Navvye/Olympiad-Problems-CSV huggingface dataset:
1993 IMO Shortlist
$a > 0$ and $b$, $c$ are integers such that $ac$ – $b^2$ is a square-free positive integer P. For exampleP could be $3*5$, but not $3^2*5$. Let $f(n)$ be the number of pairs of integers $d, e$ such that $ad^2 + 2bde + ce^2= n$.  Show that$f(n)$ is finite and that $f(n) = f(P^{k}n)$ for every positive integer $k$.



Original Statement:



Let $a,b,c$ be given integers $a > 0,$ $ac-b^2 = P = P_1 \cdots P_n$ where $P_1 \cdots P_n$ are (distinct) prime numbers. Let $M(n)$ denote the number of pairs of integers $(x,y)$ for which $$ ax^2 + 2bxy + cy^2 = n.  $$Prove that $M(n)$ is finite and $M(n) = M(P_k \cdot n)$ for every integer $k \geq 0.$ Note that the "$n$" in $P_N$ and the "$n$" in $M(n)$ do not have to be the same.
1993 IMO Shortlist
Let $c_1, \ldots, c_n \in \mathbb{R}$ with $n \geq 2$ such that $$ 0 \leq \sum^n_{i=1} c_i \leq n. $$ Show that we can find integers $k_1, \ldots, k_n$ such that $$ \sum^n_{i=1} k_i = 0 $$ and $$ 1-n \leq c_i + n \cdot k_i \leq n $$ for every $i = 1, \ldots, n.$



Another formulation:Let $x_1, \ldots, x_n,$ with $n \geq 2$ be real numbers such that $$ |x_1 + \ldots + x_n| \leq n. $$ Show that there exist integers $k_1, \ldots, k_n$ such that $$ |k_1 + \ldots + k_n| = 0. $$ and $$ |x_i + 2 \cdot n \cdot k_i| \leq 2 \cdot n -1 $$ for every $i = 1, \ldots, n.$ In order to prove this, denote $c_i = \frac{1+x_i}{2}$ for $i = 1, \ldots, n,$ etc.
1993 IMO Shortlist
Let $a,b,c,d$ be four non-negative numbers satisfying $$ a+b+c+d=1. $$ Prove the inequality $$ a \cdot b \cdot c + b \cdot c \cdot d + c \cdot d \cdot a + d \cdot a \cdot b \leq \frac{1}{27} + \frac{176}{27} \cdot a \cdot b \cdot c \cdot d. $$
1993 IMO Shortlist
a) Show that the set $ \mathbb{Q}^{ + }$ of all positive rationals can be partitioned into three disjoint subsets. $ A,B,C$ satisfying the following conditions:

$$ BA = B; \& B^2 = C; \& BC = A;

$$

where $ HK$ stands for the set $ \{hk: h \in H, k \in K\}$ for any two subsets $ H, K$ of $ \mathbb{Q}^{ + }$ and $ H^2$ stands for $ HH.$



b) Show that all positive rational cubes are in $ A$ for such a partition of $ \mathbb{Q}^{ + }.$



c) Find such a partition $ \mathbb{Q}^{ + } = A \cup B \cup C$ with the property that for no positive integer $ n \leq 34,$ both $ n$ and $ n + 1$ are in $ A,$ that is,

$$ \text{min} \{n \in \mathbb{N}: n \in A, n + 1 \in A \} > 34.

$$
1993 IMO Shortlist
Let $n,k \in \mathbb{Z}^{+}$ with $k \leq n$ and let $S$ be a set containing $n$ distinct real numbers. Let $T$ be a set of all real numbers of the form $x_1 + x_2 + \ldots + x_k$ where $x_1, x_2, \ldots, x_k$ are distinct elements of $S.$ Prove that $T$ contains at least $k(n-k)+1$ distinct elements.
1993 IMO Shortlist
Let $n \geq 2, n \in \mathbb{N}$ and $A_0 = (a_{01},a_{02}, \ldots, a_{0n})$ be any $n-$tuple of natural numbers, such that $0 \leq a_{0i} \leq i-1,$ for $i = 1, \ldots, n.$

