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Problems
1Let \(a\), \(b\), \(c\), \(d\) be positive real numbers such that \[ \frac{5a+b}{5c+d} = \frac{6a+b}{6c+d} \qquad \text{and} \qquad \frac{7a+b}{7c+d} = 8 . \] Determine the value of \(\dfrac{9a+b}{9c+d}\). …Open2Let \(a\), \(b\), \(c\) be positive numbers with \(a > c\) and \(b > c\). Prove that \[ \sqrt{c(a-c)} + \sqrt{c(b-c)} \;\leq\; \sqrt{ab} . \]3Prove that among all triangles with one and the same perimeter, the equilateral triangle has the largest area.4Find five real numbers whose pairwise sums are \[ 0,\; 2,\; 4,\; 5,\; 7,\; 9,\; 10,\; 12,\; 14,\; 17 . \] (Five numbers form exactly ten pairs, and ten values are listed; the sums may be listed in any …5(a) Factor the expression \(x^{4}+x^{2}y^{2}+y^{4}\) into a product of polynomials of lower degree. (b) Decide whether the number \(9^{1998}+3^{1998}+1\) is prime.6In a convex hexagon \(ABCDEF\), each of the two diagonals \(AD\) and \(BE\) divides the hexagon into two pieces of equal area. Prove that the quadrilateral \(BDEA\) is a trapezoid.7Let \(a_1, a_2, \dots, a_n\) and \(b_1, b_2, \dots, b_n\), where \(n > 1\), be positive integers satisfying \[ \frac{a_1}{b_1} = \frac{a_2}{b_2} = \cdots = \frac{a_n}{b_n}. \] Prove that \(a_1 + a_2 + \cdots + a_n + b_1 + b_2 + \cdots + b_n\) …8Let \(a_1, a_2, \ldots, a_n\) be positive real numbers with \(a_1 + a_2 + \cdots + a_n = 1\), and let \[ S = \sum_{i=1}^{n} \sum_{j=1}^{n} \frac{a_i a_j}{a_i + a_j} \] be the sum of all \(n^2\) expressions …9Let \(A\) be a set of \(10\) numbers chosen from \(\{1, 2, \ldots, 100\}\). Prove that \(A\) has two nonempty subsets \(S\) and \(T\) with no element in common such that the sum of the elements of \(S\) …10Can the numbers \(1, 2, \dots, 100\) be distributed into three groups so that the sum of the numbers in the first group is divisible by \(102\), the sum of the numbers in the second group is divisible …11A polynomial \(p(x)\) has integer coefficients. Divided by \(x^2 - 12x + 11\), it leaves the remainder \(990x - 889\). Prove that no integer is a zero of \(p(x)\).12An island is inhabited by \(45\) chameleons: \(17\) yellow, \(15\) grey and \(13\) blue. They wander about and meet from time to time, never more than two at a time. When two chameleons of the same colour …13The pupils of a school went to the theatre on two occasions, and every pupil of the school attended at least one of the two performances. Boys formed \(60\%\) of the audience at the first performance and …14Ali-Baba is standing in a cave full of gold and diamonds. A kilogram of gold is worth \(20\) dinars and a kilogram of diamonds is worth \(60\) dinars. He has a single chest with him. Filled with gold, …15Let \(a = 123456789\) and \(b = 987654321\). (1) Find \(\gcd(a, b)\). (2) Find the remainder left by \(\operatorname{lcm}(a, b)\) on division by \(11\).16Real numbers \(a, b, c, d\) satisfy \[ a^{2} + b^{2} = c^{2} + d^{2} , \qquad ab + cd > 0 , \qquad ac + bd > 0 . \] Prove that \(ad + bc > 0\).17The points of a plane \(\alpha\) are split between two nonempty sets \(A\) and \(B\): no point belongs to both, and every point belongs to one of them. Prove that some isosceles right triangle has all …18Let \(m\) be an arbitrary integer. Prove that there is at least one pair \((x, y)\) of integers for which \[ 2x^2 + 11xy + 12y^2 + 4x + 5y + 6 = 2m . \]19Two concentric circles \(k_1\) and \(k_2\) have radii \(a\) and \(b\). Consider all rectangles that have two vertices on \(k_1\) and the remaining two vertices on \(k_2\). Determine the rectangle of largest …20Let \(a\), \(b\), \(c\) be three distinct nonzero real numbers, and for real \(x, y \neq a\) set \[ V(x,y)=\frac{1}{(a-x)^{2}(a-y)^{2}}\Bigl((a-b)^{2}(c-x)(c-y)-(c-a)^{2}(b-x)(b-y)\Bigr). \] Prove that …21Let \(a\), \(b\), \(c\), \(d\), \(x\), \(y\) be positive real numbers such that \[ a + 2ay + y = b + 2bx + x \qquad \text{and} \qquad x + 2xd + d = y + 2yc + c . \] Prove that \[ a + 2ad + d = b + 2bc + c . \] …22Let \(x\) and \(y\) be nonnegative real numbers with \(x + y = 2\). Prove that \[ x^2 y^2 \left( x^2 + y^2 \right) \leq 2 . \] When does equality hold?23How many integers \(n\) with \(10 \le n < 100000\) are divisible by \(4\), contain no digit \(0\) in their decimal representation, and have no two adjacent digits equal?24Prove that for all real numbers \(a\) and \(b\), \[ a(1 + b^2) + b(1 + a^2) \leq (1 + a^2)(1 + b^2) . \]25A natural number \(n \geqslant 2\) is divided by each of the natural numbers \(1, 2, \ldots, n-1\) in turn, and all the remainders obtained are written down. Find every \(n\) for which the sum of the distinct …26Two vertical poles stand on level ground, at a distance of \(9\) m from each other; one pole is \(11\) m high and the other is \(15\) m high. A rope of length \(15\) m is fastened to the top of one pole …27In the expression \[ *\,1 * 3 * 3^{2} * 3^{3} * \cdots * 3^{1997} * 3^{1998} \] Arkadije and Branislav take turns replacing one of the stars by \(+\) or by \(-\), one star per move, until no star is left. …28Let \(ABCD\) be a parallelogram whose interior angle at \(A\) is acute, and let \(E\) be a point of the plane such that \(EA \perp AB\) and \(EC \perp CB\). Prove that \[ \angle AED = \angle CEB . \]29Let \(ABC\) be a triangle and let \(M\), \(N\), \(P\) be points on its sides \(AB\), \(BC\), \(AC\) respectively, chosen so that \(AMNP\) is a parallelogram. Let \(k_1\) be the circle circumscribed about …30In a group of \(20\) people, every person chooses ten of the other nineteen and sends one letter to each of them. Prove that there are two people who sent a letter to each other.31How many isosceles trapezoids with integer side lengths have perimeter \(2005\)? (A trapezoid here means a quadrilateral with exactly two parallel sides, so a parallelogram is not one. Trapezoids with …32For a natural number \(n\), let \(S(n)\) be the sum of its decimal digits and \(P(n)\) the product of its decimal digits. Find all natural numbers \(n\) for which \[ S(n) + P(n) = n . \]33Let \(A_1, A_2, \dots, A_{501}\) be arbitrary pairwise distinct points of the plane. Prove that on every circle of radius \(4\) there is a point \(M\) for which \[ MA_1 + MA_2 + \dots + MA_{501} \geq 2004 . \] …34Find every triple of integers \(x, y, z\) with \[ 3x^2 + 3y^2 + 3z^2 + 2x + 2y + 2z = 2004 . \]35Two circles \(R_1\) and \(R_2\) meet at points \(A\) and \(B\). A line through \(A\) is allowed to vary; it meets \(R_1\) again at \(P\) and \(R_2\) again at \(Q\). Prove that all the resulting perpendicular …36A circle \(k\) has radius \(31\,\mathrm{mm}\), and \(\ell\) is a broken line of length \(61\,\mathrm{mm}\) whose two endpoints both lie on \(k\). Prove that there is a line \(p\) passing through the centre …37Let \(ABCD\) be a trapezoid with \(AB \parallel CD\), and let \(P\) be a point on the extension of the diagonal \(AC\) beyond \(C\), so that \(C\) lies between \(A\) and \(P\). Let \(X\) and \(Y\) be the …38Let \(n\) be a natural number and let \(d\) be a positive divisor of \(2n^2\). Can \(n^2 + d\) be a perfect square?39A triangle \(ABC\) is given. Find every point \(M\) of its plane for which the three