ILL 1986

Let $k$ be one of the integers $2, 3,4$ and let $n = 2^k -1$ . Prove the inequality $$1+ b^k + b^{2k} + \cdots+ b^{nk} \geq (1 + b^n)^k$$ for all real $b \geq 0.$

Let $ABCD$ be a convex quadrilateral. $DA$ and $CB$ meet at $F$ and $AB$ and $DC$ meet at $E$ . The bisectors of the angles $DFC$ and $AED$ are perpendicular. Prove that these angle bisectors are parallel to the bisectors of the angles between the lines $AC$ and $BD.$

A line parallel to the side $BC$ of a triangle $ABC$ meets $AB$ in $F$ and $AC$ in $E$ . Prove that the circles on $BE$ and $CF$ as diameters intersect in a point lying on the altitude of the triangle $ABC$ dropped from $A$ to $BC.$

Find the last eight digits of the binary development of $27^{1986}.$

Let $ABC$ and $DEF$ be acute-angled triangles. Write $d = EF, e = FD, f = DE.$ Show that there exists a point $P$ in the interior of $ABC$ for which the value of the expression $X=d \cdot AP +e \cdot BP +f \cdot CP$ attains a minimum.

In an urn there are one ball marked $1$ , two balls marked $2$ , and so on, up to $n$ balls marked $n$ . Two balls are randomly drawn without replacement. Find the probability that the two balls are assigned the same number.

Let $f(n)$ be the least number of distinct points in the plane such that for each $k = 1, 2, \cdots, n$ there exists a straight line containing exactly $k$ of these points. Find an explicit expression for $f(n).$

Simplified version.

Show that $f(n)=\left[\frac{n+1}{2}\right]\left[\frac{n+2}{2}\right].$ Where $[x]$ denoting the greatest integer not exceeding $x.$

A tetrahedron $ABCD$ is given such that $AD = BC = a; AC = BD = b; AB\cdot CD = c^2$. Let $f(P) = AP + BP + CP + DP$, where $P$ is an arbitrary point in space. Compute the least value of $f(P).$

In a triangle $ABC$ , $\angle BAC = 100^{\circ}, AB = AC$ . A point $D$ is chosen on the side $AC$ such that $\angle ABD = \angle CBD$ . Prove that $AD + DB = BC.$

A set of $n$ standard dice are shaken and randomly placed in a straight line. If $n < 2r$ and $r < s$ , then the probability that there will be a string of at least $r$ , but not more than $s$ , consecutive $1$ 's can be written as $\frac{P}{6^{s+2}}$ . Find an explicit expression for $P$ .

Prove that the sum of the face angles at each vertex of a tetrahedron is a straight angle if and only if the faces are congruent triangles.

Let $O$ be an interior point of a tetrahedron $A_1A_2A_3A_4$ . Let $S_1, S_2, S_3, S_4$ be spheres with centers $A_1,A_2,A_3,A_4$ , respectively, and let $U, V$ be spheres with centers at $O$ . Suppose that for $i, j = 1, 2, 3, 4, i \neq j$ , the spheres $S_i$ and $S_j$ are tangent to each other at a point $B_{ij}$ lying on $A_iA_j$ . Suppose also that $U$ is tangent to all edges $A_iA_j$ and $V$ is tangent to the spheres $S_1, S_2, S_3, S_4$ . Prove that $A_1A_2A_3A_4$ is a regular tetrahedron.

Let $N = \{1, 2, \ldots, n\}$ , $n \geq 3$ . To each pair $i \neq j$ of elements of $N$ there is assigned a number $f_{ij} \in \{0, 1\}$ such that $f_{ij} + f_{ji} = 1$ . Let $r(i)=\sum_{i \neq j} f_{ij}$ , and write $M = \max_{i\in N} r(i)$ , $m = \min_{i\in N} r(i)$ . Prove that for any $w \in N$ with $r(w) = m$ there exist $u, v \in N$ such that $r(u) = M$ and $f_{uv}f_{vw} = 1$ .

