Navvye/Olympiad-Problems-CSV huggingface dataset:
1983 IMO Longlists
In a plane are given n points $P_i \ (i = 1, 2, \ldots , n)$ and two angles $\alpha$ and $\beta$. Over each of the segments $P_iP_{i+1}  \ (P_{n+1} = P_1)$ a point $Q_i$ is constructed such that for all $i$:

(i) upon moving from $P_i$ to $P_{i+1}, Q_i$ is seen on the same side of $P_iP_{i+1}$,

(ii) $\angle P_{i+1}P_iQ_i = \alpha,$

(iii) $\angle P_iP_{i+1}Q_i = \beta.$

Furthermore, let $g$ be a line in the same plane with the property that all the points $P_i,Q_i$ lie on the same side of $g$. Prove that

$$\sum_{i=1}^n d(P_i, g)= \sum_{i=1}^n d(Q_i, g).$$

where $d(M,g)$ denotes the distance from point $M$ to line $g.$
1983 IMO Longlists
The set $X$ has $1983$ members. There exists a family of subsets $\{S_1, S_2, \ldots , S_k \}$ such that:

(i) the union of any three of these subsets is the entire set $X$, while

(ii) the union of any two of them contains at most $1979$ members. What is the largest possible value of $k ?$
1983 IMO Longlists
The points $A_1,A_2, \ldots , A_{1983}$  are set on the circumference of a circle and each is given one of the values $\pm 1$. Show that if the number of points with the value $+1$ is greater than $1789$, then at least $1207$ of the points will have the property that the partial sums that can be formed by taking the numbers from them to any other point, in either direction, are strictly positive.
1983 IMO Longlists
Let $\{u_n \}$ be the sequence defined by its first two terms $u_0, u_1$  and the recursion formula

$$u_{n+2 }= u_n - u_{n+1}.$$

(a) Show that $u_n$ can be written in the form $u_n = \alpha a^n + \beta b^n$, where $a, b, \alpha, \beta$ are constants independent of $n$ that have to be determined.



(b) If $S_n = u_0 + u_1 + \cdots + u_n$, prove that $S_n + u_{n-1}$ is a constant independent of $n.$ Determine this constant.
1983 IMO Longlists
If $\alpha $ is the real root of the equation

$$E(x) = x^3 - 5x -50 = 0$$

such that $x_{n+1} = (5x_n + 50)^{1/3}$ and $x_1 = 5$, where $n$ is a positive integer, prove that:



(a) $x_{n+1}^3 - \alpha^3 = 5(x_n - \alpha)$



(b) $\alpha  < x_{n+1} < x_n.$
1983 IMO Longlists
Four faces of tetrahedron $ABCD$ are congruent triangles whose angles form an arithmetic progression. If the lengths of the sides of the triangles are $a < b < c$, determine the radius of the sphere circumscribed about the tetrahedron as a function on $a, b$, and $c$. What is the ratio $c/a$ if $R = a \ ?$
1983 IMO Longlists
Consider the square $ABCD$ in which a segment is drawn between each vertex and the midpoints of both opposite sides. Find the ratio of the area of the octagon determined by these segments and the area of the square $ABCD.$
1983 IMO Longlists
Given a square $ABCD$, let $P, Q, R$, and $S$ be four variable points on the sides $AB, BC, CD$, and $DA$, respectively. Determine the positions of the points $P, Q, R$, and $S$ for which the quadrilateral $PQRS$ is a parallelogram, a rectangle, a square, or a trapezoid.
1983 IMO Longlists
We are given twelve coins, one of which is a fake with a different mass from the other eleven. Determine that coin with three weighings and whether it is heavier or lighter than the others.
1983 IMO Longlists
Let two glasses, numbered $1$ and $2$, contain an equal quantity of liquid, milk in glass $1$ and coffee in glass $2$. One does the following: Take one spoon of mixture from glass $1$ and pour it into glass $2$, and then take the same spoon of the new mixture from glass $2$ and pour it back into the first glass. What happens after this operation is repeated $n$ times, and what as $n$ tends to infinity?
1983 IMO Longlists
Let $f$ be a real-valued function defined on $I = (0,+\infty)$ and having no zeros on $I$. Suppose that

