International Mathematical Olympiad

Let $d$ be the sum of the lengths of all the diagonals of a plane convex polygon with $n$ vertices $(n > 3)$, and let $p$ be its perimeter. Prove that

$$n - 3 < \frac{2d}{p} < \left\lfloor\frac{n}{2}\right\rfloor\left\lfloor\frac{n+1}{2}\right\rfloor - 2,$$

where $\lfloor x \rfloor$ denotes the greatest integer not exceeding $x$.

Let $a, b, c$ and $d$ be odd integers such that $0 < a < b < c < d$ and $ad = bc$. Prove that if $a + d = 2^k$ and $b + c = 2^m$ for some integers $k$ and $m$, then $a = 1$.

Find all functions $f$ defined on the set of positive real numbers which take positive real values and satisfy the conditions:

(i) $f(xf(y)) = yf(x)$ for all positive $x, y$;

(ii) $f(x) \rightarrow 0$ as $x \rightarrow \infty$.

Let $A$ be one of the two distinct points of intersection of two unequal coplanar circles $C_1$ and $C_2$ with centers $O_1$ and $O_2$, respectively. One of the common tangents to the circles touches $C_1$ at $P_1$ and $C_2$ at $P_2$, while the other touches $C_1$ at $Q_1$ and $C_2$ at $Q_2$. Let $M_1$ be the midpoint of $P_1Q_1$, and $M_2$ be the midpoint of $P_2Q_2$. Prove that $\angle O_1AO_2 = \angle M_1AM_2$.

Let $a, b$ and $c$ be positive integers, no two of which have a common divisor greater than 1. Show that $2abc - ab - bc - ca$ is the largest integer which cannot be expressed in the form $xbc + yca + zab$, where $x, y$ and $z$ are non-negative integers.

Let $ABC$ be an equilateral triangle and $\mathcal{E}$ the set of all points contained in the three segments $AB$, $BC$ and $CA$ (including $A$, $B$ and $C$). Determine whether, for every partition of $\mathcal{E}$ into two disjoint subsets, at least one of the two subsets contains the vertices of a right-angled triangle. Justify your answer.

Is it possible to choose 1983 distinct positive integers, all less than or equal to $10^5$, no three of which are consecutive terms of an arithmetic progression? Justify your answer.

Let $a$, $b$ and $c$ be the lengths of the sides of a triangle. Prove that $$a^2b(a - b) + b^2c(b - c) + c^2a(c - a) \geq 0.$$

Determine when equality occurs.

The function $f(n)$ is defined for all positive integers $n$ and takes on non-negative integer values. Also, for all $m, n$

$$f(m + n) - f(m) - f(n) = 0 \text{ or } 1$$

$$f(2) = 0, f(3) > 0, \text{ and } f(9999) = 3333.$$

Determine $f(1982)$.

A non-isosceles triangle $A_1A_2A_3$ is given with sides $a_1, a_2, a_3$ ($a_i$ is the side opposite $A_i$). For all $i = 1, 2, 3, M_i$ is the midpoint of side $a_i$, and $T_i$ is the point where the incircle touches side $a_i$. Denote by $S_i$ the reflection of $T_i$ in the interior bisector of angle $A_i$. Prove that the lines $M_1S_1$, $M_2S_2$, and $M_3S_3$ are concurrent.

Consider the infinite sequences $\{x_n\}$ of positive real numbers with the following properties:

$$x_0 = 1, \text{ and for all } i \geq 0, x_{i+1} \leq x_i.$$

(a) Prove that for every such sequence, there is an $n \geq 1$ such that

$$\frac{x_0^2}{x_1} + \frac{x_1^2}{x_2} + \cdots + \frac{x_{n-1}^2}{x_n} \geq 3.999.$$

(b) Find such a sequence for which

$$\frac{x_0^2}{x_1} + \frac{x_1^2}{x_2} + \cdots + \frac{x_{n-1}^2}{x_n} < 4.$$

Prove that if $n$ is a positive integer such that the equation

$$x^3 - 3xy^2 + y^3 = n$$

has a solution in integers $(x, y)$, then it has at least three such solutions.

Show that the equation has no solutions in integers when $n = 2891$.