Let $\mathbb N = B_1\cup\cdots \cup B_q$ be a partition of the set $\mathbb N$ of all positive integers and let an integer $l \in \mathbb N$ be given. Prove that there exist a set $X \subset \mathbb N$ of cardinality $l$ , an infinite set $T \subset \mathbb N$ , and an integer $k$ with $1 \leq k \leq q$ such that for any $t \in T$ and any finite set $Y \subset X$ , the sum $t+ \sum_{y \in Y} y$ belongs to $B_k.$

Given a positive integer $k$ , find the least integer $n_k$ for which there exist five sets $S_1, S_2, S_3, S_4, S_5$ with the following properties: $$|S_j|=k \text{ for } j=1, \cdots , 5 , |\bigcup_{j=1}^{5} S_j | = n_k ;$$ $$|S_i \cap S_{i+1}| = 0 = |S_5 \cap S_1|, \text{for } i=1,\cdots ,4$$

We call a tetrahedron right-faced if each of its faces is a right-angled triangle.

(a) Prove that every orthogonal parallelepiped can be partitioned into six right-faced tetrahedra.

(b) Prove that a tetrahedron with vertices $A_1,A_2,A_3,A_4$ is right-faced if and only if there exist four distinct real numbers $c_1, c_2, c_3$ , and $c_4$ such that the edges $A_jA_k$ have lengths $A_jA_k=\sqrt{|c_j-c_k|}$ for $1\leq j < k \leq 4.$

Let $f : [0, 1] \to [0, 1]$ satisfy $f(0) = 0, f(1) = 1$ and $$f(x + y) - f(x) = f(x) - f(x - y)$$ for all $x, y \geq 0$ with $x - y, x + y \in [0, 1].$ Prove that $f(x) = x$ for all $x \in [0, 1].$

For any angle α with $0 < \alpha < 180^{\circ}$ , we call a closed convex planar set an $\alpha$ -set if it is bounded by two circular arcs (or an arc and a line segment) whose angle of intersection is $\alpha$ . Given a (closed) triangle $T$ , find the greatest $\alpha$ such that any two points in $T$ are contained in an $\alpha$ -set $S \subset T .$

Let $AB$ be a segment of unit length and let $C, D$ be variable points of this segment. Find the maximum value of the product of the lengths of the six distinct segments with endpoints in the set $\{A,B,C,D\}.$

Let $(a_n)_{n \geq 0}$ be the sequence of integers defined recursively by $a_0 = 0, a_1 = 1, a_{n+2} = 4a_{n+1} + a_n$ for $n \geq 0.$ Find the common divisors of $a_{1986}$ and $a_{6891}.$

Let $I$ and $J$ be the centers of the incircle and the excircle in the angle $BAC$ of the triangle $ABC$ . For any point $M$ in the plane of the triangle, not on the line $BC$ , denote by $I_M$ and $J_M$ the centers of the incircle and the excircle (touching $BC$ ) of the triangle $BCM$ . Find the locus of points $M$ for which $II_MJJ_M$ is a rectangle.

Two families of parallel lines are given in the plane, consisting of $15$ and $11$ lines, respectively. In each family, any two neighboring lines are at a unit distance from one another; the lines of the first family are perpendicular to the lines of the second family. Let $V$ be the set of $165$ intersection points of the lines under consideration. Show that there exist not fewer than $1986$ distinct squares with vertices in the set $V .$

Let real numbers $x_1, x_2, \cdots , x_n$ satisfy $0 < x_1 < x_2 < \cdots< x_n < 1$ and set $x_0 = 0, x_{n+1} = 1$. Suppose that these numbers satisfy the following system of equations: $$\sum_{j=0, j \neq i}^{n+1} \frac{1}{x_i-x_j}=0 \text{where } i = 1, 2, . . ., n.$$

Prove that $x_{n+1-i} = 1- x_i$ for $i = 1, 2, . . . , n.$

In an urn there are n balls numbered $1, 2, \cdots, n$ . They are drawn at random one by one without replacement and the numbers are recorded. What is the probability that the resulting random permutation has only one local maximum? A term in a sequence is a local maximum if it is greater than all its neighbors.