$n-$tuples $A_1= (a_{11},a_{12}, \ldots, a_{1n}), A_2 = (a_{21},a_{22}, \ldots, a_{2n}), \ldots$ are defined by: $a_{i+1,j} = Card \{a_{i,l}| 1 \leq l \leq j-1, a_{i,l} \geq a_{i,j}\},$ for $i \in \mathbb{N}$  and $j = 1, \ldots, n.$ Prove that there exists $k \in \mathbb{N},$  such that $A_{k+2} = A_{k}.$
1993 IMO Shortlist
Let $S_n$ be the number of sequences $(a_1, a_2, \ldots, a_n),$ where $a_i \in \{0,1\},$ in which no six consecutive blocks are equal. Prove that $S_n \rightarrow \infty$ when $n \rightarrow \infty.$
1993 IMO Shortlist
Let $ABC$ be a triangle, and $I$ its incenter. Consider a circle which lies inside the circumcircle of triangle $ABC$ and touches it, and which also touches the sides $CA$ and $BC$ of triangle $ABC$ at the points $D$ and $E$, respectively. Show that the point $I$ is the midpoint of the segment $DE$.
1993 IMO Shortlist
A circle $S$ bisects a circle $S'$ if it cuts $S'$ at opposite ends of a diameter. $S_A$, $S_B$,$S_C$  are circles with distinct centers $A, B, C$ (respectively).

Show that $A, B, C$ are collinear iff there is no unique circle $S$ which bisects each of $S_A$, $S_B$,$S_C$  . Show that if there is more than one circle $S$ which bisects each of $S_A$, $S_B$,$S_C$  , then all such circles pass through two fixed points. Find these points.



Original Statement:



A circle $S$ is said to cut a circle $\Sigma$ diametrically if and only if their common chord is a diameter of $\Sigma.$

Let $S_A, S_B, S_C$ be three circles with distinct centres $A,B,C$ respectively. Prove that $A,B,C$ are collinear if and only if there is no unique circle $S$ which cuts each of $S_A, S_B, S_C$ diametrically. Prove further that if there exists more than one circle $S$ which cuts each $S_A, S_B, S_C$ diametrically, then all such circles $S$ pass through two fixed points. Locate these points in relation to the circles $S_A, S_B, S_C.$
1993 IMO Shortlist
Let triangle $ABC$ be such that its circumradius is $R = 1.$ Let $r$ be the inradius of $ABC$ and let $p$ be the inradius of the orthic triangle $A'B'C'$ of triangle $ABC.$ Prove that $$ p \leq 1 - \frac{1}{3 \cdot (1+r)^2}.  $$



Similar Problem posted by Pascual2005Let $ABC$ be a triangle with circumradius $R$ and inradius $r$. If $p$ is the inradius of the orthic triangle of triangle $ABC$, show that $\frac{p}{R} \leq 1 - \frac{\left(1+\frac{r}{R}\right)^2}{3}$.



Note. The orthic triangle of triangle $ABC$ is defined as the triangle whose vertices are the feet of the altitudes of triangle $ABC$.



SOLUTION 1 by mecrazywong:



$p=2R\cos A\cos B\cos C,1+\frac{r}{R}=1+4\sin A/2\sin B/2\sin C/2=\cos A+\cos B+\cos C$.

Thus, the ineqaulity is equivalent to $6\cos A\cos B\cos C+(\cos A+\cos B+\cos C)^2\le3$. But this is easy since $\cos A+\cos B+\cos C\le3/2,\cos A\cos B\cos C\le1/8$.



SOLUTION 2 by Virgil Nicula:



I note the inradius $r'$ of a orthic triangle.



Must prove the inequality $\frac{r'}{R}\le 1-\frac 13\left( 1+\frac rR\right)^2.$



From the wellknown relations $r'=2R\cos A\cos B\cos C$



and $\cos A\cos B\cos C\le \frac 18$ results $\frac{r'}{R}\le \frac 14.$



But $\frac 14\le 1-\frac 13\left( 1+\frac rR\right)^2\Longleftrightarrow \frac 13\left( 1+\frac rR\right)^2\le \frac 34\Longleftrightarrow$



$\left(1+\frac rR\right)^2\le \left(\frac 32\right)^2\Longleftrightarrow 1+\frac rR\le \frac 32\Longleftrightarrow \frac rR\le \frac 12\Longleftrightarrow 2r\le R$ (true).



