IMO 1987 Longlist
IMO 1987 Longlist — 55 problems.
IMO 1987 Longlist
55 problems · Source: IMO Compendium
| Problem | Origin | Statement |
|---|---|---|
| AUS1 | AUS | Let x1, x2, . . . , xn be n integers. Let n = p + q, where p and q |
| AUS2 | AUS | Suppose we have a pack of 2n cards, in the order 1, 2, . . . , 2n. A |
| AUS3 | AUS | A town has a road network that consists entirely of one-way |
| AUS4 | AUS | Let a1, a2, a3, b1, b2, b3 be positive real numbers. Prove that |
| AUS5 | AUS | Let there be given three circles K1, K2, K3 with centers |
| BEL7 | BEL | Let f : (0, +\infty) \toR be a function having the property |
| BEL8 | BEL | Determine the least possible value of the natural number n |
| BEL9 | BEL | In the set of 20 elements {1, 2, 3, 4, 5, 6, 7, 8, 9, 0, A, B, C, |
| FIN10 | FIN | In a Cartesian coordinate system, the circle C1 has center |
| FIN11 | FIN | Let S \subset[0, 1] be a set of 5 points with {0, 1} \subsetS. The graph |
| FIN13 | FIN | A be an infinite set of positive integers such that every n \inA is |
| FRA14 | FRA | Given n real numbers 0 < t1 \leqt2 \leq\cdot \cdot \cdot \leqtn < 1, prove that |
| FRA15 | FRA | Let a1, a2, a3, b1, b2, b3, c1, c2, c3 be nine strictly positive real |
| FRA16 | FRA | Let ABC be a triangle. For every point M belonging to segment |
| FRA17 | FRA | Consider the number \alpha obtained by writing one after another |
| GBR23 | GBR | A lampshade is part of the surface of a right circular cone |
| GBR24 | GBR | Prove that if the equation x4 + ax3 + bx + c = 0 has all its |
| GBR25 | GBR | Numbers d(n, m), with m, n integers, 0 \leqm \leqn, ae defined |
| GBR26 | GBR | Prove that if x, y, z are real numbers such that x2+y2+z2 = 2, |
| GBR27 | GBR | Find, with proof, the smallest real number C with the following |
| GDR28 | GDR | In a chess tournament there are n \geq5 players, and they have |
| GRE30 | GRE | Consider the regular 1987-gon A1A2 . . . A1987 with center O. |
| GRE31 | GRE | Construct a triangle ABC given its side a = BC, its circum- |
| GRE32 | GRE | Solve the equation 28x = 19y + 87z, where x, y, z are integers. |
| ICE36 | ICE | A game consists in pushing a flat stone along a sequence of |
| ICE37 | ICE | Five distinct numbers are drawn successively and at random |
| LUX39 | LUX | Let A be a set of polynomials with real coefficients and let |
| MON40 | MON | The perpendicular line issued from the center of the circum- |
| MON41 | MON | Let n points be given arbitrarily in the plane, no three of |
| MON42 | MON | Find the integer solutions of the equation |
| MON43 | MON | Let 2n + 3 points be given in the plane in such a way that |
| MOR44 | MOR | Let \theta1, \theta2, . . . , \thetan be real numbers such that sin \theta1 +… |
| MOR45 | MOR | Let us consider a variable polygon with 2n sides (n \inN) in a |
| NET47 | NET | Through a point P within a triangle ABC the lines l, m, and |
| POL49 | POL | In the coordinate system in the plane we consider a convex |
| POL50 | POL | Let P, Q, R be polynomials with real coefficients, satisfying |
| POL51 | POL | The function F is a one-to-one transformation of the plane into |
| ROM54 | ROM | Let n be a natural number. Solve in integers the equation |
| ROM55 | ROM | Two moving bodies M1, M2 are displaced uniformly on two |
| ROM57 | ROM | The bisectors of the angles B, C of a triangle ABC intersect |
| SPA58 | SPA | Find, with argument, the integer solutions of the equation |
| SPA59 | SPA | It is given that a11, a22 are real numbers, that x1, x2, a12, b1, b2 |
| TUR60 | TUR | It is given that x = −2272, y = 103 +102c+10b+a, and z = 1 |
| TUR61 | TUR | Let PQ be a line segment of constant length \lambda taken on the |
| TUR62 | TUR | Let l, l′ be two lines in 3-space and let A, B, C be three points |
| TUR63 | TUR | Compute 2n |
| USA64 | USA | Let r > 1 be a real number, and let n be the largest integer |
| USA65 | USA | The runs of a decimal number are its increasing or decreasing |
| USS67 | USS | If a, b, c, d are real numbers such that a2 + b2 + c2 + d2 \leq1, |
| USS71 | USS | To every natural number k, k \geq2, there corresponds a sequence |
| VIE72 | VIE | Is it possible to cover a rectangle of dimensions m \times n with |
| VIE73 | VIE | Let f(x) be a periodic function of period T > 0 defined over R. |
| VIE75 | VIE | Let ak be positive numbers such that a1 \geq1 and ak+1 −ak \geq1 |
| VIE76 | VIE | Given two sequences of positive numbers {ak} and {bk} (k \inN) |
| YUG77 | YUG | Find the least natural number k such that for any n \in[0, 1] |