ILL 1992

Points $D$ and $E$ are chosen on the sides $AB$ and $AC$ of the triangle $ABC$ in such a way that if $F$ is the intersection point of $BE$ and $CD$ , then $AE + EF = AD + DF$ . Prove that $AC + CF = AB + BF.$

Let $m$ be a positive integer and $x_0, y_0$ integers such that $x_0, y_0$ are relatively prime, $y_0$ divides $x_0^2+m$ , and $x_0$ divides $y_0^2+m$ . Prove that there exist positive integers $x$ and $y$ such that $x$ and $y$ are relatively prime, $y$ divides $x^2 + m$ , $x$ divides $y^2 + m$ , and $x + y \leq m+ 1.$

Let $ABC$ be a triangle, $O$ its circumcenter, $S$ its centroid, and $H$ its orthocenter. Denote by $A_1, B_1$ , and $C_1$ the centers of the circles circumscribed about the triangles $CHB, CHA$ , and $AHB$ , respectively. Prove that the triangle $ABC$ is congruent to the triangle $A_1B_1C_1$ and that the nine-point circle of $\triangle ABC$ is also the nine-point circle of $\triangle A_1B_1C_1$ .

Let $p, q$ , and $r$ be the angles of a triangle, and let $a = \sin2p, b = \sin2q$ , and $c = \sin2r$ . If $s = \frac{(a + b + c)}2$ , show that $$s(s - a)(s - b)(s -c) \geq 0.$$ When does equality hold?

Let $I,H,O$ be the incenter, centroid, and circumcenter of the nonisosceles triangle $ABC$ . Prove that $AI \parallel HO$ if and only if $\angle BAC =120^{\circ}$ .

Suppose that n numbers $x_1, x_2, . . . , x_n$ are chosen randomly from the set $\{1, 2, 3, 4, 5\}$ . Prove that the probability that $x_1^2+ x_2^2 +\cdots+ x_n^2 \equiv 0 \pmod 5$ is at least $\frac 15.$

Given two positive real numbers $a$ and $b$ , suppose that a mapping $f : \mathbb R^+ \to \mathbb R^+$ satisfies the functional equation $$f(f(x)) + af(x) = b(a + b)x.$$ Prove that there exists a unique solution of this equation.

The diagonals of a quadrilateral $ABCD$ are perpendicular: $AC \perp BD.$ Four squares, $ABEF,BCGH,CDIJ,DAKL,$ are erected externally on its sides. The intersection points of the pairs of straight lines $CL, DF, AH, BJ$ are denoted by $P_1,Q_1,R_1, S_1,$ respectively (left figure), and the intersection points of the pairs of straight lines $AI, BK, CE DG$ are denoted by $P_2,Q_2,R_2, S_2,$ respectively (right figure). Prove that $P_1Q_1R_1S_1 \cong P_2Q_2R_2S_2$ where $P_1,Q_1,R_1, S_1$ and $P_2,Q_2,R_2, S_2$ are the two quadrilaterals.

Alternative formulation: Outside a convex quadrilateral $ABCD$ with perpendicular diagonals, four squares $AEFB, BGHC, CIJD, DKLA,$ are constructed (vertices are given in counterclockwise order). Prove that the quadrilaterals $Q_1$ and $Q_2$ formed by the lines $AG, BI, CK, DE$ and $AJ, BL, CF, DH,$ respectively, are congruent.

Attachments:

Let $\phi(n,m), m \neq 1$ , be the number of positive integers less than or equal to $n$ that are coprime with $m.$ Clearly, $\phi(m,m) = \phi(m)$ , where $\phi(m)$ is Euler's phi function. Find all integers $m$ that satisfy the following inequality: $$\frac{\phi(n,m)}{n} \geq \frac{\phi(m)}{m}$$ for every positive integer $n.$

Given a triangle $ABC$ such that the circumcenter is in the interior of the incircle, prove that the triangle $ABC$ is acute-angled.

