International Competitions

$AB$ is tangent to the circles $CAMN$ and $NMBD$. $M$ lies between $C$ and $D$ on the line $CD$, and $CD$ is parallel to $AB$. The chords $NA$ and $CM$ meet at $P$; the chords $NB$ and $MD$ meet at $Q$. The rays $CA$ and $DB$ meet at $E$. Prove that $PE = QE$.

$A, B, C$ are positive reals with product 1. Prove that $(A - 1 + \frac{1}{B})(B - 1 + \frac{1}{C})(C - 1 + \frac{1}{A}) \leq 1$.

$k$ is a positive real. $N$ is an integer greater than 1. $N$ points are placed on a line, not all coincident. A move is carried out as follows. Pick any two points $A$ and $B$ which are not coincident. Suppose that $A$ lies to the right of $B$. Replace $B$ by another point $B'$ to the right of $A$ such that $AB' = kBA$. For what values of $k$ can we move the points arbitrarily far to the right by repeated moves?

100 cards are numbered 1 to 100 (each card different) and placed in 3 boxes (at least one card in each box). How many ways can this be done so that if two boxes are selected and a card is taken from each, then the knowledge of their sum alone is always sufficient to identify the third box?

Can we find $N$ divisible by just 2000 different primes, so that $N$ divides $2^N + 1$? [$N$ may be divisible by a prime power.]

$A_1A_2A_3$ is an acute-angled triangle. The foot of the altitude from $A_i$ is $K_i$ and the incircle touches the side opposite $A_i$ at $L_i$. The line $K_1K_2$ is reflected in the line $L_1L_2$. Similarly, the line $K_2K_3$ is reflected in $L_2L_3$ and $K_3K_1$ is reflected in $L_3L_1$. Show that the three new lines form a triangle with vertices on the incircle.

Determine all finite sets $S$ of at least three points in the plane which satisfy the following condition:

for any two distinct points $A$ and $B$ in $S$, the perpendicular bisector of the line segment $AB$ is an axis of symmetry for $S$.

Let $n$ be a fixed integer, with $n \geq 2$.

(a) Determine the least constant $C$ such that the inequality

$$\sum_{1 \leq i < j \leq n} x_i x_j (x_i^2 + x_j^2) \leq C \left( \sum_{1 \leq i \leq n} x_i \right)^4$$

holds for all real numbers $x_1, \ldots, x_n \geq 0$.

(b) For this constant $C$, determine when equality holds.

Consider an $n \times n$ square board, where $n$ is a fixed even positive integer. The board is divided into $n^2$ unit squares. We say that two different squares on the board are adjacent if they have a common side.

$N$ unit squares on the board are marked in such a way that every square (marked or unmarked) on the board is adjacent to at least one marked square.

Determine the smallest possible value of $N$.

Determine all pairs $(n,p)$ of positive integers such that

$p$ is a prime,

$n$ not exceeded $2p$, and

$(p-1)^n + 1$ is divisible by $n^{p-1}$.

Two circles $G_1$ and $G_2$ are contained inside the circle $G$, and are tangent to $G$ at the distinct points $M$ and $N$, respectively. $G_1$ passes through the center of $G_2$. The line passing through the two points of intersection of $G_1$ and $G_2$ meets $G$ at $A$ and $B$. The lines $MA$ and $MB$ meet $G_1$ at $C$ and $D$, respectively.

Prove that $CD$ is tangent to $G_2$.

Determine all functions $f: \mathbf{R} \longrightarrow \mathbf{R}$ such that

$$f(x - f(y)) = f(f(y)) + x f(y) + f(x) - 1$$

for all real numbers $x, y$.

In the convex quadrilateral $ABCD$, the diagonals $AC$ and $BD$ are perpendicular and the opposite sides $AB$ and $DC$ are not parallel. Suppose that the point $P$, where the perpendicular bisectors of $AB$ and $DC$ meet, is inside $ABCD$. Prove that $ABCD$ is a cyclic quadrilateral if and only if the triangles $ABP$ and $CDP$ have equal areas.

In a competition, there are $a$ contestants and $b$ judges, where $b \geq 3$ is an odd integer. Each judge rates each contestant as either "pass" or "fail". Suppose $k$ is a number such that, for any two judges, their ratings coincide for at most $k$ contestants. Prove that $k/a \geq (b - 1)/(2b)$.