Therefore, $\frac{r'}{R}\le \frac 14\le 1-\frac 13\left( 1+\frac rR\right)^2\Longrightarrow \frac{r'}{R}\le 1-\frac 13\left( 1+\frac rR\right)^2.$



SOLUTION 3 by darij grinberg:



I know this is not quite an ML reference, but the problem was discussed in Hyacinthos messages #6951, #6978, #6981, #6982, #6985, #6986 (particularly the last message).
1993 IMO Shortlist
Given a triangle $ABC$,  let $D$ and $E$ be points on the side $BC$ such that $\angle BAD = \angle CAE$. If $M$ and $N$ are, respectively, the points of tangency of the incircles of the triangles $ABD$ and $ACE$ with the line $BC$, then show that

$$\frac{1}{MB}+\frac{1}{MD}= \frac{1}{NC}+\frac{1}{NE}. $$
1993 IMO Shortlist
The vertices $D,E,F$ of an equilateral triangle lie on the sides $BC,CA,AB$ respectively of a triangle $ABC.$ If $a,b,c$ are the respective lengths of these sides, and $S$ the area of $ABC,$ prove that



$$ DE \geq \frac{2 \cdot \sqrt{2} \cdot S}{\sqrt{a^2 + b^2 + c^2 + 4 \cdot \sqrt{3} \cdot S}}. $$
1993 IMO Shortlist
A natural number $n$ is said to have the property $P,$ if,  for all $a, n^2$ divides $a^n - 1$ whenever $n$ divides $a^n - 1.$



a.) Show that every prime number $n$ has property $P.$



b.) Show that there are infinitely many composite numbers $n$ that possess property $P.$
1993 IMO Shortlist
Let $a,b,n$ be positive integers, $b > 1$ and $b^n-1\mid a.$ Show that the representation of the number $a$ in the base $b$ contains at least $n$ digits different from zero.
1993 IMO Shortlist
Show that for any finite set $S$ of distinct positive integers, we can find a set $T$ ⊇ $S$ such that every member of $T$ divides the sum of all the members of $T$.



Original Statement:



A finite set of (distinct) positive integers is called a DS-set if each of the integers divides the sum of them all. Prove that every finite set of positive integers is a subset of some DS-set.
1993 IMO Shortlist
Let $S$ be the set of all pairs $(m,n)$ of relatively prime positive integers $m,n$ with $n$ even and $m < n.$ For $s = (m,n) \in S$ write $n = 2^k \cdot n_o$ where $k, n_0$ are positive integers with $n_0$ odd and define $$ f(s) = (n_0, m + n - n_0).  $$ Prove that $f$ is a function from $S$ to $S$ and that for each $s = (m,n) \in S,$ there exists a positive integer $t \leq \frac{m+n+1}{4}$ such that $$ f^t(s) = s,  $$ where $$ f^t(s) = \underbrace{ (f \circ f \circ \cdots \circ f) }_{t \text{ times}}(s).  $$



If $m+n$ is a prime number which does not divide $2^k - 1$ for $k = 1,2, \ldots, m+n-2,$ prove that the smallest value $t$ which satisfies the above conditions is $\left [\frac{m+n+1}{4} \right ]$ where $\left[ x \right]$ denotes the greatest integer $\leq x.$
1992 IMO Shortlist
Prove that for any positive integer $ m$ there exist an infinite number of pairs of integers $ (x, y)$ such that



(i) $ x$ and $ y$ are relatively prime;

(ii) $ y$ divides $ x^2 + m$;

(iii) $ x$ divides $ y^2 + m.$

(iv) $ x + y \leq m + 1-$ (optional condition)
1992 IMO Shortlist
Let $ \mathbb{R}^+$ be the set of all non-negative real numbers. Given two positive real numbers $ a$ and $ b,$ suppose that a mapping $ f: \mathbb{R}^+ \mapsto \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 Shortlist
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.
1992 IMO Shortlist
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.$
1992 IMO Shortlist
Two circles $ \Omega_{1}$ and $ \Omega_{2}$ are externally tangent to each other at a point $ I$, and both of these circles are tangent to a third circle $ \Omega$ which encloses the two circles $ \Omega_{1}$ and $ \Omega_{2}$.