$$\lim_{x \to +\infty} \frac{f'(x)}{f(x)}=+\infty.$$For the sequence $u_n = \ln \left| \frac{f(n+1)}{f(n)} \right|$, prove that $u_n \to +\infty$  as $n \to  +\infty.$
1983 IMO Longlists
In a plane, three pairwise intersecting circles $C_1, C_2, C_3$ with centers $M_1,M_2,M_3$ are given. For $i = 1, 2, 3$, let $A_i$ be one of the points of intersection of $C_j$ and $C_k \ (\{i, j, k \} =  \{1, 2, 3 \})$. Prove that if $ \angle M_3A_1M_2 = \angle M_1A_2M_3 = \angle M_2A_3M_1 = \frac{\pi}{3}$(directed angles), then $M_1A_1, M_2A_2$, and $M_3A_3$ are concurrent.
1983 IMO Longlists
Prove that in any parallelepiped the sum of the lengths of the edges is less than or equal to twice the sum of the lengths of the four diagonals.
1983 IMO Longlists
Given positive integers $k,m, n$ with $km \leq n$ and non-negative real numbers $x_1, \ldots , x_k$, prove that

$$n \left(  \prod_{i=1}^k x_i^m -1 \right) \leq m \sum_{i=1}^k (x_i^n-1).$$
1983 IMO Longlists
Let $a \in \mathbb R$ and let $z_1, z_2, \ldots, z_n$ be complex numbers of modulus $1$ satisfying the relation

$$\sum_{k=1}^n z_k^3=4(a+(a-n)i)- 3 \sum_{k=1}^n \overline{z_k}$$

Prove that $a \in \{0, 1,\ldots, n \}$ and $z_k \in \{1, i \}$ for all $k.$
1983 IMO Longlists
For every $a \in \mathbb N$ denote by $M(a)$ the number of elements of the set

$$ \{ b \in \mathbb N | a + b \text{  is a divisor of } ab \}.$$

Find $\max_{a\leq 1983} M(a).$
1983 IMO Longlists
Consider the expansion

$$(1 + x + x^2 + x^3 + x^4)^{496} = a_0 + a_1x + \cdots + a_{1984}x^{1984}.$$

(a) Determine the greatest common divisor of the coefficients $a_3, a_8, a_{13}, \ldots , a_{1983}.$



(b) Prove that $10^{340 }< a_{992} < 10^{347}.$
1983 IMO Longlists
In the system of base $n^2 + 1$ find a number $N$ with $n$ different digits such that:



(i) $N$ is a multiple of $n$. Let $N = nN'.$



(ii) The number $N$ and $N'$ have the same number $n$ of different digits in base $n^2 + 1$, none of them being zero.



(iii) If $s(C)$ denotes the number in base $n^2 + 1$ obtained by applying the permutation $s$ to the $n$ digits of the number $C$, then for each permutation $s, s(N) = ns(N').$
1983 IMO Longlists
Solve the equation

$$\tan^2(2x) + 2 \tan(2x) \cdot  \tan(3x) -1 = 0.$$
1983 IMO Longlists
Let $a$ and $b$ be integers. Is it possible to find integers $p$ and $q$ such that the integers $p+na$ and $q +nb$ have no common prime factor no matter how the integer $n$ is chosen ?
1983 IMO Longlists
$A$ circle $\gamma$ is drawn and let $AB$ be a diameter. The point $C$ on $\gamma$ is the midpoint of the line segment $BD$. The line segments $AC$ and $DO$, where $O$ is the center of $\gamma$, intersect at $P$. Prove that there is a point $E$ on $AB$ such that $P$ is on the circle with diameter $AE.$
1983 IMO Longlists
The sum of all the face angles about all of the vertices except one of a given polyhedron is $5160$. Find the sum of all of the face angles of the polyhedron.
1983 IMO Longlists
Let $ABCD$ be a convex quadrilateral whose diagonals $AC$ and $BD$ intersect in a point $P$. Prove that