A particle moves from $(0, 0)$ to $(n, n)$ directed by a fair coin. For each head it moves one step east and for each tail it moves one step north. At $(n, y), y < n$, it stays there if a head comes up and at $(x, n), x < n$, it stays there if a tail comes up. Le t $k$ be a fixed positive integer. Find the probability that the particle needs exactly $2n+k$ tosses to reach $(n, n).$

We define a binary operation $\star$ in the plane as follows: Given two points $A$ and $B$ in the plane, $C = A \star B$ is the third vertex of the equilateral triangle ABC oriented positively. What is the relative position of three points $I, M, O$ in the plane if $I \star (M \star O) = (O \star I)\star M$ holds?

Prove that a convex polyhedron all of whose faces are equilateral triangles has at most $30$ edges.

Let $P$ and $Q$ be distinct points in the plane of a triangle $ABC$ such that $AP : AQ = BP : BQ = CP : CQ$ . Prove that the line $PQ$ passes through the circumcenter of the triangle.

Find, with proof, all solutions of the equation $\frac 1x +\frac 2y- \frac 3z = 1$ in positive integers $x, y, z.$

For each non-negative integer $n$ , $F_n(x)$ is a polynomial in $x$ of degree $n$ . Prove that if the identity $$F_n(2x)=\sum_{r=0}^{n} (-1)^{n-r} \binom nr 2^r F_r(x)$$ holds for each n, then $$F_n(tx)=\sum_{r=0}^{n} \binom nr t^r (1-t)^{n-r} F_r(x)$$

Establish the maximum and minimum values that the sum $|a| + |b| + |c|$ can have if $a, b, c$ are real numbers such that the maximum value of $|ax^2 + bx + c|$ is $1$ for $-1 \leq x \leq 1.$

Prove that the set $\{1, 2, . . . , 1986\}$ can be partitioned into $27$ disjoint sets so that no one of these sets contains an arithmetic triple (i.e., three distinct numbers in an arithmetic progression).

Let $S$ be a $k$ -element set.

(a) Find the number of mappings $f : S \to S$ such that $$\text{(i) } f(x) \neq x \text{ for } x \in S, \text{(ii) } f(f(x)) = x \text{ for }x \in S.$$

(b) The same with the condition $\text{(i)}$ left out.

Find the maximum value that the quantity $2m+7n$ can have such that there exist distinct positive integers $x_i \ (1 \leq i \leq m), y_j \ (1 \leq j \leq n)$ such that the $x_i$ 's are even, the $y_j$ 's are odd, and $\sum_{i=1}^{m} x_i +\sum_{j=1}^{n} y_j=1986.$

Let $M,N,P$ be the midpoints of the sides $BC, CA, AB$ of a triangle $ABC$ . The lines $AM, BN, CP$ intersect the circumcircle of $ABC$ at points $A',B', C'$ , respectively. Show that if $A'B'C'$ is an equilateral triangle, then so is $ABC.$

The integers $1, 2, \cdots, n^2$ are placed on the fields of an $n \times n$ chessboard $(n > 2)$ in such a way that any two fields that have a common edge or a vertex are assigned numbers differing by at most $n + 1$ . What is the total number of such placements?

Three persons $A,B,C$, are playing the following game:

A $k$ - element subset of the set $\{1, . . . , 1986\}$ is randomly chosen, with an equal probability of each choice, where $k$ is a fixed positive integer less than or equal to $1986$. The winner is $A,B$ or $C$, respectively, if the sum of the chosen numbers leaves a remainder of $0, 1$, or $2$ when divided by $3$.

For what values of $k$ is this game a fair one? (A game is fair if the three outcomes are equally probable.)

The circle inscribed in a triangle $ABC$ touches the sides $BC,CA,AB$ in $D,E, F$, respectively, and $X, Y,Z$ are the midpoints of $EF, FD,DE$, respectively. Prove that the centers of the inscribed circle and of the circles around $XYZ$ and $ABC$ are collinear.