A convex quadrilateral has equal diagonals. An equilateral triangle is constructed on the outside of each side of the quadrilateral. The centers of the triangles on opposite sides are joined. Show that the two joining lines are perpendicular.

Alternative formulation. Given a convex quadrilateral $ABCD$ with congruent diagonals $AC = BD.$ Four regular triangles are errected externally on its sides. Prove that the segments joining the centroids of the triangles on the opposite sides are perpendicular to each other.

Original formulation: Let $ABCD$ be a convex quadrilateral such that $AC = BD.$ Equilateral triangles are constructed on the sides of the quadrilateral. Let $O_1,O_2,O_3,O_4$ be the centers of the triangles constructed on $AB,BC,CD,DA$ respectively. Show that $O_1O_3$ is perpendicular to $O_2O_4.$

Integers $a_1, a_2, . . . , a_n$ satisfy $|a_k| = 1$ and $$\sum_{k=1}^{n} a_ka_{k+1}a_{k+2}a_{k+3} = 2,$$ where $a_{n+j} = a_j$ . Prove that $n \neq 1992.$

Prove that there exist $78$ lines in the plane such that they have exactly $1992$ points of intersection.

Find all triples $(x, y, z)$ of integers such that $$\frac{1}{x^2}+\frac{2}{y^2}+\frac{3}{z^2} =\frac 23$$

Fibonacci numbers are defined as follows: $F_0 = F_1 = 1, F_{n+2} = F_{n+1}+F_n, n \geq 0$ . Let $a_n$ be the number of words that consist of $n$ letters $0$ or $1$ and contain no two letters $1$ at distance two from each other. Express $a_n$ in terms of Fibonacci numbers.

Denote by $a_n$ the greatest number that is not divisible by $3$ and that divides $n$ . Consider the sequence $s_0 = 0, s_n = a_1 +a_2+\cdots+a_n, n \in \mathbb N$ . Denote by $A(n)$ the number of all sums $s_k \ (0 \leq k \leq 3^n, k \in \mathbb N_0)$ that are divisible by $3$ . Prove the formula $$A(n) = 3^{n-1} + 2 \cdot 3^{(n/2)-1} \cos \left(\frac{n\pi}{6}\right), n\in \mathbb N_0.$$

Let $X$ and $Y$ be two sets of points in the plane and $M$ be a set of segments connecting points from $X$ and $Y$ . Let $k$ be a natural number. Prove that the segments from $M$ can be painted using $k$ colors in such a way that for any point $x \in X \cup Y$ and two colors $\alpha$ and $\beta$ $(\alpha \neq \beta)$ , the difference between the number of $\alpha$ -colored segments and the number of $\beta$ -colored segments originating in $X$ is less than or equal to $1$ .

Prove that if $x,y,z >1$ and $\frac 1x +\frac 1y +\frac 1z = 2$ , then $$\sqrt{x+y+z} \geq \sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}.$$

An Egyptian number is a positive integer that can be expressed as a sum of positive integers, not necessarily distinct, such that the sum of their reciprocals is $1$ . For example, $32 = 2 + 3 + 9 + 18$ is Egyptian because $\frac 12 +\frac 13 +\frac 19 +\frac{1}{18}=1$ . Prove that all integers greater than $23$ are Egyptian.

(a) Show that there exists exactly one function $f : \mathbb Q^+ \to \mathbb Q^+$ satisfying the following conditions:(i) if $0 < q < \frac 12$ , then $f(q)=1+f \left( \frac{q}{1-2q} \right);$ (ii) if $1 < q \leq 2$ , then $f(q) = 1+f(q + 1);$ (iii) $f(q)f(1/q) = 1$ for all $q \in \mathbb Q^+.$ (b) Find the smallest rational number $q \in \mathbb Q^+$ such that $f(q) = \frac{19}{92}.$