$$\frac{AP}{PC}=\frac{\cot \angle BAC + \cot \angle DAC}{\cot \angle BCA + \cot \angle DCA}$$
1983 IMO Longlists
The altitude from a vertex of a given tetrahedron intersects the opposite face in its orthocenter. Prove that all four altitudes of the tetrahedron are concurrent.
1983 IMO Longlists
Three of the roots of the equation $x^4 -px^3 +qx^2 -rx+s = 0$ are $\tan A, \tan B$, and $\tan C$, where $A, B$, and $C$ are angles of a triangle. Determine the fourth root as a function only of $p, q, r$, and $s.$
1983 IMO Longlists
Prove that for all $x_1, x_2,\ldots , x_n \in \mathbb R$ the following inequality holds:

$$\sum_{n \geq i >j \geq 1} \cos^2(x_i - x_j ) \geq \frac{n(n-2)}{4}$$
1983 IMO Longlists
Let $ABC$ be a nonequilateral triangle. Prove that there exist two points $P$ and $Q$ in the plane of the triangle, one in the interior and one in the exterior of the circumcircle of $ABC$, such that the orthogonal projections of any of these two points on the sides of the triangle are vertices of an equilateral triangle.
1983 IMO Longlists
In a plane we are given two distinct points $A,B$ and two lines $a, b$ passing through $B$ and $A$ respectively $(a  \ni  B, b  \ni A)$ such that the line $AB$ is equally inclined to a and b. Find the locus of points $M$ in the plane such that the product of distances from $M$ to $A$ and a equals the product of distances from $M$ to $B$ and $b$ (i.e., $MA \cdot  MA' = MB \cdot MB'$, where $A'$ and $B'$ are the feet of the perpendiculars from $M$ to $a$ and $b$ respectively).
1983 IMO Longlists
Find the sum of the fiftieth powers of all sides and diagonals of a regular $100$-gon inscribed in a circle of radius $R.$
1982 IMO Longlists
(a) Prove that $\frac{1}{n+1} \cdot \binom{2n}{n}$ is an integer for $n \geq 0.$



(b) Given a positive integer $k$, determine the smallest integer $C_k$ with the property that $\frac{C_k}{n+k+1} \cdot \binom{2n}{n}$ is an integer for all $n \geq k.$
1982 IMO Longlists
Given $n$ points $X_1,X_2,\ldots, X_n$ in the interval $0 \leq X_i \leq 1, i = 1, 2,\ldots, n$, show that there is a point $y, 0 \leq y \leq 1$, such that

$$\frac{1}{n} \sum_{i=1}^{n} | y - X_i | = \frac 12.$$
1982 IMO Longlists
Among all triangles with a given perimeter, find the one with the maximal radius of its incircle.
1982 IMO Longlists
On the three distinct lines $a, b$, and $c$ three points $A, B$, and $C$ are given, respectively. Construct three collinear points $X, Y,Z$ on lines $a, b, c$, respectively, such that $\frac{BY}{AX} = 2$ and $ \frac{CZ}{AX} = 3$.
1982 IMO Longlists
Find all solutions $(x, y) \in \mathbb Z^2$ of the equation

$$x^3 - y^3 = 2xy + 8.$$
1982 IMO Longlists
Given any two real numbers $\alpha$ and $\beta , 0 \leq \alpha < \beta \leq 1$, prove that there exists a natural number $m$ such that

$$\alpha < \frac{\phi(m)}{m} < \beta.$$
1982 IMO Longlists
Let $r_1, \ldots , r_n$ be the radii of $n$ spheres. Call $S_1, S_2, \ldots , S_n$ the areas of the set of points of each sphere from which one cannot see any point of any other sphere. Prove that

$$\frac{S_1}{r_1^2} + \frac{S_2}{r_2^2}+\cdots+\frac{S_n}{r_n^2} = 4 \pi.$$
1982 IMO Longlists
A rectangular pool table has a hole at each of three of its corners. The lengths of sides of the table are the real numbers $a$ and $b$. A billiard ball is shot from the fourth corner along its angle bisector. The ball falls in one of the holes. What should the relation between $a$ and $b$ be for this to happen?
1982 IMO Longlists
Let there be $3399$ numbers arbitrarily chosen among the first $6798$ integers $1, 2, \ldots , 6798$ in such a way that none of them divides another. Prove that there are exactly $1982$ numbers in $\{1, 2, \ldots, 6798\}$ that must end up being chosen.
1982 IMO Longlists
A regular $n$-gonal truncated pyramid is circumscribed around a sphere. Denote the areas of the base and the lateral surfaces of the pyramid by $S_1, S_2$, and $S$, respectively. Let $\sigma$ be the area of the polygon whose vertices are the tangential points of the sphere and the lateral faces of the pyramid. Prove that

