Navvye/Olympiad-Problems-CSV huggingface dataset:
1976 IMO Shortlist
Let $a_0, a_1, \ldots, a_n, a_{n+1}$ be a sequence of real numbers satisfying the following conditions:



$$a_0 = a_{n+1 }= 0,$$$$ |a_{k-1} - 2a_k + a_{k+1}| \leq 1 \quad (k = 1, 2,\ldots , n).$$

Prove that $|a_k| \leq \frac{k(n+1-k)}{2} \quad (k = 0, 1,\ldots ,n + 1).$
1976 IMO Shortlist
A sequence $(u_{n})$ is defined by $$ u_{0}=2 \quad u_{1}=\frac{5}{2}, u_{n+1}=u_{n}(u_{n-1}^{2}-2)-u_{1} \quad \textnormal{for  } n=1,\ldots $$ Prove that for any positive integer $n$ we have $$ [u_{n}]=2^{\frac{(2^{n}-(-1)^{n})}{3}} $$(where [x] denotes the smallest integer $\leq$ x)$.$
1976 IMO Shortlist
Let $I = (0, 1]$ be the unit interval of the real line. For a given number $a \in (0, 1)$ we define a map $T : I \to I$ by the formula

if

$$ T (x, y) = \{\begin{array}{cc} x + (1 - a),&\mbox{ if } 0< x \leq a,\\ \text{ } \\ x - a, & \mbox{ if } a < x \leq 1.\end{array} $$



Show that for every interval $J \subset I$ there exists an integer $n > 0$ such that $T^n(J) \cap  J \neq \emptyset.$
1976 IMO Shortlist
Let $P$ be a polynomial with real coefficients such that $P(x) > 0$ if $x > 0$. Prove that there exist polynomials $Q$ and $R$ with nonnegative coefficients such that $P(x) = \frac{Q(x)}{R(x)}$ if $x > 0.$
1976 IMO Shortlist
Prove that $5^n$ has a block of $1976$ consecutive $0's$ in its decimal representation.
1976 IMO Shortlist
The polynomial $1976(x+x^2+ \cdots +x^n)$ is decomposed into a sum of polynomials of the form $a_1x + a_2x^2 + \cdots + a_nx^n$, where $a_1, a_2, \ldots , a_n$ are distinct positive integers not greater than $n$. Find all values of $n$ for which such a decomposition is possible.
1975 IMO Shortlist
There are six ports on a lake. Is it possible to organize a series of routes satisfying the following conditions ?

(i)	Every route includes exactly three ports;

(ii)	No two routes contain the same three ports;

(iii)	The series offers exactly two routes to each tourist who desires to visit two different arbitrary ports.
1975 IMO Shortlist
Let $a_1, a_2, \ldots , a_n, \ldots  $ be a sequence of real numbers such that $0 \leq a_n \leq 1$ and $a_n - 2a_{n+1} + a_{n+2} \geq  0$ for $n = 1, 2, 3, \ldots$. Prove that

$$0 \leq (n + 1)(a_n - a_{n+1}) \leq 2 \qquad \text{ for } n = 1, 2, 3, \ldots$$
1975 IMO Shortlist
Let $M$ be the set of all positive integers that do not contain the digit $9$ (base $10$). If $x_1, \ldots , x_n$ are arbitrary but distinct elements in $M$, prove that

$$\sum_{j=1}^n \frac{1}{x_j} < 80 .$$
1975 IMO Shortlist
Prove that from $x + y = 1 \  (x, y \in \mathbb R)$ it follows that