(a) Show that the set $\mathbb N$ of all positive integers can be partitioned into three disjoint subsets $A, B$ , and $C$ satisfying the following conditions: $$A^2 = A, B^2 = C, C^2 = B,$$ $$AB = B, AC = C, BC = A,$$ where $HK$ stands for $\{hk | h \in H, k \in K\}$ for any two subsets $H, K$ of $\mathbb N$ , and $H^2$ denotes $HH.$ (b) Show that for every such partition of $\mathbb N$ , $\min\{n \in N | n \in A \text{ and } n + 1 \in A\}$ is less than or equal to $77.$

Let $ABC$ be an arbitrary scalene triangle. Define $\sum$ to be the set of all circles $y$ that have the following properties:(i) $y$ meets each side of $ABC$ in two (possibly coincident) points;(ii) if the points of intersection of $y$ with the sides of the triangle are labeled by $P, Q, R, S, T , U$ , with the points occurring on the sides in orders $\mathcal B(B,P,Q,C), \mathcal B(C, R, S,A), \mathcal B(A, T,U,B)$ , then the following relations of parallelism hold: $TS \parallel BC; PU\parallel CA; RQ\parallel AB$ . (In the limiting cases, some of the conditions of parallelism will hold vacuously; e.g., if $A$ lies on the circle $y$ , then $T$ , $S$ both coincide with $A$ and the relation $TS \parallel BC$ holds vacuously.)

(a) Under what circumstances is $\sum$ nonempty?

(b) Assuming that Σ is nonempty, show how to construct the locus of centers of the circles in the set $\sum$ .

(c) Given that the set $\sum$ has just one element, deduce the size of the largest angle of $ABC.$ (d) Show how to construct the circles in $\sum$ that have, respectively, the largest and the smallest radii.

Let $P_n = (19 + 92)(19^2 +92^2) \cdots(19^n +92^n)$ for each positive integer $n$ . Determine, with proof, the least positive integer $m$ , if it exists, for which $P_m$ is divisible by $33^{33}.$

Let $S_n = \{1, 2,\cdots, n\}$ and $f_n : S_n \to S_n$ be defined inductively as follows: $f_1(1) = 1, f_n(2j) = j \ (j = 1, 2, \cdots , [n/2])$ and

• (i) if $n = 2k \ (k \geq 1)$ , then $f_n(2j - 1) = f_k(j) + k \ (j = 1, 2, \cdots, k);$
• (ii) if $n = 2k + 1 \ (k \geq 1)$ , then $f_n(2k + 1) = k + f_{k+1}(1), f_n(2j - 1) = k + f_{k+1}(j + 1) \ (j = 1, 2,\cdots , k).$

Prove that $f_n(x) = x$ if and only if $x$ is an integer of the form $$\frac{(2n + 1)(2^d - 1)}{2^{d+1} - 1}$$ for some positive integer $d.$

Let $a, b, c$ be positive real numbers and $p, q, r$ complex numbers. Let $S$ be the set of all solutions $(x, y, z)$ in $\mathbb C$ of the system of simultaneous equations $$ax + by + cz = p,$$$$ax2 + by2 + cz2 = q,$$$$ax3 + bx3 + cx3 = r.$$ Prove that $S$ has at most six elements.

Let $a, b, c$ be integers. Prove that there are integers $p_1, q_1, r_1, p_2, q_2, r_2$ such that $$a = q_1r_2 - q_2r_1, b = r_1p_2 - r_2p_1, c = p_1q_2 - p_2q_1.$$

Find all rational solutions of $$a^2 + c^2 + 17(b^2 + d^2) = 21,$$$$ab + cd = 2.$$

Let the circles $C_1, C_2$ , and $C_3$ be orthogonal to the circle $C$ and intersect each other inside $C$ forming acute angles of measures $A, B$ , and $C$ . Show that $A + B +C < \pi.$

Let $n \geq 2$ be an integer. Find the minimum $k$ for which there exists a partition of $\{1, 2, . . . , k\}$ into $n$ subsets $X_1,X_2, \cdots , X_n$ such that the following condition holds: for any $i, j, 1 \leq i < j \leq n$ , there exist $x_i \in X_1, x_j \in X_2$ such that $|x_i - x_j | = 1.$

The colonizers of a spherical planet have decided to build $N$ towns, each having area $1/1000$ of the total area of the planet. They also decided that any two points belonging to different towns will have different latitude and different longitude. What is the maximal value of $N$ ?