The diagonals $AC$ and $CE$ of the regular hexagon $ABCDEF$ are divided by the inner points $M$ and $N$, respectively, so that

$$\frac{AM}{AC} = \frac{CN}{CE} = r.$$

Determine $r$ if $B$, $M$, and $N$ are collinear.

Let $S$ be a square with sides of length 100, and let $L$ be a path within $S$ which does not meet itself and which is composed of line segments $A_0A_1, A_1A_2, \ldots, A_{n-1}A_n$ with $A_0 \neq A_n$. Suppose that for every point $P$ of the boundary of $S$ there is a point of $L$ at a distance from $P$ not greater than $1/2$. Prove that there are two points $X$ and $Y$ in $L$ such that the distance between $X$ and $Y$ is not greater than 1, and the length of that part of $L$ which lies between $X$ and $Y$ is not smaller than 198.

$P$ is a point inside a given triangle $ABC$. $D, E, F$ are the feet of the perpendiculars from $P$ to the lines $BC, CA, AB$ respectively. Find all $P$ for which $$\frac{BC}{PD} + \frac{CA}{PE} + \frac{AB}{PF}$$ is least.

Let $1 \leq r \leq n$ and consider all subsets of $r$ elements of the set $\{1, 2, \ldots, n\}$. Each of these subsets has a smallest member. Let $F(n, r)$ denote the arithmetic mean of these smallest numbers; prove that $$F(n, r) = \frac{n + 1}{r + 1}.$$

Determine the maximum value of $m^3 + n^3$, where $m$ and $n$ are integers satisfying $m, n \in \{1, 2, \ldots, 1981\}$ and $(n^2 - mn - m^2)^2 = 1$.

(a) For which values of $n > 2$ is there a set of $n$ consecutive positive integers such that the largest number in the set is a divisor of the least common multiple of the remaining $n - 1$ numbers?

(b) For which values of $n > 2$ is there exactly one set having the stated property?

Three congruent circles have a common point $O$ and lie inside a given triangle. Each circle touches a pair of sides of the triangle. Prove that the incenter and the circumcenter of the triangle and the point $O$ are collinear.

The function $f(x, y)$ satisfies

(1) $f(0, y) = y + 1$,

(2) $f(x + 1, 0) = f(x, 1)$,

(3) $f(x + 1, y + 1) = f(x, f(x + 1, y))$,

for all non-negative integers $x, y$. Determine $f(4, 1981)$.

Let $p$ and $q$ be natural numbers such that

$$\frac{p}{q}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots-\frac{1}{1318}+\frac{1}{1319}.$$

Prove that $p$ is divisible by 1979.

A prism with pentagons $A_1A_2A_3A_4A_5$ and $B_1B_2B_3B_4B_5$ as top and bottom faces is given. Each side of the two pentagons and each of the line-segments $A_iB_j$ for all $i,j=1,\ldots,5,$ is colored either red or green. Every triangle whose vertices are vertices of the prism and whose sides have all been colored has two sides of a different color. Show that all 10 sides of the top and bottom faces are the same color.

Two circles in a plane intersect. Let $A$ be one of the points of intersection. Starting simultaneously from $A$ two points move with constant speeds, each point travelling along its own circle in the same sense. The two points return to A simultaneously after one revolution. Prove that there is a fixed point $P$ in the plane such that, at any time, the distances from $P$ to the moving points are equal.

Given a plane $\pi,$ a point $P$ in this plane and a point $Q$ not in $\pi,$ find all points $R$ in $\pi$ such that the ratio $(QP+PA)/QR$ is a maximum.

