Navvye/Olympiad-Problems-CSV huggingface dataset:
1977 IMO Longlists
Prove the identity

$$ (z+a)^n=z^n+a\sum_{k=1}^n\binom{n}{k}(a-kb)^{k-1}(z+kb)^{n-k}$$
1977 IMO Longlists
Let $p$ be a prime number greater than $5.$ Let $V$ be the collection of all positive integers $n$ that can be written in the form $n = kp + 1$ or $n = kp - 1  \ (k = 1, 2, \ldots).$ A number $n \in V$ is called indecomposable in $V$ if it is impossible to find $k, l \in V$ such that $n = kl.$ Prove that there exists a number $N \in V$ that can be factorized into indecomposable factors in $V$ in more than one way.
1977 IMO Longlists
A triangle $ABC$ with $\angle A = 30^\circ$ and $\angle C = 54^\circ$ is given. On $BC$ a point $D$ is chosen such that $ \angle CAD = 12^\circ.$ On $AB$ a point $E$ is chosen such that $\angle ACE = 6^\circ.$ Let $S$ be the point of intersection of $AD$ and $CE.$ Prove that $BS = BC.$
1977 IMO Longlists
Let $f$ be a function defined on the set of pairs of nonzero rational numbers whose values are positive real numbers. Suppose that $f$ satisfies the following conditions:



(1) $f(ab,c)=f(a,c)f(b,c),\ f(c,ab)=f(c,a)f(c,b);$



(2) $f(a,1-a)=1$



Prove that $f(a,a)=f(a,-a)=1,f(a,b)f(b,a)=1$.
1977 IMO Longlists
In a room there are nine men. Among every three of them there are two mutually acquainted. Prove that some four of them are mutually acquainted.
1977 IMO Longlists
A circle $K$ centered at $(0,0)$ is given. Prove that for every vector $(a_1,a_2)$ there is a positive integer $n$ such that the circle $K$ translated by the vector $n(a_1,a_2)$ contains a lattice point (i.e., a point both of whose coordinates are integers).
1977 IMO Longlists
Find all numbers $N=\overline{a_1a_2\ldots a_n}$ for which $9\times \overline{a_1a_2\ldots a_n}=\overline{a_n\ldots a_2a_1}$ such that at most one of the digits $a_1,a_2,\ldots ,a_n$ is zero.
1977 IMO Longlists
Consider a sequence of numbers $(a_1, a_2, \ldots , a_{2^n}).$ Define the operation

$$S\biggl((a_1, a_2, \ldots , a_{2^n})\biggr) = (a_1a_2, a_2a_3, \ldots , a_{2^{n-1}a_{2^n}, a_{2^n}a_1).}$$

Prove that whatever the sequence $(a_1, a_2, \ldots , a_{2^n})$ is, with $a_i \in \{-1, 1\}$ for $i = 1, 2, \ldots , 2^n,$ after finitely many applications of the operation we get the sequence $(1, 1, \ldots, 1).$
1977 IMO Longlists
Let $A_1,A_2,\ldots ,A_{n+1}$ be positive integers such that $(A_i,A_{n+1})=1$ for every $i=1,2,\ldots ,n$. Show that the equation

$$x_1^{A_1}+x_2^{A_2}+\ldots + x_n^{A_n}=x_{n+1}^{A_{n+1} }$$

has an infinite set of solutions $(x_1,x_2,\ldots , x_{n+1})$ in positive integers.
1977 IMO Longlists
Let $m_j > 0$ for $j = 1, 2,\ldots, n$ and $a_1 \leq \cdots \leq a_n < b_1 \leq \cdots \leq b_n < c_1 \leq \cdots \leq c_n$ be real numbers. Prove that