Let $S$ be a set of positive integers $n_1, n_2, \cdots, n_6$ and let $n(f)$ denote the number $n_1n_{f(1)} +n_2n_{f(2)} +\cdots+n_6n_{f(6)}$ , where $f$ is a permutation of $\{1, 2, . . . , 6\}$ . Let $$\Omega=\{n(f) | f \text{ is a permutation of } \{1, 2, . . . , 6\} \}$$ Give an example of positive integers $n_1, \cdots, n_6$ such that $\Omega$ contains as many elements as possible and determine the number of elements of $\Omega$ .

Find the number of positive integers $n$ satisfying $\phi(n) | n$ such that $$\sum_{m=1}^{\infty} \left( \left[ \frac nm \right] - \left[\frac{n-1}{m} \right] \right) = 1992$$ What is the largest number among them? As usual, $\phi(n)$ is the number of positive integers less than or equal to $n$ and relatively prime to $n.$

Prove that $\frac{5^{125}-1}{5^{25}-1}$ is a composite number.

Let $n$ be a positive integer. Prove that the number of ways to express $n$ as a sum of distinct positive integers (up to order) and the number of ways to express $n$ as a sum of odd positive integers (up to order) are the same.

Prove that the sequence $5, 12, 19, 26, 33,\cdots$ contains no term of the form $2^n -1.$

Evaluate $$\left \lfloor \ \prod_{n=1}^{1992} \frac{3n+2}{3n+1} \ \right \rfloor$$

Find all the functions $f : \mathbb R^+ \to \mathbb R$ satisfying the identity $$f(x)f(y)=y^{\alpha}f\left(\frac x2 \right) + x^{\beta} f\left(\frac y2 \right) \forall x,y \in \mathbb R^+$$ Where $\alpha,\beta$ are given real numbers.

Given real numbers $x_i \ (i = 1, 2, \cdots, 4k + 2)$ such that $$\sum_{i=1}^{4k +2} (-1)^{i+1} x_ix_{i+1} = 4m ( \ x_1=x_{4k+3} \ )$$ prove that it is possible to choose numbers $x_{k_{1}}, \cdots, x_{k_{6}}$ such that $$\sum_{i=1}^{6} (-1)^{i} k_i k_{i+1} > m ( \ x_{k_{1}} = x_{k_{7}} \ )$$

Let $n$ be an integer $> 1$ . In a circular arrangement of $n$ lamps $L_0, \cdots, L_{n-1}$ , each one of which can be either ON or OFF, we start with the situation that all lamps are ON, and then carry out a sequence of steps, $Step_0, Step_1, \cdots$ . If $L_{j-1}$ ( $j$ is taken mod n) is ON, then $Step_j$ changes the status of $L_j$ (it goes from ON to OFF or from OFF to ON) but does not change the status of any of the other lamps. If $L_{j-1}$ is OFF, then $Step_j$ does not change anything at all. Show that:

(a) There is a positive integer $M(n)$ such that after $M(n)$ steps all lamps are ON again.

(b) If $n$ has the form $2^k$ , then all lamps are ON after $n^2 - 1$ steps.

*(c)*If $n$ has the form $2^k +1$ , then all lamps are ON after $n^2 -n+1$ steps.