Find all real numbers $a$ for which there exist non-negative real numbers $x_1,x_2,x_3,x_4,x_5$ satisfying the relations

$$\sum_{k=1}^{5}kx_k=a,\quad\sum_{k=1}^{5}k^3x_k=a^2,\quad\sum_{k=1}^{5}k^5x_k=a^3.$$

Let $A$ and $E$ be opposite vertices of a regular octagon. A frog starts jumping at vertex $A$. From any vertex of the octagon except $E,$ it may jump to either of the two adjacent vertices. When it reaches vertex $E,$ the frog stops and stays there. Let $a_n$ be the number of distinct paths of exactly $n$ jumps ending at $E.$ Prove that $a_{2n-1}=0,$

$$a_{2n}=\frac{1}{\sqrt{2}}(x^{n-1}-y^{n-1}),\quad n=1,2,3,\cdots,$$

where $x=2+\sqrt{2}$ and $y=2-\sqrt{2}.$

Note. A path of $n$ jumps is a sequence of vertices $(P_0,\ldots,P_n)$ such that

(i) $P_0=A, P_n=E;$

(ii) for every $i, 0\leq i\leq n-1, P_i$ is distinct from $E;$

(iii) for every $i, 0\leq i\leq n-1, P_i$ and $P_{i+1}$ are adjacent.

$m$ and $n$ are natural numbers with $1 \leq m < n$. In their decimal representations, the last three digits of $1978^m$ are equal, respectively, to the last three digits of $1978^n$. Find $m$ and $n$ such that $m + n$ has its least value.

$P$ is a given point inside a given sphere. Three mutually perpendicular rays from $P$ intersect the sphere at points $U, V$, and $W$; $Q$ denotes the vertex diagonally opposite to $P$ in the parallelepiped determined by $PU$, $PV$, and $PW$. Find the locus of $Q$ for all such triads of rays from $P$.

The set of all positive integers is the union of two disjoint subsets $\{f(1), f(2), \ldots, f(n), \ldots\}$, $\{g(1), g(2), \ldots, g(n), \ldots\}$, where

$$f(1) < f(2) < \cdots < f(n) < \cdots,$$ $$g(1) < g(2) < \cdots < g(n) < \cdots,$$

and

$$g(n) = f(f(n)) + 1 \text{ for all } n \geq 1.$$

Determine $f(240)$.

In triangle $ABC$, $AB = AC$. A circle is tangent internally to the circumcircle of triangle $ABC$ and also to sides $AB$, $AC$ at $P$, $Q$, respectively. Prove that the midpoint of segment $PQ$ is the center of the incircle of triangle $ABC$.

Let $\{a_k\}$ $(k = 1, 2, 3, \ldots, n, \ldots)$ be a sequence of distinct positive integers. Prove that for all natural numbers $n$,

$$\sum_{k=1}^n \frac{a_k}{k^2} \geq \sum_{k=1}^n \frac{1}{k}.$$

An international society has its members from six different countries. The list of members contains 1978 names, numbered $1, 2, \ldots, 1978$. Prove that there is at least one member whose number is the sum of the numbers of two members from his own country, or twice as large as the number of one member from his own country.

Equilateral triangles $ABK$, $BCL$, $CDM$, $DAN$ are constructed inside the square $ABCD$. Prove that the midpoints of the four segments $KL$, $LM$, $MN$, $NK$ and the midpoints of the eight segments $AK$, $BK$, $BL$, $CL$, $CM$, $DM$, $DN$, $AN$ are the twelve vertices of a regular dodecagon.

In a finite sequence of real numbers the sum of any seven successive terms is negative, and the sum of any eleven successive terms is positive. Determine the maximum number of terms in the sequence.

Let $n$ be a given integer $> 2$, and let $V_n$ be the set of integers $1 + kn$, where $k = 1, 2, \ldots$. A number $m \in V_n$ is called indecomposable in $V_n$ if there do not exist numbers $p, q \in V_n$ such that $pq = m$. Prove that there exists a number $r \in V_n$ that can be expressed as the product of elements indecomposable in $V_n$ in more than one way. (Products which differ only in the order of their factors will be considered the same.)

Four real constants $a$, $b$, $A$, $B$ are given, and $$f(\theta) = 1 - a\cos\theta - b\sin\theta - A\cos 2\theta - B\sin 2\theta.$$ Prove that if $f(\theta) \geq 0$ for all real $\theta$, then $$a^2 + b^2 \leq 2 \text{ and } A^2 + B^2 \leq 1.$$

Let $a$ and $b$ be positive integers. When $a^2 + b^2$ is divided by $a + b$, the quotient is $q$ and the remainder is $r$. Find all pairs $(a, b)$ such that $q^2 + r = 1977$.