$$\sigma S = 4S_1S_2 \cos^2 \frac{\pi}{n}.$$
1982 IMO Longlists
Show that the set $S$ of natural numbers $n$ for which $\frac{3}{n}$ cannot be written as the sum of two reciprocals of natural numbers ($S =\left\{n |\frac{3}{n} \neq \frac{1}{p} + \frac{1}{q} \text{  for any  } p, q \in \mathbb N \right\}$) is not the union of finitely many arithmetic progressions.
1982 IMO Longlists
You are given an algebraic system admitting addition and multiplication for which all the laws of ordinary arithmetic are valid except commutativity of multiplication. Show that

$$(a + ab^{-1} a)^{-1}+ (a + b)^{-1} = a^{-1},$$

where $x^{-1}$ is the element for which $x^{-1}x = xx^{-1} = e$, where $e$ is the element of the system such that for all $a$ the equality $ea = ae = a$ holds.
1982 IMO Longlists
Consider a cube $C$ and two planes $\sigma, \tau$, which divide Euclidean space into several regions. Prove that the interior of at least one of these regions meets at least three faces of the cube.
1982 IMO Longlists
All edges and all diagonals of regular hexagon $A_1A_2A_3A_4A_5A_6$ are colored blue or red such that each triangle $A_jA_kA_m, 1 \leq j < k < m\leq 6$ has at least one red edge. Let $R_k$ be the number of red segments $A_kA_j, (j \neq k)$. Prove the inequality

$$\sum_{k=1}^6 (2R_k-7)^2 \leq 54.$$
1982 IMO Longlists
Determine the sum of all positive integers whose digits (in base ten) form either a strictly increasing or a strictly decreasing sequence.
1982 IMO Longlists
Prove that if a person a has infinitely many descendants (children, their children, etc.), then a has an infinite sequence $a_0, a_1, \ldots$ of descendants (i.e., $a = a_0$ and for all $n \geq 1, a_{n+1}$ is always a child of $a_n$). It is assumed that no-one can have infinitely many children.



Variant 1. Prove that if $a$ has infinitely many ancestors, then $a$ has an infinite descending sequence of ancestors (i.e., $a_0, a_1, \ldots$ where $a = a_0$ and $a_n$ is always a child of $a_{n+1}$).



Variant 2. Prove that if someone has infinitely many ancestors, then all people cannot descend from $A(dam)$ and $E(ve)$.
1982 IMO Longlists
Let $(a_n)_{n\geq0}$ and $(b_n)_{n \geq 0}$ be two sequences of natural numbers. Determine whether there exists a pair $(p, q)$ of natural numbers that satisfy

$$p < q \quad \text{ and } \quad  a_p \leq a_q, b_p \leq b_q.$$
1982 IMO Longlists
Let $(u_1, \ldots, u_n)$ be an ordered $n$tuple. For each $k, 1 \leq k \leq n$, define $v_k=\sqrt[k]{u_1u_2 \cdots u_k}$. Prove that

$$\sum_{k=1}^n v_k \leq e \cdot \sum_{k=1}^n u_k.$$

($e$ is the base of the natural logarithm).
1982 IMO Longlists
Let $f : \mathbb R \to \mathbb R$ be a continuous function. Suppose that the restriction of $f$ to the set of irrational numbers is injective. What can we say about $f$? Answer the analogous question if $f$ is restricted to rationals.
1982 IMO Longlists
Let $M$ be the set of all functions $f$ with the following properties:



(i) $f$ is defined for all real numbers and takes only real values.