For any positive integer $n$, let $d(n)$ denote the number of positive divisors of $n$ (including 1 and $n$ itself). Determine all positive integers $k$ such that $d(n^2)/d(n) = k$ for some $n$.

Determine all pairs $(a,b)$ of positive integers such that $ab^2 + b + 7$ divides $a^2b + a + b$.

Let $I$ be the incenter of triangle $ABC$. Let the incircle of $ABC$ touch the sides $BC$, $CA$, and $AB$ at $K$, $L$, and $M$, respectively. The line through $B$ parallel to $MK$ meets the lines $LM$ and $LK$ at $R$ and $S$, respectively. Prove that angle $RIS$ is acute.

Consider all functions $f$ from the set $N$ of all positive integers into itself satisfying $f(t^2f(s)) = s(f(t))^2$ for all $s$ and $t$ in $N$. Determine the least possible value of $f(1998)$.

In the plane the points with integer coordinates are the vertices of unit squares. The squares are colored alternately black and white (as on a chessboard).

For any pair of positive integers $m$ and $n$, consider a right-angled triangle whose vertices have integer coordinates and whose legs, of lengths $m$ and $n$, lie along edges of the squares.

Let $S_1$ be the total area of the black part of the triangle and $S_2$ be the total area of the white part. Let

$$f(m,n) = |S_1 - S_2|.$$

(a) Calculate $f(m,n)$ for all positive integers $m$ and $n$ which are either both even or both odd.

(b) Prove that $f(m,n) \leq \frac{1}{2}\max\{m,n\}$ for all $m$ and $n$.

(c) Show that there is no constant $C$ such that $f(m,n) < C$ for all $m$ and $n$.

The angle at $A$ is the smallest angle of triangle $ABC$. The points $B$ and $C$ divide the circumcircle of the triangle into two arcs. Let $U$ be an interior point of the arc between $B$ and $C$ which does not contain $A$. The perpendicular bisectors of $AB$ and $AC$ meet the line $AU$ at $V$ and $W$, respectively. The lines $BV$ and $CW$ meet at $T$. Show that

$$AU = TB + TC.$$

Let $x_1, x_2, \ldots, x_n$ be real numbers satisfying the conditions

$$|x_1 + x_2 + \cdots + x_n| = 1$$

and

$$|x_i| \leq \frac{n+1}{2} \qquad i = 1, 2, \ldots, n.$$

Show that there exists a permutation $y_1, y_2, \ldots, y_n$ of $x_1, x_2, \ldots, x_n$ such that

$$|y_1 + 2y_2 + \cdots + ny_n| \leq \frac{n+1}{2}.$$

An $n \times n$ matrix whose entries come from the set $S = \{1, 2, \ldots, 2n - 1\}$ is called a silver matrix if, for each $i = 1, 2, \ldots, n$, the $i$th row and the $i$th column together contain all elements of $S$. Show that

(a) there is no silver matrix for $n = 1997$;

(b) silver matrices exist for infinitely many values of $n$.

Find all pairs $(a, b)$ of integers $a, b \geq 1$ that satisfy the equation

$$a^{b^2} = b^a.$$

For each positive integer $n$, let $f(n)$ denote the number of ways of representing $n$ as a sum of powers of 2 with nonnegative integer exponents. Representations which differ only in the ordering of their summands are considered to be the same. For instance, $f(4) = 4$, because the number 4 can be represented in the following four ways:

$$4; 2 + 2; 2 + 1 + 1; 1 + 1 + 1 + 1.$$

Prove that, for any integer $n \geq 3$,

$$2^{n^2/4} < f(2^n) < 2^{n^2/2}.$$

We are given a positive integer $r$ and a rectangular board $ABCD$ with dimensions $|AB| = 20$, $|BC| = 12$. The rectangle is divided into a grid of $20 \times 12$ unit squares. The following moves are permitted on the board: one can move from one square to another only if the distance between the centers of the two squares is $\sqrt{r}$. The task is to find a sequence of moves leading from the square with $A$ as a vertex to the square with $B$ as a vertex.

