, such that \(\overrightarrow{O G} = a+\) \(b + c\) . Now let \(A\in O x\) \(B\in O y\) \(C\in O z\) and let \(\overrightarrow{O A} = \alpha a\) \(\overrightarrow{O B} = \beta b\) \(\overrightarrow{O C} = \gamma c\) where \(\alpha ,\beta ,\gamma >0\) . ... |
IMOSL-1973-10 | Let \(a_1, a_2, \ldots, a_n\) be positive numbers and \(q\) a given real number, \(0 < q < 1\) . Find \(n\) real numbers \(b_1, b_2, \ldots, b_n\) that satisfy:
(1) \(a_k < b_k\) for all \(k = 1, 2, \ldots , n\) ;
(2) \(q < \frac{b_{k+1}}{b_k} < \frac{1}{q}\) for all \(k = 1, 2, \ldots , n - 1\) ;
(3) \(b_1 + b_2 ... | Let
\[b_{k} = a_{1}q^{k - 1} + \dots +a_{k - 1}q + a_{k} + a_{k + 1}q + \dots +a_{n}q^{n - k},\quad k = 1,2,\dots ,n.\]
We show that these numbers satisfy the required conditions. Obviously \(b_{k} > a_{k}\) . Further,
\[b_{k + 1} - q b_{k} = -[(q^{2} - 1)a_{k + 1} + \dots +q^{n - k - 1}(q^{2} - 1)a_{n}] > 0;\... |
IMOSL-1973-11 | Determine the minimum of \(a^2 + b^2\) if \(a\) and \(b\) are real numbers for which the equation
\[x^{4} + ax^{3} + bx^{2} + ax + 1 = 0\]
has at least one real solution. | Putting \(x + \frac{1}{x} = t\) we also get \(x^{2} + \frac{1}{x^{2}} = t^{2} - 2\) , and the given equation reduces to \(t^{2} + at + b - 2 = 0\) . Since \(x = \frac{t\pm\sqrt{t^{2} - 4}}{2}\) , \(x\) will be real if and only if \(|t|\geq 2\) , \(t\in \mathbb{R}\) . Thus we need the minimum value of \(a^{2} + b^{2}\) ... |
IMOSL-1973-12 | Consider the two square matrices
\[
A =
\begin{bmatrix}
1 & 1 & 1 & 1 & 1 \\
1 & 1 & 1 & -1 & -1 \\
1 & -1 & -1 & 1 & 1 \\
1 & -1 & 1 & -1 & 1 \\
1 & 1 & -1 & 1 & -1
\end{bmatrix}
\quad \text{and} \quad
B =
\begin{bmatrix}
1 & 1 & 1 & 1 & 1 \\
1 & 1 & 1 & -1 & -1 \\
1 & 1 & -1 & 1 & -1 \\
1 & -1 & 1 & 1 & -1 \\
1 & ... | Observe that the absolute values of the determinants of the given matrices are invariant under all the admitted operations. The statement follows from \(\operatorname *{det}A = 16 \neq \operatorname *{det}B = 0\) . |
IMOSL-1973-13 | Find the sphere of maximal radius that can be placed inside every tetrahedron that has all altitudes of length greater than or equal to 1. | Let \(S_{1}, S_{2}, S_{3}, S_{4}\) denote the areas of the faces of the tetrahedron, \(V\) its volume, \(h_{1}, h_{2}, h_{3}, h_{4}\) its altitudes, and \(r\) the radius of its inscribed sphere. Since
\[3V = S_{1}h_{1} = S_{2}h_{2} = S_{3}h_{3} = S_{4}h_{4} = (S_{1} + S_{2} + S_{3} + S_{4})r,\]
it follows that
... |
IMOSL-1973-14 | A soldier has to investigate whether there are mines in an area that has the form of an equilateral triangle. The radius of his detector is equal to one-half of an altitude of the triangle. The soldier starts from one vertex of the triangle. Determine the shortest path that the soldier has to traverse in order to check... | Suppose that the soldier starts at the vertex \(A\) of the equilateral triangle \(ABC\) of side length \(a\) . Let \(\phi , \psi\) be the arcs of circles with centers \(B\) and \(C\) and radii \(a\sqrt{3} /4\) respectively, that lie inside the triangle. In order to check the vertices \(B, C\) , he must visit some point... |
IMOSL-1973-15 | Prove that for all \(n \in \mathbb{N}\) the following is true:
\[2^{n}\prod_{k = 1}^{n}\sin {\frac{k\pi}{2n + 1}} = \sqrt{2n + 1}.\] | If \(z = \cos \theta + i\sin \theta\) , then \(z - z^{-1} = 2i\sin \theta\) . Now put \(z = \cos \frac{\pi}{2n + 1} + i\sin \frac{\pi}{2n + 1}\) . Using de Moivre's formula we transform the required equality into
\[A = \prod_{k = 1}^{n}(z^{k} - z^{-k}) = i^{n}\sqrt{2n + 1}. \quad (1)\]
On the other hand, the comp... |
IMOSL-1973-16 | Given \(a, \theta \in \mathbb{R}\) , \(m \in \mathbb{N}\) , and \(P(x) = x^{2m} - 2|a|^{m}x^{m}\cos \theta + a^{2m}\) , factorize \(P(x)\) as a product of \(m\) real quadratic polynomials. | First, we have \(P(x) = Q(x)R(x)\) for \(Q(x) = x^{m} - |a|^{m}e^{i\theta}\) and \(R(x) = x^{m} - |a|^{m}e^{-i\theta}\) , where \(e^{i\phi}\) means of course \(\cos \phi +i\sin \phi\) . It remains to factor both \(Q\) and \(R\) . Suppose that \(Q(x) = (x - q_{1})\cdot \cdot \cdot (x - q_{m})\) and \(R(x) = (x - r_{1})\... |
IMOSL-1973-17 | Let \(\mathcal{F}\) be a nonempty set of functions \(f:\mathbb{R}\to \mathbb{R}\) of the form \(f(x) = ax + b\) , where \(a\) and \(b\) are real numbers and \(a\neq 0\) . Suppose that \(\mathcal{F}\) satisfies the following conditions:
(1) If \(f,g\in \mathcal{F}\) , then \(g\circ f\in \mathcal{F}\) , where \((g\cir... | Let \(f_{1}(x) = ax + b\) and \(f_{2}(x) = cx + d\) be two functions from \(\mathcal{F}\) . We define
\[g(x) = f_{1}\circ f_{2}(x) = acx + (ad + b)\quad \mathrm{and}\quad h(x) = f_{2}\circ f_{1}(x) = acx + (bc + d).\]
By the condition for \(\mathcal{F}\) , both \(g(x)\) and \(h(x)\) belong to \(\mathcal{F}\) . Mo... |
IMOSL-1974-1 | Alice, Betty, and Carol took the same series of examinations. There was one grade of \(A\) , one grade of \(B\) , and one grade of \(C\) for each examination, where \(A, B, C\) are different positive integers. The final test scores were
\[\frac{\mathrm{Alice}}{\mathrm{20}}\qquad \frac{\mathrm{Betty}}{\mathrm{10}}\qq... | Denote by \(n\) the number of exams. We have \(n(A + B + C) = 20 + 10 + 9 = 39\) , and since \(A, B, C\) are distinct, their sum is at least 6; therefore \(n = 3\) and \(A + B + C = 13\) .
Assume w.l.o.g. that \(A > B > C\) . Since Betty gained \(A\) points in arithmetic, but fewer than 13 points in total, she had \... |
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