(ii) For all $x, y \in \mathbb R$ the following equality holds: $f(x)f(y) = f(x + y) + f(x - y).$



(iii) $f(0) \neq 0.$



Determine all functions $f \in M$ such that



(a) $f(1)=\frac 52$,



(b) $f(1)= \sqrt 3$.
1982 IMO Longlists
If the inradius of a triangle is half of its circumradius, prove that the triangle is equilateral.
1982 IMO Longlists
Numbers $u_{n,k} \ (1\leq k \leq n)$ are defined as follows

$$u_{1,1}=1, \quad u_{n,k}=\binom{n}{k} - \sum_{d \mid n, d \mid k, d>1} u_{n/d, k/d}.$$

(the empty sum is defined to be equal to zero). Prove that $n \mid u_{n,k}$ for every natural number $n$ and for every $k \ (1 \leq k \leq n).$
1982 IMO Longlists
Let $S$ be the unit circle with center $O$ and let $P_1, P_2,\ldots, P_n$ be points of $S$ such that the sum of vectors $v_i=\stackrel{\longrightarrow}{OP_i}$ is the zero vector. Prove that the inequality $\sum_{i=1}^n XP_i \geq n$ holds for every point $X$.
1982 IMO Longlists
We consider a game on an infinite chessboard similar to that of solitaire: If two adjacent fields are occupied by pawns and the next field is empty (the three fields lie on a vertical or horizontal line), then we may remove these two pawns and put one of them on the third field. Prove that if in the initial position pawns fill a $3k \times n$ rectangle, then it is impossible to reach a position with only one pawn on the board.
1982 IMO Longlists
Let $\mathfrak F$ be the family of all $k$-element subsets of the set $\{1, 2, \ldots, 2k + 1\}$. Prove that there exists a bijective function $f :\mathfrak F \to \mathfrak F$ such that for every $A \in \mathfrak F$, the sets $A$ and $f(A)$ are disjoint.
1982 IMO Longlists
(a) What is the maximal number of acute angles in a convex polygon?



(b) Consider $m$ points in the interior of a convex $n$-gon. The $n$-gon is partitioned into triangles whose vertices are among the $n + m$ given points (the vertices of the $n$-gon and the given points). Each of the $m$ points in the interior is a vertex of at least one triangle. Find the number of triangles obtained.
1982 IMO Longlists
Let $A$ and $B$ be positions of two ships $M$ and $N$, respectively, at the moment when $N$ saw $M$ moving with constant speed $v$ following the line $Ax$. In search of help, $N$ moves with speed $kv$ ($k < 1$) along the line $By$ in order to meet $M$ as soon as possible. Denote by $C$ the point of meeting of the two ships, and set

$$AB = d, \angle BAC = \alpha, 0 \leq \alpha < \frac{\pi}{2}.$$

Determine the angle $\angle ABC = \beta$ and time $t$ that $N$ needs in order to meet $M$.
1982 IMO Longlists
Prove that if a diagonal is drawn in a quadrilateral inscribed in a circle, the sum of the radii of the circles inscribed in the two triangles thus formed is the same, no matter which diagonal is drawn.
1982 IMO Longlists
Evaluate $\sec'' \frac{\pi}4 +\sec'' \frac{3\pi}4+\sec'' \frac{5\pi}4+\sec'' \frac{7\pi}4$. (Here $\sec''$ means the second derivative of $\sec$).
1982 IMO Longlists
Given a finite sequence of complex numbers $c_1, c_2, \ldots , c_n$, show that there exists an integer $k$ ($1 \leq k \leq n$) such that for every finite sequence $a_1, a_2, \ldots, a_n$ of real numbers with $1 \geq a_1 \geq a_2 \geq \cdots \geq a_n \geq 0$, the following inequality holds:

$$\left| \sum_{m=1}^n a_mc_m \right| \leq \left| \sum_{m=1}^k c_m \right|.$$
1982 IMO Longlists
Simplify

$$\sum_{k=0}^n \frac{(2n)!}{(k!)^2((n-k)!)^2}.$$
1982 IMO Longlists
Let $O$ be the midpoint of the axis of a right circular cylinder. Let $A$ and $B$ be diametrically opposite points of one base, and $C$ a point of the other base circle that does not belong to the plane $OAB$. Prove that the sum of dihedral angles of the trihedral $OABC$ is equal to $2\pi$.
1982 IMO Longlists
Let n numbers $x_1, x_2, \ldots, x_n$ be chosen in such a way that $1 \geq x_1 \geq x_2 \geq \cdots \geq x_n \geq 0$. Prove that

$$(1 + x_1 + x_2 + \cdots + x_n)^\alpha \leq 1 + x_1^\alpha+ 2^{\alpha-1}x_2^\alpha+ \cdots+ n^{\alpha-1}x_n^\alpha$$

if $0 \leq \alpha \leq 1$.
1982 IMO Longlists
We are given $2n$ natural numbers

$$1, 1, 2, 2, 3, 3, \ldots, n - 1, n - 1, n, n.$$

Find all $n$ for which these numbers can be arranged in a row such that for each $k \leq n$, there are exactly $k$ numbers between the two numbers $k$.
1982 IMO Longlists
Let $f(x) = ax^2 + bx+ c$ and $g(x) = cx^2 + bx + a$. If $|f(0)| \leq 1, |f(1)| \leq 1, |f(-1)| \leq 1$, prove that for $|x| \leq 1$,