$$\Biggl( \sum_{j=1}^{n} m_j(a_j+b_j+c_j) \Biggr)^2 > 3 \Biggl( \sum_{j=1}^{n} m_j \Biggr) \Biggl( \sum_{j=1}^{n} m_j(a_jb_j+b_jc_j+c_ja_j) \Biggr).$$
1977 IMO Longlists
Consider $37$ distinct points in space, all with integer coordinates. Prove that we may find among them three distinct points such that their barycentre has integers coordinates.
1977 IMO Longlists
The numbers $1, 2, 3,\ldots , 64$ are placed on a chessboard, one number in each square. Consider all squares on the chessboard of size $2 \times 2.$ Prove that there are at least three such squares for which the sum of the $4$ numbers contained exceeds $100.$
1977 IMO Longlists
A wheel consists of a fixed circular disk and a mobile circular ring. On the disk the numbers $1, 2, 3, \ldots ,N$ are marked, and on the ring $N$ integers $a_1,a_2,\ldots ,a_N$ of sum $1$ are marked. The ring can be turned into $N$ different positions in which the numbers on the disk and on the ring match each other. Multiply every number on the ring with the corresponding number on the disk and form the sum of $N$ products. In this way a sum is obtained for every position of the ring. Prove that the $N$ sums are different.
1977 IMO Longlists
The sequence $a_{n,k} \ , k = 1, 2, 3,\ldots, 2^n \ , n = 0, 1, 2,\ldots,$ is defined by the following recurrence formula:

$$a_1 = 2,\qquad  a_{n,k} = 2a_{n-1,k}^3, \qquad , a_{n,k+2^{n-1}} =\frac 12 a_{n-1,k}^3$$$$\text{for} \quad k = 1, 2, 3,\ldots, 2^{n-1} \ , n = 0, 1, 2,\ldots$$

Prove that the numbers $a_{n,k}$ are all different.
1977 IMO Longlists
Evaluate

$$S=\sum_{k=1}^n k(k+1)\ldots (k+p),  $$

where $n$ and $p$ are positive integers.
1977 IMO Longlists
Let $E$ be a finite set of points in space such that $E$ is not contained in a plane and no three points of $E$ are collinear. Show that $E$ contains the vertices of a tetrahedron $T = ABCD$ such that $T \cap E = \{A,B,C,D\}$ (including interior points of $T$ ) and such that the projection of $A$ onto the plane $BCD$ is inside a triangle that is similar to the triangle $BCD$ and whose sides have midpoints $B,C,D.$
1977 IMO Longlists
Let $f$ be a strictly increasing function defined on the set of real numbers. For $x$ real and $t$ positive, set$$g(x,t)=\frac{f(x+t)-f(x)}{f(x) - f(x - t)}.$$

Assume that the inequalities$$2^{-1} < g(x, t) < 2$$

hold for all positive t if $x = 0$, and for all $t \leq |x|$ otherwise.

Show that$$ 14^{-1} < g(x, t) < 14$$

for all real $x$ and positive $t.$
1977 IMO Longlists
A square $ABCD$ is given. A line passing through $A$ intersects $CD$ at $Q$. Draw a line parallel to $AQ$ that intersects the boundary of the square at points $M$ and $N$ such that the area of the quadrilateral $AMNQ$ is maximal.
1977 IMO Longlists
The intersection of a plane with a regular tetrahedron with edge $a$ is a quadrilateral with perimeter $P.$ Prove that $2a \leq P \leq 3a.$
1977 IMO Longlists
Find all pairs of integers $(p,q)$ for which all roots of the trinomials $x^2+px+q$ and $x^2+qx+p$ are integers.
1977 IMO Longlists
Determine all positive integers $n$ for which there exists a polynomial $P_n(x)$ of degree $n$ with integer coefficients that is equal to $n$ at $n$ different integer points and that equals zero at zero.
1977 IMO Longlists
Several segments, which we shall call white, are given, and the sum of their lengths is $1$. Several other segments, which we shall call black, are given, and the sum of their lengths is $1$. Prove that every such system of segments can be distributed on the segment that is $1.51$ long in the following way: Segments of the same colour are disjoint, and segments of different colours are either disjoint or one is inside the other. Prove that there exists a system that cannot be distributed in that way on the segment that is $1.49$ long.
1977 IMO Longlists
Two perpendicular chords are drawn through a given interior point $P$ of a circle with radius $R.$ Determine, with proof, the maximum and the minimum of the sum of the lengths of these two chords if the distance from $P$ to the center of the circle is $kR.$
1977 IMO Longlists
Find all pairs of integers $a$ and $b$ for which

$$7a+14b=5a^2+5ab+5b^2$$
1977 IMO Longlists
If $0 \leq a \leq b \leq c \leq d,$ prove that