Given $n$ real numbers $a_1 \leq a_2 \leq \cdots \leq a_n$ , define $$M_1=\frac 1n \sum_{i=1}^{n} a_i , M_2=\frac{2}{n(n-1)} \sum_{1 \leq i<j \leq n} a_ia_j, Q=\sqrt{M_1^2-M_2}$$ Prove that $$a_1 \leq M_1 - Q \leq M_1 + Q \leq a_n$$ and that equality holds if and only if $a_1 = a_2 = \cdots = a_n.$

We wish to construct a matrix with $19$ rows and $86$ columns, with entries $x_{ij} \in \{0, 1, 2\} \ (1 \leq i \leq 19, 1 \leq j \leq 86)$ , such that:

(i) in each column there are exactly $k$ terms equal to $0$ ;

(ii) for any distinct $j, k \in \{1, . . . , 86\}$ there is $i \in \{1, . . . , 19\}$ with $x_{ij} + x_{ik} = 3.$ For what values of $k$ is this possible?

Let $P$ be a convex $1986$ -gon in the plane. Let $A,D$ be interior points of two distinct sides of P and let $B,C$ be two distinct interior points of the line segment $AD$ . Starting with an arbitrary point $Q_1$ on the boundary of $P$ , define recursively a sequence of points $Q_n$ as follows: given $Q_n$ extend the directed line segment $Q_nB$ to meet the boundary of $P$ in a point $R_n$ and then extend $R_nC$ to meet the boundary of $P$ again in a point, which is defined to be $Q_{n+1}$ . Prove that for all $n$ large enough the points $Q_n$ are on one of the sides of $P$ containing $A$ or $D$ .

Let $C_1, C_2$ be circles of radius $1/2$ tangent to each other and both tangent internally to a circle $C$ of radius $1$ . The circles $C_1$ and $C_2$ are the first two terms of an infinite sequence of distinct circles $C_n$ defined as follows: $C_{n+2}$ is tangent externally to $C_n$ and $C_{n+1}$ and internally to $C$ . Show that the radius of each $C_n$ is the reciprocal of an integer.

Let $D$ be the point on the side $BC$ of the triangle $ABC$ such that $AD$ is the bisector of $\angle CAB$ . Let $I$ be the incenter of $ABC.$ (a) Construct the points $P$ and $Q$ on the sides $AB$ and $AC$ , respectively, such that $PQ$ is parallel to $BC$ and the perimeter of the triangle $APQ$ is equal to $k \cdot BC$ , where $k$ is a given rational number.

*(b)*Let $R$ be the intersection point of $PQ$ and $AD$ . For what value of $k$ does the equality $AR = RI$ hold?

(c) In which case do the equalities $AR = RI = ID$ hold?

Let $a, b, c, d$ be the lengths of the sides of a quadrilateral circumscribed about a circle and let $S$ be its area. Prove that $S \leq \sqrt{abcd}$ and find conditions for equality.

Solve the system of equations $$\tan x_1 +\cot x_1=3 \tan x_2,$$$$\tan x_2 +\cot x_2=3 \tan x_3,$$$$\vdots$$$$\tan x_n +\cot x_n=3 \tan x_1$$

For given positive integers $r, v, n$ let $S(r, v, n)$ denote the number of $n$ -tuples of non-negative integers $(x_1, \cdots, x_n)$ satisfying the equation $x_1 +\cdots+ x_n = r$ and such that $x_i \leq v$ for $i = 1, \cdots , n$ . Prove that $$S(r, v, n)=\sum_{k=0}^{m} (-1)^k \binom nk \binom{r - (v + 1)k + n - 1}{n-1}$$ Where $m=\left\{n,\left[\frac{r}{v+1}\right]\right\}.$

Find the least integer $n$ with the following property: For any set $V$ of $8$ points in the plane, no three lying on a line, and for any set $E$ of n line segments with endpoints in $V$ , one can find a straight line intersecting at least $4$ segments in $E$ in interior points.