(a) $|f(x)| \leq 5/4$,



(b) $|g(x)| \leq 2$.
1979 IMO Longlists
For a finite set $E$ of cardinality $n \geq 3$, let $f(n)$ denote the maximum number of $3$-element subsets of $E$, any two of them having exactly one common element. Calculate $f(n)$.
1979 IMO Longlists
Is it possible to partition $3$-dimensional Euclidean space into $1979$ mutually isometric subsets?
1979 IMO Longlists
Describe which positive integers do not belong to the set

$$E = \left\{ \lfloor n+ \sqrt n +\frac 12 \rfloor | n \in \mathbb N\right\}.$$
1979 IMO Longlists
Prove that $\frac 12 \cdot \sqrt{4\sin^2 36^{\circ} - 1}=\cos 72^\circ$.
1979 IMO Longlists
$M = (a_{i,j} ), \ i, j = 1, 2, 3, 4$, is a square matrix of order four. Given that:



(i) for each $i = 1, 2, 3,4$ and for each $k = 5, 6, 7$,

$$a_{i,k} = a_{i,k-4};$$$$P_i = a_{1,}i + a_{2,i+1} + a_{3,i+2} + a_{4,i+3};$$$$S_i = a_{4,i }+ a_{3,i+1} + a_{2,i+2} + a_{1,i+3};$$$$L_i = a_{i,1} + a_{i,2} + a_{i,3} + a_{i,4};$$$$C_i = a_{1,i} + a_{2,i} + a_{3,i} + a_{4,i},$$



(ii) for each $i, j = 1, 2, 3, 4$, $P_i = P_j , S_i = S_j , L_i = L_j , C_i = C_j$, and

(iii) $a_{1,1} = 0, a_{1,2} = 7, a_{2,1} = 11, a_{2,3} = 2$, and $a_{3,3} = 15$.





find the matrix M.
1979 IMO Longlists
The sequence $(a_n)$ of real numbers is defined as follows:

$$a_1=1, \qquad a_2=2, \quad \text{and} \quad a_n=3a_{n-1}-a_{n-2} , \ \ n \geq 3.$$

Prove that for $n \geq 3$, $a_n=\left[ \frac{a_{n-1}^2}{a_{n-2}} \right] +1$, where $[x]$ denotes the integer $p$ such that $p \leq  x < p + 1$.
1979 IMO Longlists
The real numbers $\alpha_1 , \alpha_2, \alpha_3, \ldots, \alpha_n$ are positive. Let us denote by $h = \frac{n}{1/\alpha_1 + 1/\alpha_2 + \cdots + 1/\alpha_n}$ the harmonic mean, $g=\sqrt[n]{\alpha_1\alpha_2\cdots \alpha_n}$ the geometric mean, and $a=\frac{\alpha_1+\alpha_2+\cdots + \alpha_n}{n}$ the arithmetic mean. Prove that $h \leq g \leq a$, and that each of the equalities implies the other one.
1979 IMO Longlists
Prove that a pyramid $A_1A_2 \ldots A_{2k+1}S$ with equal lateral edges and equal space angles between adjacent lateral walls is regular.
1979 IMO Longlists
The plane is divided into equal squares by parallel lines; i.e., a square net is given. Let $M$ be an arbitrary set of $n$ squares of this net. Prove that it is possible to choose no fewer than $n/4$ squares of $M$ in such a way that no two of them have a common point.
1979 IMO Longlists
Let $S$ be a set of $n^2 + 1$ closed intervals ($n$ a positive integer). Prove that at least one of the following assertions holds:



(i) There exists a subset $S'$ of $n+1$ intervals from $S$ such that the intersection of the intervals in $S'$ is nonempty.