$$a^bb^cc^dd^a \geq b^ac^bd^ca^d.$$
1977 IMO Longlists
Through a point $O$ on the diagonal $BD$ of a parallelogram $ABCD$, segments $MN$ parallel to $AB$, and $PQ$ parallel to $AD$, are drawn, with $M$ on $AD$, and $Q$ on $AB$. Prove that diagonals $AO,BP,DN$ (extended if necessary) will be concurrent.
1977 IMO Longlists
The four circumcircles of the four faces of a tetrahedron have equal radii. Prove that the four faces of the tetrahedron are congruent triangles.
1977 IMO Longlists
Prove that for every triangle the following inequality holds:

$$\frac{ab+bc+ca}{4S} \geq \cot \frac{\pi}{6}.$$

where $a, b, c$ are lengths of the sides and $S$ is the area of the triangle.
1977 IMO Longlists
Suppose $x_0, x_1, \ldots , x_n$ are integers and $x_0 > x_1 > \cdots > x_n.$ Prove that at least one of the numbers $|F(x_0)|, |F(x_1)|, |F(x_2)|, \ldots, |F(x_n)|,$ where

$$F(x) = x^n + a_1x^{n-1} + \cdots+ a_n,  \quad a_i \in  \mathbb R, \quad  i = 1, \ldots , n,$$

is greater than $\frac{n!}{2^n}.$
1976 IMO Longlists
Let $P$ be a set of $n$ points and $S$ a set of $l$ segments. It is known that:

$(i)$ No four points of $P$ are coplanar.

$(ii)$ Any segment from $S$ has its endpoints at $P$.

$(iii)$ There is a point, say $g$, in $P$ that is the endpoint of a maximal number of segments from $S$ and that is not a vertex of a tetrahedron having all its edges in $S$.

Prove that $l \leq \frac{n^2}{3}$
1976 IMO Longlists
Find all pairs of natural numbers $(m, n)$ for which $2^m3^n +1$ is the square of some integer.
1976 IMO Longlists
Let $ABCDS$ be a pyramid with four faces and with $ABCD$ as a base, and let a plane $\alpha$ through the vertex $A$ meet its edges $SB$ and $SD$ at points $M$ and $N$, respectively. Prove that if the intersection of the plane $\alpha$ with the pyramid $ABCDS$ is a parallelogram, then $SM \cdot SN > BM \cdot DN$.
1976 IMO Longlists
For each point $X$ of a given polytope, denote by $f(X)$ the sum of the distances of the point $X$ from all the planes of the faces of the polytope. Prove that if $f$ attains its maximum at an interior point of the polytope, then $f$ is constant.
1976 IMO Longlists
Let $P$ be a fixed point and $T$ a given triangle that contains the point $P$. Translate the triangle $T$ by a given vector $\bold{v}$ and denote by $T'$ this new triangle. Let $r, R$, respectively, be the radii of the smallest disks centered at $P$ that contain the triangles $T , T'$, respectively. Prove that $r + |\bold{v}| \leq 3R$ and find an example to show that equality can occur.
1976 IMO Longlists
Find all (real) solutions of the system

$$3x_1-x_2-x_3-x_5 = 0,$$$$-x_1+3x_2-x_4-x_6= 0,$$$$-x_1 + 3x_3 - x_4 - x_7 = 0,$$$$-x_2 - x_3 + 3x_4 - x_8 = 0,$$$$-x_1 + 3x_5 - x_6 - x_7 = 0,$$$$-x_2 - x_5 + 3x_6 - x_8 = 0,$$$$-x_3 - x_5 + 3x_7 - x_8 = 0,$$$$-x_4 - x_6 - x_7 + 3x_8 = 0.$$
1976 IMO Longlists
Show that the reciprocal of any number of the form $2(m^2+m+1)$, where $m$ is a positive integer, can be represented as a sum of consecutive terms in the sequence $(a_j)_{j=1}^{\infty}$

$$ a_j = \frac{1}{j(j + 1)(j + 2)}$$
1976 IMO Longlists
Five points lie on the surface of a ball of unit radius. Find the maximum of the smallest distance between any two of them.
1976 IMO Longlists
A sequence $\{ u_n \}$ of integers is defined by