(a) Show that the task cannot be done if $r$ is divisible by 2 or 3.

(b) Prove that the task is possible when $r = 73$.

(c) Can the task be done when $r = 97$?

Let $P$ be a point inside triangle $ABC$ such that

$$\angle APB - \angle ACB = \angle APC - \angle ABC.$$

Let $D, E$ be the incenters of triangles $APB, APC$, respectively. Show that $AP, BD, CE$ meet at a point.

Let $S$ denote the set of nonnegative integers. Find all functions $f$ from $S$ to itself such that

$$f(m + f(n)) = f(f(m)) + f(n) \qquad \forall m, n \in S.$$

The positive integers $a$ and $b$ are such that the numbers $15a + 16b$ and $16a - 15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?

Let $ABCDEF$ be a convex hexagon such that $AB$ is parallel to $DE$, $BC$ is parallel to $EF$, and $CD$ is parallel to $FA$. Let $R_A, R_C, R_E$ denote the circumradii of triangles $FAB, BCD, DEF$, respectively, and let $P$ denote the perimeter of the hexagon. Prove that

$$R_A + R_C + R_E \geq \frac{P}{2}.$$

Let $p, q, n$ be three positive integers with $p + q < n$. Let $(x_0, x_1, \ldots, x_n)$ be an $(n + 1)$-tuple of integers satisfying the following conditions:

(a) $x_0 = x_n = 0$.

(b) For each $i$ with $1 \leq i \leq n$, either $x_i - x_{i-1} = p$ or $x_i - x_{i-1} = -q$.

Show that there exist indices $i < j$ with $(i, j) \neq (0, n)$, such that $x_i = x_j$.

Let $A, B, C, D$ be four distinct points on a line, in that order. The circles with diameters $AC$ and $BD$ intersect at $X$ and $Y$. The line $XY$ meets $BC$ at $Z$. Let $P$ be a point on the line $XY$ other than $Z$. The line $CP$ intersects the circle with diameter $AC$ at $C$ and $M$, and the line $BP$ intersects the circle with diameter $BD$ at $B$ and $N$. Prove that the lines $AM, DN, XY$ are concurrent.

Let $a, b, c$ be positive real numbers such that $abc = 1$. Prove that $$\frac{1}{a^3(b + c)} + \frac{1}{b^3(c + a)} + \frac{1}{c^3(a + b)} \geq \frac{3}{2}.$$

Determine all integers $n > 3$ for which there exist $n$ points $A_1, \ldots, A_n$ in the plane, no three collinear, and real numbers $r_1, \ldots, r_n$ such that for $1 \leq i < j < k \leq n$, the area of $\triangle A_i A_j A_k$ is $r_i + r_j + r_k$.

Find the maximum value of $x_0$ for which there exists a sequence $x_0, x_1, \ldots, x_{1995}$ of positive reals with $x_0 = x_{1995}$, such that for $i = 1, \ldots, 1995$, $$x_{i-1} + \frac{2}{x_{i-1}} = 2x_i + \frac{1}{x_i}.$$

Let $ABCDEF$ be a convex hexagon with $AB = BC = CD$ and $DE = EF = FA$, such that $\angle BCD = \angle EFA = \pi/3$. Suppose $G$ and $H$ are points in the interior of the hexagon such that $\angle AGB = \angle DHE = 2\pi/3$. Prove that $AG + GB + GH + DH + HE \geq CF$.

Let $p$ be an odd prime number. How many $p$-element subsets $A$ of $\{1, 2, \ldots, 2p\}$ are there, the sum of whose elements is divisible by $p$?

Let $m$ and $n$ be positive integers. Let $a_1, a_2, \ldots, a_m$ be distinct elements of $\{1, 2, \ldots, n\}$ such that whenever $a_i + a_j \leq n$ for some $i, j$, $1 \leq i \leq j \leq m$, there exists $k$, $1 \leq k \leq m$, with $a_i + a_j = a_k$. Prove that

$$\frac{a_1 + a_2 + \cdots + a_m}{m} \geq \frac{n + 1}{2}.$$

$ABC$ is an isosceles triangle with $AB = AC$. Suppose that

1. $M$ is the midpoint of $BC$ and $O$ is the point on the line $AM$ such that $OB$ is perpendicular to $AB$;
2. $Q$ is an arbitrary point on the segment $BC$ different from $B$ and $C$;
3. $E$ lies on the line $AB$ and $F$ lies on the line $AC$ such that $E, Q, F$ are distinct and collinear.