(ii) There exists a subset $S''$ of $n + 1$ intervals from $S$ such that any two of the intervals in $S''$ are disjoint.
1979 IMO Longlists
Let $Q$ be a square with side length $6$. Find the smallest integer $n$ such that in $Q$ there exists a set $S$ of $n$ points with the property that any square with side $1$ completely contained in $Q$ contains in its interior at least one point from $S$.
1979 IMO Longlists
Show that for no integers $a \geq 1, n \geq 1$ is the sum

$$1+\frac{1}{1+a}+\frac{1}{1+2a}+\cdots+\frac{1}{1+na}$$

an integer.
1979 IMO Longlists
For $k = 1, 2, \ldots$ consider the $k$-tuples $(a_1, a_2, \ldots, a_k)$ of positive integers such that

$$a_1 + 2a_2 + \cdots + ka_k = 1979.$$

Show that there are as many such $k$-tuples with odd $k$ as there are with even $k$.
1979 IMO Longlists
Let $E$ be the set of all bijective mappings from $\mathbb R$ to $\mathbb R$ satisfying

$$f(t) + f^{-1}(t) = 2t, \qquad \forall t \in \mathbb R,$$

where $f^{-1}$ is the mapping inverse to $f$. Find all elements of $E$ that are monotonic mappings.
1979 IMO Longlists
Consider two quadrilaterals $ABCD$ and $A'B'C'D'$ in an affine Euclidian plane such that $AB = A'B', BC = B'C', CD = C'D'$, and $DA = D'A'$. Prove that the following two statements are true:



(a) If the diagonals $BD$ and $AC$ are mutually perpendicular, then the diagonals $B'D'$ and $A'C'$ are also mutually perpendicular.



(b) If the perpendicular bisector of $BD$ intersects $AC$ at $M$, and that of $B'D'$ intersects $A'C'$ at $M'$, then $\frac{\overline{MA}}{\overline{MC}}=\frac{\overline{M'A'}}{\overline{M'C'}}$ (if $MC = 0$ then $M'C' = 0$).
1979 IMO Longlists
Consider the set $E$ consisting of pairs of integers $(a, b)$, with $a \geq 1$ and $b \geq 1$, that satisfy in the decimal system the following properties:



(i) $b$ is written with three digits, as $\overline{\alpha_2\alpha_1\alpha_0}$, $\alpha_2 \neq 0$;



(ii) $a$ is written as $\overline{\beta_p \ldots \beta_1\beta_0}$ for some $p$;



(iii) $(a + b)^2$ is written as $\overline{\beta_p\ldots \beta_1 \beta_0 \alpha_2 \alpha_1 \alpha_0}.$



Find the elements of $E$.
1979 IMO Longlists
Let $a$ and $b$ be coprime integers, greater than or equal to $1$. Prove that all integers $n$ greater than or equal to $(a - 1)(b - 1)$ can be written in the form:

$$n = ua + vb, \qquad \text{with} (u, v) \in \mathbb N \times \mathbb N.$$
1979 IMO Longlists
Let $n$ be a positive integer. If $4^n + 2^n + 1$ is a prime, prove that $n$ is a power of three.
1979 IMO Longlists
Let $M$ be a set of points in a plane with at least two elements. Prove that if $M$ has two axes of symmetry $g_1$ and $g_2$ intersecting at an angle $\alpha = q\pi$, where $q$ is irrational, then $M$ must be infinite.
1979 IMO Longlists
Let $n, k \ge 1$ be natural numbers. Find the number $A(n, k)$ of solutions in integers of the equation

$$|x_1| + |x_2| +\cdots + |x_k| = n$$
1979 IMO Longlists
Notice that in the fraction $\frac{16}{64}$ we can perform a simplification as $\cancel{\frac{16}{64}}=\frac 14$ obtaining a correct equality. Find all fractions whose numerators and denominators are two-digit positive integers for which such a simplification is correct.
1979 IMO Longlists
Given a sequence $(a_n)$, with $a_1 = 4$ and $a_{n+1} = a_n^2-2 (\forall n \in\mathbb{N})$, prove that there is a triangle with side lengths $a_{n-1}, a_n, a_{n+1},$ and that its area is equal to an integer.