$$u_1 = 2, u_2 = u_3 = 7,$$

$$u_{n+1} = u_nu_{n-1} - u_{n-2}, \text{  for  }n \geq 3$$

Prove that for each $n \geq 1$, $u_n$ differs by $2$ from an integral square.
1976 IMO Longlists
Let $ABC$ and $A'B'C'$ be any two coplanar triangles. Let $L$ be a point such that $AL || BC, A'L || B'C'$ , and $M,N$ similarly defined. The line $BC$ meets $B'C'$ at $P$, and similarly defined are $Q$ and $R$. Prove that $PL, QM, RN$ are concurrent.
1976 IMO Longlists
Prove that there is a positive integer $n$ such that the decimal representation of $7^n$ contains a block of at least $m$ consecutive zeros, where $m$ is any given positive integer.
1976 IMO Longlists
Show that there exists a convex polyhedron with all its vertices on the surface of a sphere and with all its faces congruent isosceles triangles whose ratio of sides are $\sqrt{3} :\sqrt{3} :2$.
1976 IMO Longlists
Prove that the number $19^{1976} + 76^{1976}$:

$(a)$ is divisible by the (Fermat) prime number $F_4 = 2^{2^4} + 1$;

$(b)$ is divisible by at least four distinct primes other than $F_4$.
1976 IMO Longlists
For a positive integer $n$, let $6^{(n)}$ be the natural number whose decimal representation consists of $n$ digits $6$. Let us define, for all natural numbers $m$, $k$ with $1 \leq k \leq m$

$$\left[\begin{array}{ccc}m\\ k\end{array}\right] =\frac{ 6^{(m)} 6^{(m-1)}\cdots 6^{(m-k+1)}}{6^{(1)} 6^{(2)}\cdots 6^{(k)}} .$$

Prove that for all $m, k$, $ \left[\begin{array}{ccc}m\\ k\end{array}\right] $ is a natural number whose decimal representation consists of exactly $k(m + k - 1) - 1$ digits.
1976 IMO Longlists
Let $(a_n), n = 0, 1, . . .,$ be a sequence of real numbers such that $a_0 = 0$ and

$$a^3_{n+1} = \frac{1}{2} a^2_n -1, n= 0, 1,\cdots$$

Prove that there exists a positive number $q, q < 1$, such that for all $n = 1, 2, \ldots ,$

$$|a_{n+1} - a_n| \leq q|a_n - a_{n-1}|,$$

and give one such $q$ explicitly.
1976 IMO Longlists
Find the largest positive real number $p$ (if it exists) such that the inequality

$$x^2_1+ x_2^2+ \cdots + x^2_n\ge p(x_1x_2 + x_2x_3 + \cdots + x_{n-1}x_n)$$

is satisfied for all real numbers $x_i$, and $(a) n = 2; (b) n = 5.$

Find the largest positive real number $p$ (if it exists) such that the inequality holds for all real numbers $x_i$ and all natural numbers $n, n \ge 2.$
1976 IMO Longlists
A regular pentagon $A_1A_2A_3A_4A_5$ with side length $s$ is given. At each point $A_i$, a sphere $K_i$ of radius $\frac{s}{2}$ is constructed. There are two spheres $K_1$ and $K_2$ each of radius $\frac{s}{2}$ touching all the five spheres $K_i.$ Decide whether $K_1$ and $K_2$ intersect each other, touch each other, or have no common points.
1976 IMO Longlists
Prove that in a Euclidean plane there are infinitely many concentric circles $C$ such that all triangles inscribed in $C$ have at least one irrational side.
1976 IMO Longlists
Let $0 \le x_1 \le x_2\le\cdots\le x_n \le 1$. Prove that for all $A \ge 1$, there exists an interval $I$ of length $2\sqrt[n]{A}$ such that for all $x \in I$,

$$|(x - x_1)(x - x_2) \cdots (x -x_n)| \le A.$$
1976 IMO Longlists
In a plane three points $P,Q,R,$ not on a line, are given. Let $k, l, m$ be positive numbers. Construct a triangle $ABC$ whose sides pass through $P, Q,$ and $R$ such that