Given an integer $n \geq 2$ , determine all $n$ -digit numbers $M_0 = \overline{a_1a_2 \cdots a_n} \ (a_i \neq 0, i = 1, 2, . . ., n)$ divisible by the numbers $M_1 = \overline{a_2a_3 \cdots a_na_1}$ , $M_2 = \overline{a_3a_4 \cdots a_na_1 a_2}$ , $\cdots$ , $M_{n-1} = \overline{a_na_1a_2 . . .a_{n-1}}.$

Let $A_1A_2A_3A_4A_5A_6$ be a hexagon inscribed into a circle with center $O$ . Consider the circular arc with endpoints $A_1,A_6$ not containing $A_2$ . For any point $M$ of that arc denote by $h_i$ the distance from $M$ to the line $A_iA_{i+1} \ (1 \leq i \leq 5)$ . Construct $M$ such that the sum $h_1 + \cdots + h_5$ is maximal.

In a triangle $ABC$ , the incircle touches the sides $BC, CA, AB$ in the points $A',B', C'$ , respectively; the excircle in the angle $A$ touches the lines containing these sides in $A_1,B_1, C_1$ , and similarly, the excircles in the angles $B$ and $C$ touch these lines in $A_2,B_2, C_2$ and $A_3,B_3, C_3$ . Prove that the triangle $ABC$ is right-angled if and only if one of the point triples $(A',B_3, C'),$ $(A_3,B', C_3), (A',B', C_2), (A_2,B_2, C'), (A_2,B_1, C_2), (A_3,B_3, C_1),$ $(A_1,B_2, C_1), (A_1,B_1, C_3)$ is collinear.

Find four positive integers each not exceeding $70000$ and each having more than $100$ divisors.

Let $ABCD$ be a convex quadrilateral whose vertices do not lie on a circle. Let $A'B'C'D'$ be a quadrangle such that $A',B', C',D'$ are the centers of the circumcircles of triangles $BCD,ACD,ABD$, and $ABC$. We write $T (ABCD) = A'B'C'D'$. Let us define $A''B''C''D'' = T (A'B'C'D') = T (T (ABCD)).$

(a) Prove that $ABCD$ and $A''B''C''D''$ are similar.

(b) The ratio of similitude depends on the size of the angles of $ABCD$. Determine this ratio.

Prove the inequality $$(-a+b+c)^2(a-b+c)^2(a+b-c)^2 \geq (-a^2+b^2+c^2)(a^2-b^2+c^2)(a^2+b^2-c^2)$$ for all real numbers $a, b, c.$

Given a positive integer $n$ , find the greatest integer $p$ with the property that for any function $f : \mathbb P(X) \to C$ , where $X$ and $C$ are sets of cardinality $n$ and $p$ , respectively, there exist two distinct sets $A,B \in \mathbb P(X)$ such that $f(A) = f(B) = f(A \cup B)$ . ( $\mathbb P(X)$ is the family of all subsets of $X$ .)

Determine all pairs of positive integers $(x, y)$ satisfying the equation $p^x - y^3 = 1$ , where $p$ is a given prime number.

Let $AA',BB', CC'$ be the bisectors of the angles of a triangle $ABC \ (A' \in BC, B' \in CA, C' \in AB)$ . Prove that each of the lines $A'B', B'C', C'A'$ intersects the incircle in two points.

Let $(a_n)_{n\in \mathbb N}$ be the sequence of integers defined recursively by $a_1 = a_2 = 1, a_{n+2} = 7a_{n+1} - a_n - 2$ for $n \geq 1$ . Prove that $a_n$ is a perfect square for every $n.$

Let $A_1A_2A_3A_4$ be a quadrilateral inscribed in a circle $C$ . Show that there is a point $M$ on $C$ such that $MA_1 -MA_2 +MA_3 -MA_4 = 0.$

One hundred red points and one hundred blue points are chosen in the plane, no three of them lying on a line. Show that these points can be connected pairwise, red ones with blue ones, by disjoint line segments.

Let $f(x) = x^n$ where $n$ is a fixed positive integer and $x =1, 2, \cdots .$ Is the decimal expansion $a = 0.f (1)f(2)f(3) . . .$ rational for any value of $n$?