$$x^{m+1} \sum_{j=0}^n \binom{m+j}{j} y^j + y^{n+1} \sum_{i=0}^m \binom{n+i}{i} x^i = 1 \qquad (m, n = 0, 1, 2, \ldots ).$$
1975 IMO Shortlist
Let $f(x)$ be a continuous function defined on the closed interval $0 \leq x \leq 1$. Let $G(f)$ denote the graph of $f(x): G(f) = \{(x, y) \in \mathbb R^2 | 0 \leq$$ x \leq 1, y = f(x) \}$. Let $G_a(f)$ denote the graph of the translated function $f(x - a)$ (translated over a distance $a$), defined by $G_a(f) = \{(x, y) \in \mathbb R^2 | a \leq x \leq a + 1, y = f(x - a) \}$. Is it possible to find for every $a, \  0 < a < 1$, a continuous function $f(x)$, defined on $0 \leq x \leq 1$, such that $f(0) = f(1) = 0$ and $G(f)$ and $G_a(f)$ are disjoint point sets ?
1975 IMO Shortlist
Consider on the first quadrant of the trigonometric circle the arcs $AM_1 = x_1,AM_2 = x_2,AM_3 = x_3, \ldots , AM_v = x_v$ , such that $x_1 < x_2 < x_3 < \cdots < x_v$. Prove that

$$\sum_{i=0}^{v-1} \sin 2x_i - \sum_{i=0}^{v-1} \sin (x_i- x_{i+1}) < \frac{\pi}{2} + \sum_{i=0}^{v-1} \sin (x_i + x_{i+1})$$
1975 IMO Shortlist
Let $A_0,A_1, \ldots , A_n$ be points in a plane such that

(i)	$A_0A_1 \leq \frac{1}{ 2} A_1A_2  \leq  \cdots  \leq  \frac{1}{2^{n-1} } A_{n-1}A_n$ and

(ii) $0 < \measuredangle A_0A_1A_2 < \measuredangle A_1A_2A_3 < \cdots < \measuredangle A_{n-2}A_{n-1}A_n < 180^\circ,$

where all these angles have the same orientation. Prove that the segments $A_kA_{k+1},A_mA_{m+1}$ do not intersect for each $k$ and $n$ such that $0 \leq k \leq m - 2 < n- 2.$
1975 IMO Shortlist
Let $x_0 = 5$ and $x_{n+1} = x_n + \frac{1}{x_n} \ (n = 0, 1, 2, \ldots )$. Prove that

$$45 < x_{1000} < 45. 1.$$
1974 IMO Shortlist
Prove that the squares with sides $\frac{1}{1}, \frac{1}{2}, \frac{1}{3},\ldots$ may be put into the square with side $\frac{3}{2} $ in such a way that no two of them have any interior point in common.
1974 IMO Shortlist
The sum of the squares of five real numbers $a_1, a_2, a_3, a_4, a_5$ equals $1$. Prove that the least of the numbers $(a_i - a_j)^2$, where $i, j = 1, 2, 3, 4,5$ and $i \neq j$, does not exceed $\frac{1}{10}.$
1974 IMO Shortlist
Let $A_r,B_r, C_r$ be points on the circumference of a given circle $S$. From the triangle $A_rB_rC_r$, called $\Delta_r$, the triangle $\Delta_{r+1}$ is obtained by constructing the points $A_{r+1},B_{r+1}, C_{r+1} $on $S$ such that $A_{r+1}A_r$ is parallel to $B_rC_r$, $B_{r+1}B_r$ is parallel to $C_rA_r$, and $C_{r+1}C_r$ is parallel to $A_rB_r$. Each angle of $\Delta_1$ is an integer number of degrees and those integers are not multiples of $45$. Prove that at least two of the triangles $\Delta_1,\Delta_2, \ldots ,\Delta_{15}$  are congruent.
1974 IMO Shortlist
Let $a_i, b_i$ be coprime positive integers for $i = 1, 2, \ldots , k$, and $m$ the least common multiple of $b_1, \ldots , b_k$. Prove that the greatest common divisor of $a_1 \frac{m}{b_1} , \ldots, a_k \frac{m}{b_k}$ equals the greatest common divisor of $a_1, \ldots , a_k.$
1974 IMO Shortlist
Let $x, y, z$ be real numbers each of whose absolute value is different from $\frac{1}{\sqrt 3}$ such that $x + y + z = xyz$. Prove that

$$\frac{3x - x^3}{1-3x^2} + \frac{3y - y^3}{1-3y^2} + \frac{3z -z^3}{1-3z^2} = \frac{3x - x^3}{1-3x^2} \cdot \frac{3y - y^3}{1-3y^2} \cdot \frac{3z - z^3}{1-3z^2}$$
1974 IMO Shortlist
We consider the division of a chess board $8 \times 8$ in p disjoint rectangles which satisfy the conditions:



a) every rectangle is formed from a number of full squares (not partial) from the 64 and the number of white squares is equal to the number of black squares.



b) the numbers $\ a_{1}, \ldots, a_{p}$ of white squares from $p$ rectangles satisfy $a_1, , \ldots, a_p.$ Find the greatest value of $p$ for which there exists such a division and then for that value of $p,$ all the sequences $a_{1}, \ldots, a_{p}$ for which we can have such a division.