$P$ divides the segment $AB$ in the ratio $1 : k$,

$Q$ divides the segment $BC$ in the ratio $1 : l$, and

$R$ divides the segment $CA$ in the ratio $1 : m.$
1976 IMO Longlists
Let $Q$ be a unit square in the plane: $Q = [0, 1] \times [0, 1]$. Let $T :Q \longrightarrow Q$ be defined as follows:

$$T(x, y) =\left\{\begin{array}{cc}(2x, \frac{y}{2}) &\mbox{ if } 0 \le x \le \frac{1}{2};\\ \text{ } \\(2x - 1, \frac{y}{2}+ \frac{1}{2})&\mbox{ if } \frac{1}{2} < x \le 1.\end{array}\right.$$

Show that for every disk $D \subset Q$ there exists an integer $n > 0$ such that $T^n(D) \cap D \neq \emptyset.$
1976 IMO Longlists
Prove that if $P(x) = (x-a)^kQ(x)$, where $k$ is a positive integer, $a$ is a nonzero real number, $Q(x)$ is a nonzero polynomial, then $P(x)$ has at least $k + 1$ nonzero coefficients.
1976 IMO Longlists
Into every lateral face of a quadrangular pyramid a circle is inscribed. The circles inscribed into adjacent faces are tangent (have one point in common). Prove that the points of contact of the circles with the base of the pyramid lie on a circle.
1976 IMO Longlists
We consider the infinite chessboard covering the whole plane. In every field of the chessboard there is a nonnegative real number. Every number is the arithmetic mean of the numbers in the four adjacent fields of the chessboard. Prove that the numbers occurring in the fields of the chessboard are all equal.
1976 IMO Longlists
A finite set of points $P$ in the plane has the following property: Every line through two points in $P$ contains at least one more point belonging to $P$. Prove that all points in $P$ lie on a straight line.



Remark.This may be a well known theorem called "Sylvester Gallai", but I didn't find this problem (I mean, exactly this one) using search function. So please discuss about the problem here, in this topic. Thanks :)
1976 IMO Longlists
Let $\{a_n\}^{\infty}_0$ and $\{b_n\}^{\infty}_0$ be two sequences determined by the recursion formulas

$$a_{n+1} = a_n + b_n,$$

$$ b_{n+1} = 3a_n + b_n, n= 0, 1, 2, \cdots,$$

and the initial values $a_0 = b_0 = 1$. Prove that there exists a uniquely determined constant $c$ such that $n|ca_n-b_n| < 2$ for all nonnegative integers $n$.
1976 IMO Longlists
Three concentric circles with common center $O$ are cut by a common chord in successive points $A, B, C$. Tangents drawn to the circles at the points $A, B, C$ enclose a triangular region. If the distance from point $O$ to the common chord is equal to $p$, prove that the area of the region enclosed by the tangents is equal to

$$\frac{AB \cdot BC \cdot CA}{2p}$$
1976 IMO Longlists
From a square board $11$ squares long and $11$ squares wide, the central square is removed. Prove that the remaining $120$ squares cannot be covered by $15$ strips each $8$ units long and one unit wide.
1976 IMO Longlists
Let $x =\sqrt{a}+\sqrt{b}$, where $a$ and $b$ are natural numbers, $x$ is not an integer, and $x < 1976$. Prove that the fractional part of $x$ exceeds $10^{-19.76}$.
1976 IMO Longlists
In $ ABC$, the inscribed circle is tangent to side $BC$ at$ X$. Segment $ AX$ is drawn. Prove that the line joining the midpoint of $ AX$ to the midpoint of side $ BC$ passes through center $ I$ of the inscribed circle.
1976 IMO Longlists
Let $g(x)$ be a fixed polynomial with real coefficients and define $f(x)$ by $f(x) =x^2 + xg(x^3)$. Show that $f(x)$ is not divisible by $x^2 - x + 1$.
1976 IMO Longlists
For a point $O$ inside a triangle $ABC$, denote by $A_1,B_1, C_1,$ the respective intersection points of $AO, BO, CO$ with the corresponding sides. Let

$$n_1 =\frac{AO}{A_1O}, n_2 = \frac{BO}{B_1O}, n_3 = \frac{CO}{C_1O}.$$

What possible values of $n_1, n_2, n_3$ can all be positive integers?
1976 IMO Longlists
Prove that if for a polynomial $P(x, y)$, we have