Suppose that $n > m \geq 1$ are integers such that the string of digits $143$ occurs somewhere in the decimal representation of the fraction $\frac{m}{n}$ . Prove that $n > 125.$

A directed graph (any two distinct vertices joined by at most one directed line) has the following property: If $x, u,$ and $v$ are three distinct vertices such that $x \to u$ and $x \to v$ , then $u \to w$ and $v \to w$ for some vertex $w$ . Suppose that $x \to u \to y \to\cdots \to z$ is a path of length $n$ , that cannot be extended to the right (no arrow goes away from $z$ ). Prove that every path beginning at $x$ arrives after $n$ steps at $z.$

For positive numbers $a, b, c$ define $A = \frac{(a + b + c)}{3}$ , $G = \sqrt[3]{abc}$ , $H = \frac{3}{(a^{-1} + b^{-1} + c^{-1})}.$ Prove that $$\left( \frac AG \right)^3 \geq \frac 14 + \frac 34 \cdot \frac AH.$$

Let a regular $7$ -gon $A_0A_1A_2A_3A_4A_5A_6$ be inscribed in a circle. Prove that for any two points $P, Q$ on the arc $A_0A_6$ the following equality holds: $$\sum_{i=0}^6 (-1)^{i} PA_i = \sum_{i=0}^6 (-1)^{i} QA_i .$$

There are a board with $2n \cdot 2n \ (= 4n^2)$ squares and $4n^2-1$ cards numbered with different natural numbers. These cards are put one by one on each of the squares. One square is empty. We can move a card to an empty square from one of the adjacent squares (two squares are adjacent if they have a common edge). Is it possible to exchange two cards on two adjacent squares of a column (or a row) in a finite number of movements?

Let $c_1, \cdots, c_n \ (n \geq 2)$ be real numbers such that $0 \leq \sum c_i \leq n$ . Prove that there exist integers $k_1, \cdots , k_n$ such that $\sum k_i=0$ and $1-n \leq c_i + nk_i \leq n$ for every $i = 1, \cdots , n.$

Let $a$ and $b$ be integers. Prove that $\frac{2a^2-1}{b^2+2}$ is not an integer.

For any positive integer $n$ consider all representations $n = a_1 + \cdots+ a_k$ , where $a_1 > a_2 > \cdots > a_k > 0$ are integers such that for all $i \in \{1, 2, \cdots , k - 1\}$ , the number $a_i$ is divisible by $a_{i+1}$ . Find the longest such representation of the number $1992.$

If $A, B, C$ , and $D$ are four distinct points in space, prove that there is a plane $P$ on which the orthogonal projections of $A, B, C$ , and $D$ form a parallelogram (possibly degenerate).

A circle of radius $\rho$ is tangent to the sides $AB$ and $AC$ of the triangle $ABC$, and its center $K$ is at a distance $p$ from $BC$.

(a) Prove that $a(p - \rho) = 2s(r - \rho)$, where $r$ is the inradius and $2s$ the perimeter of $ABC$. (b) Prove that if the circle intersect $BC$ at $D$ and $E$, then $$DE=\frac{4\sqrt{rr_1(\rho-r)(r_1-\rho)}}{r_1-r}$$

where $r_1$ is the exradius corresponding to the vertex $A.$

In a triangle, a symmedian is a line through a vertex that is symmetric to the median with the respect to the internal bisector (all relative to the same vertex). In the triangle $ABC$, the median $m_a$ meets $BC$ at $A'$ and the circumcircle again at $A_1$. The symmedian $s_a$ meets $BC$ at $M$ and the circumcircle again at $A_2$. Given that the line $A_1A_2$ contains the circumcenter $O$ of the triangle, prove that:

(a) $\frac{AA'}{AM} = \frac{b^2+c^2}{2bc} ;$

(b) $1+4b^2c^2 = a^2(b^2+c^2)$

Show that the numbers $\tan \left(\frac{r \pi }{15}\right)$ , where $r$ is a positive integer less than $15$ and relatively prime to $15$ , satisfy $$x^8 - 92x^6 + 134x^4 - 28x^2 + 1 = 0.$$