Moderator says: see
1974 IMO Shortlist
In a certain language words are formed using an alphabet of three letters. Some words of two or more letters are not allowed, and any two such distinct words are of different lengths. Prove that one can form a word of arbitrary length that does not contain any non-allowed word.
1973 IMO Shortlist
Let a tetrahedron $ABCD$ be inscribed in a sphere $S$. Find the locus of points $P$ inside the sphere $S$ for which the equality

$$\frac{AP}{PA_1}+\frac{BP}{PB_1}+\frac{CP}{PC_1}+\frac{DP}{PD_1}=4$$

holds, where $A_1,B_1, C_1$, and $D_1$ are the intersection points of $S$ with the lines $AP,BP,CP$, and $DP$, respectively.
1973 IMO Shortlist
Given a circle $K$, find the locus of vertices $A$ of parallelograms $ABCD$ with diagonals $AC \leq BD$, such that $BD$ is inside $K$.
1973 IMO Shortlist
Let $P$ be a set of $7$ different prime numbers and $C$ a set of $28$ different composite numbers each of which is a product of two (not necessarily different) numbers from $P$. The set $C$ is divided into $7$ disjoint four-element subsets such that each of the numbers in one set has a common prime divisor with at least two other numbers in that set. How many such partitions of $C$ are there ?
1973 IMO Shortlist
A circle of radius 1 is located in a right-angled trihedron and touches all its faces. Find the locus of centers of such circles.
1973 IMO Shortlist
Given a tetrahedron $ABCD$, let $x = AB \cdot CD$, $y = AC \cdot BD$, and $z = AD \cdot  BC$. Prove that there exists a triangle with edges $x, y, z.$
1973 IMO Shortlist
Prove that there are exactly $\binom{k}{[k/2]}$ arrays $a_1, a_2, \ldots , a_{k+1}$ of nonnegative integers such that $a_1 = 0$ and $|a_i-a_{i+1}| = 1$ for $i = 1, 2, \ldots , k.$
1973 IMO Shortlist
Let $Ox, Oy, Oz$ be three rays, and $G$ a point inside the trihedron $Oxyz$. Consider all planes passing through $G$ and cutting $Ox, Oy, Oz$ at points $A,B,C$, respectively. How is the plane to be placed in order to yield a tetrahedron $OABC$ with minimal perimeter ?
1973 IMO Shortlist
Consider the two square matrices

$$A=\begin{bmatrix} +1 & +1 &+1& +1 &+1 \\+1 &+1 &+1&-1 &-1 \\ +1 &-1&-1 &+1& +1 \\ +1 & -1 & -1 & -1 & +1 \\ +1 &+1&-1 &+1&-1  \end{bmatrix} \quad \text{ and } \quad B=\begin{bmatrix} +1 & +1 &+1& +1 &+1 \\+1 &+1 &+1&-1 &-1 \\ +1 &+1&-1& +1&-1 \\ +1 &-1& -1& +1& +1 \\ +1 & -1& +1&-1 &+1 \end{bmatrix}$$



with entries $+1$ and $-1$. The following operations will be called elementary:



(1) Changing signs of all numbers in one row;



(2) Changing signs of all numbers in one column;



(3) Interchanging two rows (two rows exchange their positions);



(4) Interchanging two columns.