Prove that $OQ$ is perpendicular to $EF$ if and only if $QE = QF$.

For any positive integer $k$, let $f(k)$ be the number of elements in the set $\{k + 1, k + 2, \ldots, 2k\}$ whose base 2 representation has precisely three 1s.

• (a) Prove that, for each positive integer $m$, there exists at least one positive integer $k$ such that $f(k) = m$.
• (b) Determine all positive integers $m$ for which there exists exactly one $k$ with $f(k) = m$.

Determine all ordered pairs $(m, n)$ of positive integers such that

$$\frac{n^3 + 1}{mn - 1}$$

is an integer.

Let $S$ be the set of real numbers strictly greater than $-1$. Find all functions $f: S \to S$ satisfying the two conditions:

1. $f(x + f(y) + xf(y)) = y + f(x) + yf(x)$ for all $x$ and $y$ in $S$;
2. $\frac{f(x)}{x}$ is strictly increasing on each of the intervals $-1 < x < 0$ and $0 < x$.

Show that there exists a set $A$ of positive integers with the following property: For any infinite set $S$ of primes there exist two positive integers $m \in A$ and $n \notin A$ each of which is a product of $k$ distinct elements of $S$ for some $k \geq 2$.

Let $f(x) = x^n + 5x^{n-1} + 3$, where $n > 1$ is an integer. Prove that $f(x)$ cannot be expressed as the product of two nonconstant polynomials with integer coefficients.

Let $D$ be a point inside acute triangle $ABC$ such that $\angle ADB = \angle ACB + \pi/2$ and $AC \cdot BD = AD \cdot BC$.

(a) Calculate the ratio $(AB \cdot CD)/(AC \cdot BD)$.

(b) Prove that the tangents at $C$ to the circumcircles of $\triangle ACD$ and $\triangle BCD$ are perpendicular.

On an infinite chessboard, a game is played as follows. At the start, $n^2$ pieces are arranged on the chessboard in an $n$ by $n$ block of adjoining squares, one piece in each square. A move in the game is a jump in a horizontal or vertical direction over an adjacent occupied square to an unoccupied square immediately beyond. The piece which has been jumped over is removed.

Find those values of $n$ for which the game can end with only one piece remaining on the board.

For three points $P, Q, R$ in the plane, we define $m(PQR)$ as the minimum length of the three altitudes of $\triangle PQR$. (If the points are collinear, we set $m(PQR) = 0$.)

Prove that for points $A, B, C, X$ in the plane, $$m(ABC) \leq m(ABX) + m(AXC) + m(XBC).$$

Does there exist a function $f: \mathbf{N} \to \mathbf{N}$ such that $f(1) = 2$, $f(f(n)) = f(n) + n$ for all $n \in \mathbf{N}$, and $f(n) < f(n + 1)$ for all $n \in \mathbf{N}$?

There are $n$ lamps $L_0, \ldots, L_{n-1}$ in a circle ($n > 1$), where we denote $L_{n+k} = L_k$. (A lamp at all times is either on or off.) Perform steps $s_0, s_1, \ldots$ as follows: at step $s_i$, if $L_{i-1}$ is lit, switch $L_i$ from on to off or vice versa, otherwise do nothing. Initially all lamps are on. Show that:

(a) There is a positive integer $M(n)$ such that after $M(n)$ steps all the lamps are on again;

(b) If $n = 2^k$, we can take $M(n) = n^2 - 1$;

(c) If $n = 2^k + 1$, we can take $M(n) = n^2 - n + 1$.

Find all integers $a, b, c$ with $1 < a < b < c$ such that

$(a - 1)(b - 1)(c - 1)$ is a divisor of $abc - 1$.

Let $\mathbf{R}$ denote the set of all real numbers. Find all functions $f: \mathbf{R} \to \mathbf{R}$ such that

$$f \left(x^2 + f(y)\right) = y + \left(f(x)\right)^2 \quad \text{for all } x, y \in R.$$