$$P(x - 1, y - 2x + 1) = P(x, y),$$

then there exists a polynomial $\Phi(x)$ with $P(x, y) = \Phi(y - x^2).$
1976 IMO Longlists
A circle of radius $1$ rolls around a circle of radius $\sqrt{2}$. Initially, the tangent point is colored red. Afterwards, the red points map from one circle to another by contact. How many red points will be on the bigger circle when the center of the smaller one has made $n$ circuits around the bigger one?
1976 IMO Longlists
We are given $n (n \ge 5)$ circles in a plane. Suppose that every three of them have a common point. Prove that all $n$ circles have a common point.
1976 IMO Longlists
Let $ a,b,c,d$ be nonnegative real numbers. Prove that

$$ a^4+b^4+c^4+d^4+2abcd \ge a^2b^2+a^2c^2+a^2d^2+b^2c^2+b^2d^2+c^2d^2.$$
1976 IMO Longlists
Determine whether there exist $1976$ nonsimilar triangles with angles $\alpha, \beta, \gamma,$ each of them satisfying the relations

$$\frac{\sin \alpha + \sin\beta + \sin\gamma}{\cos \alpha + \cos \beta + \cos \gamma}=\frac{12}{7}\text{  and  }\sin \alpha \sin \beta \sin \gamma =\frac{12}{25}$$
1976 IMO Longlists
Find a function $f(x)$ defined for all real values of $x$ such that for all $x$,

$$f(x+ 2) - f(x) = x^2 + 2x + 4,$$

and if $x \in [0, 2)$, then $f(x) = x^2.$
1976 IMO Longlists
Four swallows are catching a fly. At first, the swallows are at the four vertices of a tetrahedron, and the fly is in its interior. Their maximal speeds are equal. Prove that the swallows can catch the fly.
1974 IMO Longlists
Let ${u_n}$ be the Fibonacci sequence, i.e., $u_0=0,u_1=1,u_n=u_{n-1}+u_{n-2}$ for $n>1$. Prove that there exist infinitely many prime numbers $p$ that divide $u_{p-1}$.
1974 IMO Longlists
Let $ABCD$ be an arbitrary quadrilateral. Let squares $ABB_1A_2, BCC_1B_2, CDD_1C_2, DAA_1D_2$ be constructed in the exterior of the quadrilateral. Furthermore, let $AA_1PA_2$ and $CC_1QC_2$ be parallelograms. For any arbitrary point $P$ in the interior of $ABCD$, parallelograms $RASC$ and $RPTQ$ are constructed. Prove that these two parallelograms have two vertices in common.
1974 IMO Longlists
Let $K_a,K_b,K_c$ with centres $O_a,O_b,O_c$ be the excircles of a triangle $ABC$, touching the interiors of the sides $BC,CA,AB$ at points $T_a,T_b,T_c$ respectively.

Prove that the lines $O_aT_a,O_bT_b,O_cT_c$ are concurrent in a point $P$ for which $PO_a=PO_b=PO_c=2R$ holds, where $R$ denotes the circumradius of $ABC$. Also prove that the circumcentre $O$ of $ABC$ is the midpoint of the segment $PI$, where $I$ is the incentre of $ABC$.
1974 IMO Longlists
A straight cone is given inside a rectangular parallelepiped $B$, with the apex at one of the vertices, say $T$ , of the parallelepiped, and the base touching the three faces opposite to $T .$ Its axis lies at the long diagonal through $T.$ If $V_1$ and $V_2$ are the volumes of the cone and the parallelepiped respectively, prove that

$$V_1 \leq \frac{\sqrt 3 \pi V_2}{27}.$$
1974 IMO Longlists
Prove that the product of two natural numbers with their sum cannot be the third power of a natural number.
1974 IMO Longlists
Let $p$ be a prime number and $n$ a positive integer. Prove that the product

$${N=\frac{1}{p^{n^2}}} \prod_{i=1;2 \nmid i}^{2n-1} \biggl[ \left( (p-1)! \right) \binom{p^2 i}{pi}\biggr]$$