Prove that the matrix $B$ cannot be obtained from the matrix $A$ using these operations.
1973 IMO Shortlist
Find the sphere of maximal radius that can be placed inside every tetrahedron that has all altitudes of length greater than or equal to $1.$
1973 IMO Shortlist
Prove that for all $n \in  \mathbb N$ the following is true:

$$2^n \prod_{k=1}^n \sin \frac{k \pi}{2n+1} = \sqrt{2n+1}$$
1973 IMO Shortlist
Given $a, \theta \in \mathbb R, m \in \mathbb N$, and $P(x) = x^{2m}- 2|a|^mx^m \cos \theta +a^{2m}$, factorize $P(x)$ as a product of $m$ real quadratic polynomials.
1972 IMO Shortlist
We are given $3n$ points $A_1,A_2, \ldots , A_{3n}$ in the plane, no three of them collinear. Prove that one can construct $n$ disjoint triangles with vertices at the points $A_i.$
1972 IMO Shortlist
The least number is $m$ and the greatest number is $M$ among $ a_1 ,a_2 ,\ldots,a_n$ satisfying $ a_1 +a_2 +...+a_n =0$. Prove that

$$  a_1^2 +\cdots +a_n^2 \le-nmM$$
1972 IMO Shortlist
Let $n_1, n_2$ be positive integers. Consider in a plane $E$ two disjoint sets of points $M_1$ and $M_2$ consisting of $2n_1$ and $2n_2$ points, respectively, and such that no three points of the union $M_1  \cup M_2$  are collinear. Prove that there exists a straightline $g$ with the following property: Each of the two half-planes determined by $g$ on $E$ ($g$ not being included in either) contains exactly half of the points of $M_1$ and exactly half of the points of $M_2.$
1972 IMO Shortlist
Prove the following assertion: The four altitudes of a tetrahedron $ABCD$ intersect in a point if and only if

$$AB^2 + CD^2 = BC^2 + AD^2 = CA^2 + BD^2.$$
1972 IMO Shortlist
Show that for any $n \not \equiv 0 \pmod{10}$ there exists a multiple of $n$ not containing the digit $0$ in its decimal expansion.
1972 IMO Shortlist
Consider a sequence of circles $K_1,K_2,K_3,K_4, \ldots$ of radii $r_1, r_2, r_3, r_4, \ldots$ , respectively, situated inside a triangle $ABC$. The circle $K_1$ is tangent to $AB$ and $AC$; $K_2$ is tangent to $K_1$, $BA$, and $BC$; $K_3$ is tangent to $K_2$, $CA$, and $CB$; $K_4$ is tangent to $K_3$, $AB$, and $AC$; etc.

(a) Prove the relation

$$r_1  \cot \frac 12 A+ 2 \sqrt{r_1r_2} + r_2  \cot \frac 12 B = r \left(\cot \frac 12 A + \cot \frac 12 B \right) $$

where $r$ is the radius of the incircle of the triangle $ABC$. Deduce the existence of a $t_1$ such that

$$r_1=r \cot \frac 12 B \cot \frac 12 C \sin^2 t_1$$

(b) Prove that the sequence of circles $K_1,K_2, \ldots $ is periodic.
1971 IMO Shortlist
Consider a sequence of polynomials $P_0(x), P_1(x), P_2(x), \ldots, P_n(x), \ldots$, where $P_0(x) = 2, P_1(x) = x$ and for every $n \geq 1$ the following equality holds:

$$P_{n+1}(x) + P_{n-1}(x) = xP_n(x).$$

Prove that there exist three real numbers $a, b, c$ such that for all $n \geq 1,$

$$(x^2 - 4)[P_n^2(x) - 4] = [aP_{n+1}(x) + bP_n(x) + cP_{n-1}(x)]^2.$$
1971 IMO Shortlist
Prove that for every positive integer $m$ we can find a finite set $S$ of points in the plane, such that given any point $A$ of $S$, there are exactly $m$ points in $S$ at unit distance from $A$.
1971 IMO Shortlist
Knowing that the system