triangles \(ABM\), \(BCM\) and \(CAM\) have equal areas.40Let \(k > 3\) and consider the number \(2^k\). Prove that no rearrangement of the decimal digits of \(2^k\) can produce the number \(2^n\) with \(n > k\).41The number \(1\) is written on a board \(2005\) times. A move consists of erasing two of the numbers written on the board and writing, in their place, one quarter of their sum. The move is repeated until …42Two fixed points \(A\) and \(B\) are given in the plane. A point \(M\) is chosen and then travels along the straight segment from \(M\) to \(A\). Determine all positions of \(M\) for which the distance …43Circles \(k_1, k_2, \ldots, k_{1999}\) lie in the plane and touch one another externally in a closed chain: \(k_1\) touches \(k_2\) at \(A_1\), \(k_2\) touches \(k_3\) at \(A_2\), and so on, and finally …44A \(2004 \times 2004\) board is completely tiled by pieces of size \(1 \times 4\); each piece covers four cells of a single row (call it horizontal) or four cells of a single column (vertical). Can the …45A pentagon \(ABCDE\) is inscribed in a circle of radius \(r\), and three of its sides have length \(r\): \[ AB = BC = DE = r . \] Let \(G\) and \(F\) be the midpoints of the sides \(CD\) and \(EA\). Prove …46A set \(\mathcal{A}\) of \(2000\) points in the plane contains no three collinear points. Prove that these points can be joined by \(1000\) blue, \(1000\) red and \(1000\) yellow segments in such a way …47A point \(P\) inside a triangle \(ABC\) satisfies \[ \angle BPC = \angle BAC + 60^\circ, \qquad \angle CPA = \angle CBA + 60^\circ, \qquad \angle APB = \angle ACB + 60^\circ. \] The lines \(AP\), \(BP\), …48An octagon has all of its interior angles equal, and the lengths of all of its sides are rational numbers. Prove that it has a centre of symmetry.49A triangle \(ABC\) is given. Consider all lines which cut the side \(AC\) at a point \(M\) and the side \(BC\) at a point \(N\) in such a way that \(MN = AM + BN\). Prove that there is a circle \(k\) which …50On the sides \(BC\), \(CA\) and \(AB\) of a triangle \(ABC\) points \(A_1\), \(B_1\) and \(C_1\) are marked, respectively. Let \(T\) be the centroid of the triangle \(ABC\) and \(T_1\) the centroid of …51How many triples \((a, b, c)\) of positive integers are there such that \(2a + 1\) is divisible by \(b\), \(2b + 1\) is divisible by \(c\), and \(2c + 1\) is divisible by \(a\)?52Let \(ABC\) be an isosceles triangle with \(AB = AC\). Let \(D\) be the point of the side \(AC\) for which \(CD = 2 \, AD\), and let \(P\) be a point of the segment \(BD\) with \(\angle APC = 90^\circ\). …53An acute triangle \(ABC\) is given. Construct, with ruler and compass, a point \(P\) inside the triangle such that the rays \(AP\), \(BP\) and \(CP\) meet the circumcircle of \(ABC\) again in the three …54Let \(ABC\) be an acute triangle. The bisector of its interior angle at \(B\) meets \(AC\) at \(K\), and \(CD\) is the altitude from \(C\), with \(D\) on \(AB\). Let \(N\) be the point of \(CD\) for which …55For some integer \(n > 3\), all the digits of the number \[ 1 + 2 + \cdots + n \] are equal to one another. Which digit can this be?56The circle inscribed in a triangle \(ABC\) touches the sides \(AB\) and \(AC\) at the points \(M\) and \(N\) respectively. Let \(P\) be the point in which the bisector of the angle \(ABC\) meets the line …57In an equilateral triangle \(ABC\) the side has length \(|AB| = 2\). Let \(M\) and \(N\) be interior points of the side \(AB\) with \(|MN| = 1\). Prove that \[ \angle MCN > 30^\circ . \]
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