Is a positive integer that is not divisible by $p.$
1974 IMO Longlists
Solve the following system of linear equations with unknown $x_1,x_2 \ldots, x_n \ (n \geq 2)$ and parameters $c_1,c_2, \ldots , c_n:$



$$2x_1 -x_2 = c_1;$$$$-x_1 +2x_2 -x_3 = c_2;$$$$-x_2 +2x_3 -x_4 = c_3;$$$$\cdots \qquad \cdots \qquad \cdots \qquad$$$$-x_{n-2} +2x_{n-1} -x_n = c_{n-1};$$$$-x_{n-1} +2x_n = c_n.$$
1974 IMO Longlists
A regular octagon $P$ is given whose incircle $k$ has diameter $1$. About $k$ is circumscribed a regular $16$-gon, which is also inscribed in $P$, cutting from $P$ eight isosceles triangles. To the octagon $P$, three of these triangles are added so that exactly two of them are adjacent and no two of them are opposite to each other. Every $11$-gon so obtained is said to be $P'$. Prove the following statement: Given a finite set $M$ of points lying in $P$ such that every two points of this set have a distance not exceeding $1$, one of the $11$-gons $P'$ contains all of $M$.
1974 IMO Longlists
Given a line $p$ and a triangle $\Delta$  in the plane, construct an equilateral triangle one of whose vertices lies on the line $p$, while the other two halve the perimeter of $\Delta.$
1974 IMO Longlists
A circle $K$ with radius $r$, a point $D$ on $K$, and a convex angle with vertex $S$ and rays $a$ and $b$ are given in the plane. Construct a parallelogram $ABCD$ such that $A$ and $B$ lie on $a$ and $b$ respectively, $SA+SB=r$, and $C$ lies on $K$.
1974 IMO Longlists
Prove that $2^{147} - 1$ is divisible by $343.$
1974 IMO Longlists
Let $n$ and $k$ be natural numbers and $a_1,a_2,\ldots ,a_n$ be positive real numbers satisfying $a_1+a_2+\cdots +a_n=1$. Prove that

$$\dfrac {1} {a_1^{k}}+\dfrac {1} {a_2^{k}}+\cdots +\dfrac {1} {a_n^{k}} \ge n^{k+1}.$$
1974 IMO Longlists
A pack of $2n$ cards contains $n$ different pairs of cards. Each pair consists of two identical cards, either of which is called the twin of the other. A game is played between two players $A$ and $B$. A third person called the dealer shuffles the pack and deals the cards one by one face upward onto the table. One of the players, called the receiver, takes the card dealt, provided he does not have already its twin. If he does already have the twin, his opponent takes the dealt card and becomes the receiver.

$A$ is initially the receiver and takes the first card dealt. The player who first obtains a complete set of $n$ different cards wins the game. What fraction of all possible arrangements of the pack lead to $A$ winning? Prove the correctness of your answer.
1974 IMO Longlists
Show that there exists a set $S$ of $15$ distinct circles on the surface of a sphere, all having the same radius and such that $5$ touch exactly $5$ others, $5$ touch exactly $4$ others, and $5$ touch exactly $3$ others.



[General Problem:
1974 IMO Longlists
Prove that there exists, for $n \geq 4$, a set $S$ of $3n$ equal circles in space that can be partitioned into three subsets $s_5, s_4$, and $s_3$, each containing $n$ circles, such that each circle in $s_r$ touches exactly $r$ circles in $S.$
1974 IMO Longlists
For which natural numbers $n$ do there exist $n$ natural numbers $a_i\ (1\le i\le n)$ such that $\sum_{i=1}^n a_i^{-2}=1$?
1974 IMO Longlists
Let $M$ be a nonempty subset of $\mathbb Z^+$ such that for every element $x$ in $M,$ the numbers $4x$ and $\lfloor \sqrt x \rfloor$ also belong to $M.$ Prove that $M = \mathbb Z^+.$
1974 IMO Longlists
Let $f : \mathbb R \to \mathbb R$ be of the form $f(x) = x + \epsilon \sin x,$ where $0 < |\epsilon| \leq 1.$ Define for any $x \in \mathbb R,$

$$x_n=\underbrace{f \ o \ \ldots \ o \ f}_{n \text{ times}}  (x).$$

Show that for every $x \in \mathbb R$ there exists an integer $k$ such that $\lim_{n\to \infty } x_n = k\pi.$