$$x + y + z = 3,$$$$x^3 + y^3 + z^3 = 15,$$$$x^4 + y^4 + z^4 = 35,$$

has a real solution $x, y, z$ for which $x^2 + y^2 + z^2 < 10$, find the value of $x^5 + y^5 + z^5$ for that solution.
1971 IMO Shortlist
We are given two mutually tangent circles in the plane, with radii $r_1, r_2$. A line intersects these circles in four points, determining three segments of equal length. Find this length as a function of $r_1$ and $r_2$ and the condition for the solvability of the problem.
1971 IMO Shortlist
Let $$ E_n=(a_1-a_2)(a_1-a_3)\ldots(a_1-a_n)+(a_2-a_1)(a_2-a_3)\ldots(a_2-a_n)+\ldots+(a_n-a_1)(a_n-a_2)\ldots(a_n-a_{n-1}). $$ Let $S_n$ be the proposition that $E_n\ge0$ for all real $a_i$. Prove that $S_n$ is true for $n=3$ and $5$, but for no other $n>2$.
1971 IMO Shortlist
Let $n \geq 2$ be a natural number. Find a way to assign natural numbers to the vertices of a regular $2n$-gon such that the following conditions are satisfied:



(1) only digits $1$ and $2$ are used;



(2) each number consists of exactly $n$ digits;



(3) different numbers are assigned to different vertices;



(4) the numbers assigned to two neighboring vertices differ at exactly one digit.
1971 IMO Shortlist
All faces of the tetrahedron $ABCD$ are acute-angled. Take a point $X$ in the interior of the segment $AB$, and similarly $Y$ in $BC, Z$ in $CD$ and $T$ in $AD$.



a.) If $\angle DAB+\angle BCD\ne\angle CDA+\angle ABC$, then prove none of the closed paths $XYZTX$ has minimal length;



b.) If $\angle DAB+\angle BCD=\angle CDA+\angle ABC$, then there are infinitely many shortest paths $XYZTX$, each with length $2AC\sin k$, where $2k=\angle BAC+\angle CAD+\angle DAB$.
1971 IMO Shortlist
Determine whether there exist distinct real numbers $a, b, c, t$ for which:



(i) the equation $ax^2 + btx + c = 0$ has two distinct real roots $x_1, x_2,$



(ii) the equation $bx^2 + ctx + a = 0$ has two distinct real roots $x_2, x_3,$



(iii) the equation $cx^2 + atx + b = 0$ has two distinct real roots $x_3, x_1.$
1971 IMO Shortlist
Let $T_k = k - 1$ for $k = 1, 2, 3,4$ and

$$T_{2k-1} = T_{2k-2} + 2^{k-2}, T_{2k} = T_{2k-5} + 2^k \qquad  (k \geq 3).$$

Show that for all $k$,

$$1 + T_{2n-1} = \left[ \frac{12}{7}2^{n-1} \right] \quad \text{and} \quad 1 + T_{2n} = \left[ \frac{17}{7}2^{n-1} \right],$$

where $[x]$ denotes the greatest integer not exceeding $x.$
1971 IMO Shortlist
Prove that we can find an infinite set of positive integers of the from $2^n-3$ (where $n$ is a positive integer) every pair of which are relatively prime.
1971 IMO Shortlist
The matrix

$$A=\begin{pmatrix} a_{11} & \ldots & a_{1n} \\ \vdots & \ldots & \vdots  \\ a_{n1} & \ldots & a_{nn}  \end{pmatrix}$$

satisfies the inequality $\sum_{j=1}^n |a_{j1}x_1 + \cdots+ a_{jn}x_n| \leq M$ for each choice of numbers $x_i$ equal to $\pm 1$. Show that

$$|a_{11} + a_{22} + \cdots+ a_{nn}| \leq M.$$
1971 IMO Shortlist
Two congruent equilateral triangles $ABC$ and $A'B'C'$ in the plane are given. Show that the midpoints of the segments $AA',BB', CC'$ either are collinear or form an equilateral triangle.
1971 IMO Shortlist
Let $ A = (a_{ij})$, where $ i,j = 1,2,\ldots,n$, be a square matrix with all $ a_{ij}$ non-negative integers. For each $ i,j$ such that $ a_{ij} = 0$, the sum of the elements in the $ i$th row and the $ j$th column is at least $ n$. Prove that the sum of all the elements in the matrix is at least $ \frac {n^2}{2}$.
1971 IMO Shortlist
A broken line $A_1A_2 \ldots A_n$ is drawn in a $50 \times 50$ square, so that the distance from any point of the square to the broken line is less than $1$. Prove that its total length is greater than $1248.$
1971 IMO Shortlist
Natural numbers from $1$ to $99$ (not necessarily distinct) are written on $99$ cards. It is given that the sum of the numbers on any subset of cards (including the set of all cards) is not divisible by $100$. Show that all the cards contain the same number.
1971 IMO Shortlist
Let $P_1$ be a convex polyhedron with vertices $A_1,A_2,\ldots,A_9$. Let $P_i$ be the polyhedron obtained from $P_1$ by a translation that moves $A_1$ to $A_i$. Prove that at least two of the polyhedra $P_1,P_2,\ldots,P_9$ have an interior point in common.
1971 IMO Shortlist
Prove the inequality