Let $f(n)$ be a function defined on the set of all positive integers and having all its values in the same set. Prove that if $$f(n + 1) > f(f(n))$$ for each positive integer $n$, then $$f(n) = n \text{ for each } n.$$

In a plane convex quadrilateral of area 32, the sum of the lengths of two opposite sides and one diagonal is 16. Determine all possible lengths of the other diagonal.

Let $P_1(x) = x^2 - 2$ and $P_j(x) = P_1(P_{j-1}(x))$ for $j = 2, 3, \cdots$. Show that, for any positive integer $n$, the roots of the equation $P_n(x) = x$ are real and distinct.

A rectangular box can be filled completely with unit cubes. If one places as many cubes as possible, each with volume 2, in the box, so that their edges are parallel to the edges of the box, one can fill exactly 40% of the box. Determine the possible dimensions of all such boxes.

Determine, with proof, the largest number which is the product of positive integers whose sum is $1976$.

Consider the system of $p$ equations in $q = 2p$ unknowns $x_1, x_2, \cdots, x_q$: $$\begin{aligned} a_{11}x_1 + a_{12}x_2 + \cdots + a_{1q}x_q &= 0\\ a_{21}x_1 + a_{22}x_2 + \cdots + a_{2q}x_q &= 0\\ &\cdots\\ a_{p1}x_1 + a_{p2}x_2 + \cdots + a_{pq}x_q &= 0 \end{aligned}$$ with every coefficient $a_{ij}$ member of the set $\{-1, 0, 1\}$. Prove that the system has a solution $(x_1, x_2, \cdots, x_q)$ such that

• (a) all $x_j$ $(j = 1, 2, \ldots, q)$ are integers,

• (b) there is at least one value of $j$ for which $x_j \neq 0$,

• (c) $|x_j| \leq q$ $(j = 1, 2, \ldots, q)$.

A sequence $\{u_n\}$ is defined by $$u_0 = 2, \quad u_1 = 5/2, \quad u_{n+1} = u_n(u_{n-1}^2 - 2) - u_1 \text{ for } n = 1, 2, \cdots$$

Prove that for positive integers $n$, $$[u_n] = 2^{[2^n - (-1)^n]/3}$$ where $[x]$ denotes the greatest integer $\leq x$.

Let $x_i, y_i$ $(i = 1, 2, \ldots, n)$ be real numbers such that

$$x_1 \geq x_2 \geq \cdots \geq x_n \text{ and } y_1 \geq y_2 \geq \cdots \geq y_n.$$

Prove that, if $z_1, z_2, \cdots, z_n$ is any permutation of $y_1, y_2, \cdots, y_n$, then

$$\sum_{i=1}^{n}(x_i - y_i)^2 \leq \sum_{i=1}^{n}(x_i - z_i)^2.$$

Let $a_1, a_2, a_3, \cdots$ be an infinite increasing sequence of positive integers. Prove that for every $p \geq 1$ there are infinitely many $a_m$ which can be written in the form

$$a_m = xa_p + ya_q$$

with $x, y$ positive integers and $q > p$.

On the sides of an arbitrary triangle $ABC$, triangles $ABR, BCP, CAQ$ are constructed externally with $\angle CBP = \angle CAQ = 45°$, $\angle BCP = \angle ACQ = 30°$, $\angle ABR = \angle BAR = 15°$. Prove that $\angle QRP = 90°$ and $QR = RP$.

When $4444^{4444}$ is written in decimal notation, the sum of its digits is $A$. Let $B$ be the sum of the digits of $A$. Find the sum of the digits of $B$. ($A$ and $B$ are written in decimal notation.)

Determine, with proof, whether or not one can find 1975 points on the circumference of a circle with unit radius such that the distance between any two of them is a rational number.

Find all polynomials $P$ in two variables, with the following properties: (i) for a positive integer $n$ and all real $t, x, y$

$$P(tx, ty) = t^n P(x, y)$$

(that is, $P$ is homogeneous of degree $n$), (ii) for all real $a, b, c$,

$$P(b + c, a) + P(c + a, b) + P(a + b, c) = 0,$$

(iii) $P(1, 0) = 1$.