The common tangent to the two circles $ \Omega_{1}$ and $ \Omega_{2}$ at the point $ I$ meets the circle $ \Omega$ at a point $ A$. One common tangent to the circles $ \Omega_{1}$ and $ \Omega_{2}$ which doesn't pass through $ I$ meets the circle $ \Omega$ at the points $ B$ and $ C$ such that the points $ A$ and $ I$ lie on the same side of the line $ BC$.

Prove that the point $ I$ is the incenter of triangle $ ABC$.



Alternative formulation. Two circles touch externally at a point $ I$. The two circles lie inside a large circle and both touch it. The chord $ BC$ of the large circle touches both smaller circles (not at $ I$). The common tangent to the two smaller circles at the point $ I$ meets the large circle at a point $ A$, where the points $ A$ and $ I$ are on the same side of the chord $ BC$. Show that the point $ I$ is the incenter of triangle $ ABC$.
1992 IMO Shortlist
Show that in the plane there exists a convex polygon of 1992 sides satisfying the following conditions:



(i) its side lengths are $ 1, 2, 3, \ldots, 1992$ in some order;

(ii) the polygon is circumscribable about a circle.



Alternative formulation: Does there exist a 1992-gon with side lengths $ 1, 2, 3, \ldots, 1992$ circumscribed about a circle? Answer the same question for a 1990-gon.
1992 IMO Shortlist
Let $ f(x)$ be a polynomial with rational coefficients and $ \alpha$ be a real number such that $$ \alpha^3 - \alpha = [f(\alpha)]^3 - f(\alpha) = 33^{1992}.$$ Prove that for each $ n \geq 1,$ $$ \left [ f^{n}(\alpha) \right]^3 - f^{n}(\alpha) = 33^{1992},$$ where $ f^{n}(x) = f(f(\cdots f(x))),$ and $ n$ is a positive integer.
1992 IMO Shortlist
In a triangle $ ABC,$ let $ D$ and $ E$ be the intersections of the bisectors of $ \angle ABC$ and $ \angle ACB$ with the sides $ AC,AB,$ respectively. Determine the angles $ \angle A,\angle B, \angle C$ if $ \angle BDE = 24 ^{\circ},$ $ \angle CED = 18 ^{\circ}.$
1992 IMO Shortlist
Let $ f, g$ and $ a$ be polynomials with real coefficients, $ f$ and $ g$ in one variable and $ a$ in two variables. Suppose



$$ f(x) - f(y) = a(x, y)(g(x) - g(y)) \forall x,y \in \mathbb{R}$$



Prove that there exists a polynomial $ h$ with $ f(x) = h(g(x)) \text{ } \forall x \in \mathbb{R}.$
1992 IMO Shortlist
For any positive integer $ x$ define $ g(x)$ as greatest odd divisor of $ x,$ and



$$ f(x) = \begin{cases} \frac {x}{2} + \frac {x}{g(x)} & \text{if \ \(x\) is even}, \\
 2^{\frac {x + 1}{2}} & \text{if \ \(x\) is odd}. \end{cases}

$$



Construct the sequence $ x_1 = 1, x_{n + 1} = f(x_n).$ Show that the number 1992 appears in this sequence, determine  the least $ n$ such that $ x_n = 1992,$ and determine whether $ n$ is unique.
1992 IMO Shortlist
Does there exist a set $ M$ with the following properties?



(i) The set $ M$ consists of 1992 natural numbers.