The decimal expansion of a is defined as follows: If $f(x) = d_1(x)d_2(x) \cdots d_{r(x)}(x)$ is the decimal expansion of $f(x)$, then $a = 0.1d_1(2)d_2(2) \cdots d_{r(2)}(2)d_1(3) . . . d_{r(3)}(3)d_1(4) \cdots .$

Consider the equation $x^4 + ax^3 + bx^2 + ax + 1 = 0$ with real coefficients $a, b$ . Determine the number of distinct real roots and their multiplicities for various values of $a$ and $b$ . Display your result graphically in the $(a, b)$ plane.

Let $AX,BY,CZ$ be three cevians concurrent at an interior point $D$ of a triangle $ABC$. Prove that if two of the quadrangles $DY AZ,DZBX,DXCY$ are circumscribable, so is the third.

Let $ABCD$ be a tetrahedron having each sum of opposite sides equal to $1$. Prove that $$r_A + r_B + r_C + r_D \leq \frac{\sqrt 3}{3}$$

where $r_A, r_B, r_C, r_D$ are the inradii of the faces, equality holding only if $ABCD$ is regular.

Two straight lines perpendicular to each other meet each side of a triangle in points symmetric with respect to the midpoint of that side. Prove that these two lines intersect in a point on the nine-point circle.

A one-person game with two possible outcomes is played as follows: After each play, the player receives either $a$ or $b$ points, where $a$ and $b$ are integers with $0 < b < a < 1986$ . The game is played as many times as one wishes and the total score of the game is defined as the sum of points received after successive plays. It is observed that every integer $x \geq 1986$ can be obtained as the total score whereas $1985$ and $663$ cannot. Determine $a$ and $b.$

Let $(a_i)_{i\in \mathbb N}$ be a strictly increasing sequence of positive real numbers such that $\lim_{i \to \infty} a_i = +\infty$ and $a_{i+1}/a_i \leq 10$ for each $i$ . Prove that for every positive integer $k$ there are infinitely many pairs $(i, j)$ with $10^k \leq a_i/a_j \leq 10^{k+1}.$

From a collection of $n$ persons $q$ distinct two-member teams are selected and ranked $1, \cdots, q$ (no ties). Let $m$ be the least integer larger than or equal to $2q/n$. Show that there are $m$ distinct teams that may be listed so that : (i) each pair of consecutive teams on the list have one member in common and (ii) the chain of teams on the list are in rank order.

Alternative formulation. Given a graph with $n$ vertices and $q$ edges numbered $1, \cdots , q$, show that there exists a chain of $m$ edges, $m \geq \frac{2q}{n}$, each two consecutive edges having a common vertex, arranged monotonically with respect to the numbering.

The incenter of a triangle is the midpoint of the line segment of length $4$ joining the centroid and the orthocenter of the triangle. Determine the maximum possible area of the triangle.

Let $A, B$, and $C$ be three points on the edge of a circular chord such that $B$ is due west of $C$ and $ABC$ is an equilateral triangle whose side is $86$ meters long. A boy swam from $A$ directly toward $B$. After covering a distance of $x$ meters, he turned and swam westward, reaching the shore after covering a distance of $y$ meters. If $x$ and $y$ are both positive integers, determine $y.$

Find all integers $x,y,z$ such that $$x^3+y^3+z^3=x+y+z=8$$

If $T$ and $T_1$ are two triangles with angles $x, y, z$ and $x_1, y_1, z_1$ , respectively, prove the inequality $$\frac{\cos x_1}{\sin x}+\frac{\cos y_1}{\sin y}+\frac{\cos z_1}{\sin z} \leq \cot x+\cot y+\cot z.$$

Let $AA_1,BB_1, CC_1$ be the altitudes in an acute-angled triangle $ABC$ , $K$ and $M$ are points on the line segments $A_1C_1$ and $B_1C_1$ respectively. Prove that if the angles $MAK$ and $CAA_1$ are equal, then the angle $C_1KM$ is bisected by $AK.$

Let $ABCD$ be a tetrahedron and $O$ its incenter, and let the line $OD$ be perpendicular to $AD$ . Find the angle between the planes $DOB$ and $DOC.$