Let two circles $A$ and $B$ with unequal radii $r$ and $R$ , respectively, be tangent internally at the point $A_0$ . If there exists a sequence of distinct circles $(C_n)$ such that each circle is tangent to both $A$ and $B$ , and each circle $C_{n+1}$ touches circle $C_{n}$ at the point $A_n$ , prove that $$\sum_{n=1}^{\infty} |A_{n+1}A_n| < \frac{4 \pi Rr}{R+r}.$$

Let $P_1(x, y)$ and $P_2(x, y)$ be two relatively prime polynomials with complex coefficients. Let $Q(x, y)$ and $R(x, y)$ be polynomials with complex coefficients and each of degree not exceeding $d$ . Prove that there exist two integers $A_1, A_2$ not simultaneously zero with $|A_i| \leq d + 1 \ (i = 1, 2)$ and such that the polynomial $A_1P_1(x, y) + A_2P_2(x, y)$ is coprime to $Q(x, y)$ and $R(x, y).$

In a school six different courses are taught: mathematics, physics, biology, music, history, geography. The students were required to rank these courses according to their preferences, where equal preferences were allowed. It turned out that:

• (i) mathematics was ranked among the most preferred courses by all students;
• (ii) no student ranked music among the least preferred ones;
• **(iii)**all students preferred history to geography and physics to biology; and
• (iv) no two rankings were the same.

Find the greatest possible value for the number of students in this school.

Let $\{A_n | n = 1, 2, \cdots \}$ be a set of points in the plane such that for each $n$ , the disk with center $A_n$ and radius $2^n$ contains no other point $A_j$ . For any given positive real numbers $a < b$ and $R$ , show that there is a subset $G$ of the plane satisfying:(i) the area of $G$ is greater than or equal to $R$ ;**(ii)**for each point $P$ in $G$ , $a < \sum_{n=1}^{\infty} \frac{1}{|A_nP|} <b.$

Let $S = \{\frac{\pi^n}{1992^m} | m,n \in \mathbb Z \}.$ Show that every real number $x \geq 0$ is an accumulation point of $S.$

A sequence $\{an\}$ of positive integers is defined by $$a_n=\left[ n +\sqrt n + \frac 12 \right] , \forall n \in \mathbb N$$ Determine the positive integers that occur in the sequence.

Given any triangle $ABC$ and any positive integer $n$ , we say that $n$ is a decomposable number for triangle $ABC$ if there exists a decomposition of the triangle $ABC$ into $n$ subtriangles with each subtriangle similar to $\triangle ABC$ . Determine the positive integers that are decomposable numbers for every triangle.

Show that if $994$ integers are chosen from $1, 2,\cdots , 1992$ and one of the chosen integers is less than $64$ , then there exist two among the chosen integers such that one of them is a factor of the other.

Let $F_n$ be the nth Fibonacci number, defined by $F_1 = F_2 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for $n > 2$ . Let $A_0, A_1, A_2,\cdots$ be a sequence of points on a circle of radius $1$ such that the minor arc from $A_{k-1}$ to $A_k$ runs clockwise and such that $$\mu(A_{k-1}A_k)=\frac{4F_{2k+1}}{F_{2k+1}^2+1}$$ for $k \geq 1$ , where $\mu(XY )$ denotes the radian measure of the arc $XY$ in the clockwise direction. What is the limit of the radian measure of arc $A_0A_n$ as $n$ approaches infinity?

Given a graph with $n$ vertices and a positive integer $m$ that is less than $n$ , prove that the graph contains a set of $m+1$ vertices in which the difference between the largest degree of any vertex in the set and the smallest degree of any vertex in the set is at most $m-1.$

Suppose that points $X, Y,Z$ are located on sides $BC, CA$, and $AB$, respectively, of triangle $ABC$ in such a way that triangle $XY Z$ is similar to triangle $ABC$. Prove that the orthocenter of triangle $XY Z$ is the circumcenter of triangle $ABC.$