(ii) Every element in $ M$ and the sum of any number of elements have the form $ m^k$ $ (m, k \in \mathbb{N}, k \geq 2).$
1992 IMO Shortlist
Prove that $\frac{5^{125}-1}{5^{25}-1}$ is a composite number.
1992 IMO Shortlist
Let $ \alpha(n)$ be the number of digits equal to one in the binary representation of a positive integer $ n.$ Prove that:



(a) the inequality $ \alpha(n) (n^2 ) \leq \frac{1}{2} \alpha(n)(\alpha(n) + 1)$ holds;

(b) the above inequality is an equality for infinitely many positive integers, and

(c) there exists a sequence $ (n_i )^{\infty}_1$ such that $ \frac{\alpha ( n^2_i )}{\alpha (n_i }$

goes to zero as $ i$ goes to $ \infty.$





Alternative problem: Prove that there exists a sequence a sequence $ (n_i )^{\infty}_1$ such that $ \frac{\alpha ( n^2_i )}{\alpha (n_i )}$



(d) $ \infty;$

(e) an arbitrary real number $ \gamma \in (0,1)$;

(f) an arbitrary real number $ \gamma \geq 0$;



as $ i$ goes to $ \infty.$
1992 IMO Shortlist
Let $ \lfloor x \rfloor$ denote the greatest integer less than or equal to $ x.$ Pick any $ x_1$ in $ [0, 1)$ and define the sequence $ x_1, x_2, x_3, \ldots$ by $ x_{n+1} = 0$ if $ x_n = 0$ and $ x_{n+1} = \frac{1}{x_n} - \left \lfloor \frac{1}{x_n} \right \rfloor$ otherwise. Prove that



$$ x_1 + x_2 + \ldots + x_n < \frac{F_1}{F_2} + \frac{F_2}{F_3} + \ldots + \frac{F_n}{F_{n+1}},$$



where $ F_1 = F_2 = 1$ and $ F_{n+2} = F_{n+1} + F_n$ for $ n \geq 1.$
1992 IMO Shortlist
Let $ f(x) = x^8 + 4x^6 + 2x^4 + 28x^2 + 1.$ Let $ p > 3$ be a prime and suppose there exists an integer $ z$ such that $ p$ divides $ f(z).$ Prove that there exist integers $ z_1, z_2, \ldots, z_8$ such that if $$ g(x) = (x - z_1)(x - z_2) \cdot \ldots \cdot (x - z_8),$$ then all coefficients of $ f(x) - g(x)$ are divisible by $ p.$
1991 IMO Shortlist
Given a point $ P$ inside a triangle $ \triangle ABC$. Let $ D$, $ E$, $ F$ be the orthogonal projections of the point $ P$ on the sides $ BC$, $ CA$, $ AB$, respectively. Let the orthogonal projections of the point $ A$ on the lines $ BP$ and $ CP$ be $ M$ and $ N$, respectively. Prove that the lines $ ME$, $ NF$, $ BC$ are concurrent.





Original formulation:



Let $ ABC$ be any triangle and $ P$ any point in its interior. Let $ P_1, P_2$ be the feet of the perpendiculars from $ P$ to the two sides $ AC$ and $ BC.$ Draw $ AP$ and $ BP,$ and from $ C$ drop perpendiculars to $ AP$ and $ BP.$ Let $ Q_1$ and $ Q_2$ be the feet of these perpendiculars. Prove that the lines $ Q_1P_2,Q_2P_1,$ and $ AB$ are concurrent.
1991 IMO Shortlist
$ ABC$ is an acute-angled triangle. $ M$ is the midpoint of $ BC$ and $ P$ is the point on $ AM$ such that $ MB = MP$. $ H$ is the foot of the perpendicular from $ P$ to $ BC$. The lines through $ H$ perpendicular to $ PB$, $ PC$ meet $ AB, AC$ respectively at $ Q, R$. Show that $ BC$ is tangent to the circle through $ Q, H, R$ at $ H$.