$$ \frac{a_1+ a_3}{a_1 + a_2} + \frac{a_2 + a_4}{a_2 + a_3} + \frac{a_3 + a_1}{a_3 + a_4} + \frac{a_4 + a_2}{a_4 + a_1} \geq 4, $$

where $a_i > 0, i = 1, 2, 3, 4.$
1992 IMO Longlists
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 $AC + CF = AB + BF.$
1992 IMO Longlists
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 $x + y \leq m+ 1.$
1992 IMO Longlists
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  $\triangle A_1B_1C_1$.
1992 IMO Longlists
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?
1992 IMO Longlists
Let $I,H,O$ be the incenter, centroid, and circumcenter of the nonisosceles triangle $ABC$. Prove that $AI \parallel HO$ if and only if $\angle BAC =120^{\circ}$.
1992 IMO Longlists
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 $\frac 15.$
1992 IMO Longlists
Let $X$ be a bounded, nonempty set of points in the Cartesian plane. Let $f(X)$ be the set of all points that are at a distance of at most $1$ from some point in $X$. Let $f_n(X) = f(f(\cdots(f(X))\cdots))$ ($n$ times). Show that $f_n(X)$ becomes “more circular” as $n$ gets larger.

In other words, if $r_n = \sup\{\text{radii of circles contained in } f_n(X) \}$ and $R_n = \inf \{\text{radii of circles containing } f_n(X)\}$, then show that $R_n/r_n$ gets arbitrarily close to $1$ as $n$ becomes arbitrarily large.



Click to reveal hidden textI'm not sure that I'm posting this in a right forum. If it's in a wrong forum, please mods move it.
1992 IMO Longlists
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.
1992 IMO Longlists
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 $CL,DF; DF,AH; AH,BJ; BJ,CL$ are denoted by $P_1,Q_1,R_1, S_1$, respectively, and the intersection points of the pairs of straight lines $AI,BK; BK,CE;$ $ CE,DG; DG,AI$ are denoted by $P_2,Q_2,R_2, S_2$, respectively. Prove that $P_1Q_1R_1S_1 \cong P_2Q_2R_2S_2.$
1992 IMO Longlists
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 $n.$
1992 IMO Longlists
Given a triangle $ABC$ such that the circumcenter is in the interior of the incircle, prove that the triangle $ABC$ is acute-angled.
1992 IMO Longlists
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 $O_2O_4.$
1992 IMO Longlists
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 $n \neq 1992.$
1992 IMO Longlists
Prove that there exist $78$ lines in the plane such that they have exactly $1992$ points of intersection.
1992 IMO Longlists
Find all triples $(x, y, z)$ of integers such that

$$\frac{1}{x^2}+\frac{2}{y^2}+\frac{3}{z^2} =\frac 23$$
1992 IMO Longlists
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.
1992 IMO Longlists
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

$$A(n) = 3^{n-1} + 2 \cdot 3^{(n/2)-1} \cos \left(\frac{n\pi}{6}\right), \qquad n\in \mathbb N_0.$$
1992 IMO Longlists
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 $1$.
1992 IMO Longlists
Prove that if $x,y,z >1$ and $\frac 1x +\frac 1y +\frac 1z = 2$, then

$$\sqrt{x+y+z} \geq \sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}.$$
1992 IMO Longlists
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.
1992 IMO Longlists
(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 $f(q) = \frac{19}{92}.$
1992 IMO Longlists
(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 $77.$