Original Formulation: 



For an acute triangle $ ABC, M$ is the midpoint of the segment $ BC, P$ is a point on the segment $ AM$ such that $ PM = BM, H$ is the foot of the perpendicular line from $ P$ to $ BC, Q$ is the point of intersection of segment $ AB$ and the line passing through $ H$ that is perpendicular to $ PB,$ and finally, $ R$ is the point of intersection of the segment $ AC$ and the line passing through $ H$ that is perpendicular to $ PC.$ Show that the circumcircle of $ QHR$ is tangent to the side $ BC$ at point $ H.$
1991 IMO Shortlist
Let $ S$ be any point on the circumscribed circle of $ PQR.$ Then the feet of the perpendiculars from S to the three sides of the triangle lie on the same straight line. Denote this line by $ l(S, PQR).$ Suppose that the hexagon $ ABCDEF$ is inscribed in a circle. Show that the four lines $ l(A,BDF),$ $ l(B,ACE),$ $ l(D,ABF),$ and $ l(E,ABC)$ intersect at one point if and only if $ CDEF$ is a rectangle.
1991 IMO Shortlist
In the triangle $ ABC,$ with $ \angle A = 60 ^{\circ},$ a parallel $ IF$ to $ AC$ is drawn through the incenter $ I$ of the triangle, where $ F$ lies on the side $ AB.$ The point $ P$ on the side $ BC$ is such that $ 3BP = BC.$ Show that $ \angle BFP = \frac{\angle B}{2}.$
1991 IMO Shortlist
$ ABCD$ is a terahedron: $ AD+BD=AC+BC,$ $ BD+CD=BA+CA,$ $ CD+AD=CB+AB,$ $ M,N,P$ are the mid points of $ BC,CA,AB.$ $ OA=OB=OC=OD.$ Prove that  $ \angle MOP = \angle NOP =\angle NOM.$
1991 IMO Shortlist
$ S$ be a set of $ n$ points in the plane. No three points of $ S$ are collinear. Prove that there exists a set $ P$ containing $ 2n - 5$ points satisfying the following condition: In the interior of every triangle whose three vertices are elements of $ S$ lies a point that is an element of $ P.$
1991 IMO Shortlist
In the plane we are given a set $ E$ of 1991 points, and certain pairs of these points are joined with a path. We suppose that for every point of $ E,$ there exist at least 1593 other points of $ E$ to which it is joined by a path. Show that there exist six points of $ E$ every pair of which are joined by a path.



Alternative version: Is it possible to find a set $ E$ of 1991 points in the plane and paths joining certain pairs of the points in $ E$ such that every point of $ E$ is joined with a path to at least 1592 other points of $ E,$ and in every subset of six points of $ E$ there exist at least two points that are not joined?
1991 IMO Shortlist
Prove that $ \sum_{k = 0}^{995} \frac {( - 1)^k}{1991 - k} {1991 - k \choose k} = \frac {1}{1991}$
1991 IMO Shortlist
Given any integer $ n \geq 2,$ assume that the integers $ a_1, a_2, \ldots, a_n$ are not divisible by $ n$ and, moreover, that $ n$ does not divide $ \sum^n_{i=1} a_i.$ Prove that there exist at least $ n$ different sequences $ (e_1, e_2, \ldots, e_n)$ consisting of zeros or ones such $ \sum^n_{i=1} e_i \cdot a_i$ is divisible by $ n.$
1991 IMO Shortlist
Let $ a, b, c$ be integers and $ p$ an odd prime number. Prove that if $ f(x) = ax^2 + bx + c$ is a perfect square for $ 2p - 1$ consecutive integer values of $ x,$ then $ p$ divides $ b^2 - 4ac.$
1991 IMO Shortlist
Let $ a_n$ be the last nonzero digit in the decimal representation of the number $ n!.$ Does the sequence $ a_1, a_2, \ldots, a_n, \ldots$ become periodic after a finite number of terms?
1991 IMO Shortlist
Find all positive integer solutions $ x, y, z$ of the equation $ 3^x + 4^y = 5^z.$
1991 IMO Shortlist
Find the highest degree $ k$ of $ 1991$ for which $ 1991^k$ divides the number $$ 1990^{1991^{1992}} + 1992^{1991^{1990}}.$$
1991 IMO Shortlist
Let $ \alpha$ be a rational number with $ 0 < \alpha < 1$ and $ \cos (3 \pi \alpha) + 2\cos(2 \pi \alpha) = 0$. Prove that $ \alpha = \frac {2}{3}$.
1991 IMO Shortlist
Let $ \alpha$ be the positive root of the equation $ x^{2} = 1991x + 1$. For natural numbers $ m$ and $ n$ define

$$ m*n = mn + \lfloor\alpha m \rfloor \lfloor \alpha n\rfloor.

$$

Prove that for all natural numbers $ p$, $ q$, and $ r$,

$$ (p*q)*r = p*(q*r).

$$
1991 IMO Shortlist
Let $ f(x)$ be a monic polynomial of degree $ 1991$ with integer coefficients. Define $ g(x) = f^2(x) - 9.$ Show that the number of distinct integer solutions of $ g(x) = 0$ cannot exceed $ 1995.$
1991 IMO Shortlist
Real constants $ a, b, c$ are such that there is exactly one square all of whose vertices lie on the cubic curve $ y = x^3 + ax^2 + bx + c.$ Prove that the square has sides of length $ \sqrt[4]{72}.$
1991 IMO Shortlist
Let $ f$ and $ g$ be two integer-valued functions defined on the set of all integers such that



(a) $ f(m + f(f(n))) = -f(f(m+ 1) - n$ for all integers $ m$ and $ n;$

(b) $ g$ is a polynomial function with integer coefficients and g(n) = $ g(f(n))$ $ \forall n \in \mathbb{Z}.$
1991 IMO Shortlist
An odd integer $ n \ge 3$ is said to be nice if and only if there is at least one permutation $ a_{1}, \cdots, a_{n}$ of $ 1, \cdots, n$ such that the $ n$ sums $ a_{1} - a_{2} + a_{3} - \cdots - a_{n - 1} + a_{n}$, $ a_{2} - a_{3} + a_{3} - \cdots - a_{n} + a_{1}$, $ a_{3} - a_{4} + a_{5} - \cdots - a_{1} + a_{2}$, $ \cdots$, $ a_{n} - a_{1} + a_{2} - \cdots - a_{n - 2} + a_{n - 1}$ are all positive. Determine the set of all `nice' integers.
1991 IMO Shortlist
Suppose that $ n \geq 2$ and $ x_1, x_2, \ldots, x_n$ are real numbers between 0 and 1 (inclusive). Prove that for some index $ i$ between $ 1$ and $ n - 1$ the

inequality



$$ x_i (1 - x_{i+1}) \geq \frac{1}{4} x_1 (1 - x_{n})$$
1991 IMO Shortlist
Let $ n \geq 2, n \in \mathbb{N}$ and let $ p, a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n \in \mathbb{R}$ satisfying $ \frac{1}{2} \leq p \leq 1,$ $ 0 \leq a_i,$ $ 0 \leq b_i \leq p,$ $ i = 1, \ldots, n,$ and $$ \sum^n_{i=1} a_i = \sum^n_{i=1} b_i.$$ Prove the inequality: $$ \sum^n_{i=1} b_i \prod^n_{j = 1, j \neq i} a_j \leq \frac{p}{(n-1)^{n-1}}.$$
1991 IMO Shortlist
Determine the maximum value of the sum

$$ \sum_{i < j} x_ix_j (x_i + x_j)

$$

over all $ n -$tuples $ (x_1, \ldots, x_n),$ satisfying $ x_i \geq 0$ and $ \sum^n_{i = 1} x_i = 1.$
1991 IMO Shortlist
We call a set $ S$ on the real line $ \mathbb{R}$ superinvariant if for any stretching $ A$ of the set by the transformation taking $ x$ to $ A(x) = x_0 + a(x - x_0), a > 0$ there exists a translation $ B,$ $ B(x) = x+b,$ such that the images of $ S$ under $ A$ and $ B$ agree; i.e., for any $ x \in S$ there is a $ y \in S$ such that $ A(x) = B(y)$ and for any $ t \in S$ there is a $ u \in S$ such that $ B(t) = A(u).$ Determine all superinvariant sets.