Found problems: 504
1990 IMO Longlists, 18
Find, with proof, the least positive integer $n$ having the following property: in the binary representation of $\frac 1n$, all the binary representations of $1, 2, \ldots, 1990$ (each consist of consecutive digits) are appeared after the decimal point.
1982 IMO Longlists, 8
A box contains $p$ white balls and $q$ black balls. Beside the box there is a pile of black balls. Two balls are taken out of the box. If they have the same color, a black ball from the pile is put into the box. If they have different colors, the white ball is put back into the box. This procedure is repeated until the last two balls are removed from the box and one last ball is put in. What is the probability that this last ball is white?
1988 IMO Longlists, 30
In the triangle $ABC$ let $D,E$ and $F$ be the mid-points of the three sides, $X,Y$ and $Z$ the feet of the three altitudes, $H$ the orthocenter, and $P,Q$ and $R$ the mid-points of the line segment joining $H$ to the three vertices. Show that the nine points $D,E,F,P,Q,R,X,Y,Z$ lie on a circle.
1969 IMO Longlists, 30
$(GDR 2)^{IMO1}$ Prove that there exist infinitely many natural numbers $a$ with the following property: The number $z = n^4 + a$ is not prime for any natural number $n.$
1967 IMO Longlists, 33
In what case does the system of equations
$\begin{matrix} x + y + mz = a \\ x + my + z = b \\ mx + y + z = c \end{matrix}$
have a solution? Find conditions under which the unique solution of the above system is an arithmetic progression.
1966 IMO Shortlist, 45
An alphabet consists of $n$ letters. What is the maximal length of a word if we know that any two consecutive letters $a,b$ of the word are different and that the word cannot be reduced to a word of the kind $abab$ with $a\neq b$ by removing letters.
1979 IMO Shortlist, 3
Find all polynomials $f(x)$ with real coefficients for which
\[f(x)f(2x^2) = f(2x^3 + x).\]
1967 IMO Longlists, 49
Let $n$ and $k$ be positive integers such that $1 \leq n \leq N+1$, $1 \leq k \leq N+1$. Show that: \[ \min_{n \neq k} |\sin n - \sin k| < \frac{2}{N}. \]
1969 IMO Shortlist, 40
$(MON 1)$ Find the number of five-digit numbers with the following properties: there are two pairs of digits such that digits from each pair are equal and are next to each other, digits from different pairs are different, and the remaining digit (which does not belong to any of the pairs) is different from the other digits.
1966 IMO Longlists, 29
A given natural number $N$ is being decomposed in a sum of some consecutive integers.
[b]a.)[/b] Find all such decompositions for $N=500.$
[b]b.)[/b] How many such decompositions does the number $N=2^{\alpha }3^{\beta }5^{\gamma }$ (where $\alpha ,$ $\beta $ and $\gamma $ are natural numbers) have? Which of these decompositions contain natural summands only?
[b]c.)[/b] Determine the number of such decompositions (= decompositions in a sum of consecutive integers; these integers are not necessarily natural) for an arbitrary natural $N.$
[b]Note by Darij:[/b] The $0$ is not considered as a natural number.
1966 IMO Shortlist, 46
Let $a,b,c$ be reals and
\[f(a, b, c) = \left| \frac{ |b-a|}{|ab|} +\frac{b+a}{ab} -\frac 2c \right| +\frac{ |b-a|}{|ab|} +\frac{b+a}{ab} +\frac 2c\]
Prove that $f(a, b, c) = 4 \max \{\frac 1a, \frac 1b,\frac 1c \}.$
1969 IMO Longlists, 9
$(BUL 3)$ One hundred convex polygons are placed on a square with edge of length $38 cm.$ The area of each of the polygons is smaller than $\pi cm^2,$ and the perimeter of each of the polygons is smaller than $2\pi cm.$ Prove that there exists a disk with radius $1$ in the square that does not intersect any of the polygons.
1969 IMO Longlists, 1
$(BEL 1)$ A parabola $P_1$ with equation $x^2 - 2py = 0$ and parabola $P_2$ with equation $x^2 + 2py = 0, p > 0$, are given. A line $t$ is tangent to $P_2.$ Find the locus of pole $M$ of the line $t$ with respect to $P_1.$
1969 IMO Shortlist, 31
$(GDR 3)$ Find the number of permutations $a_1, \cdots, a_n$ of the set $\{1, 2, . . ., n\}$ such that $|a_i - a_{i+1}| \neq 1$ for all $i = 1, 2, . . ., n - 1.$ Find a recurrence formula and evaluate the number of such permutations for $n \le 6.$
1966 IMO Longlists, 28
In the plane, consider a circle with center $S$ and radius $1.$ Let $ABC$ be an arbitrary triangle having this circle as its incircle, and assume that $SA\leq SB\leq SC.$ Find the locus of
[b]a.)[/b] all vertices $A$ of such triangles;
[b]b.)[/b] all vertices $B$ of such triangles;
[b]c.)[/b] all vertices $C$ of such triangles.
1967 IMO Shortlist, 1
Prove that all numbers of the sequence \[ \frac{107811}{3}, \quad \frac{110778111}{3}, \frac{111077781111}{3}, \quad \ldots \] are exact cubes.
1967 IMO Shortlist, 4
Suppose medians $m_a$ and $m_b$ of a triangle are orthogonal. Prove that:
a.) Using medians of that triangle it is possible to construct a rectangular triangle.
b.) The following inequality: \[5(a^2+b^2-c^2) \geq 8ab,\] is valid, where $a,b$ and $c$ are side length of the given triangle.
1969 IMO Shortlist, 39
$(HUN 6)$ Find the positions of three points $A,B,C$ on the boundary of a unit cube such that $min\{AB,AC,BC\}$ is the greatest possible.
1979 IMO Longlists, 46
Let $K$ denote the set $\{a, b, c, d, e\}$. $F$ is a collection of $16$ different subsets of $K$, and it is known that any three members of $F$ have at least one element in common. Show that all $16$ members of $F$ have exactly one element in common.
1969 IMO Shortlist, 60
$(SWE 3)$ Find the natural number $n$ with the following properties:
$(1)$ Let $S = \{P_1, P_2, \cdots\}$ be an arbitrary finite set of points in the plane, and $r_j$ the distance from $P_j$ to the origin $O.$ We assign to each $P_j$ the closed disk $D_j$ with center $P_j$ and radius $r_j$. Then some $n$ of these disks contain all points of $S.$
$(2)$ $n$ is the smallest integer with the above property.
1992 IMO Longlists, 72
In a school six different courses are taught: mathematics, physics, biology, music, history, geography. The students were required to rank these courses according to their preferences, where equal preferences were allowed. It turned out that:
[list]
[*][b](i)[/b] mathematics was ranked among the most preferred courses by all students;
[*][b](ii)[/b] no student ranked music among the least preferred ones;
[*][b](iii) [/b]all students preferred history to geography and physics to biology; and
[*][b](iv)[/b] no two rankings were the same. [/list]
Find the greatest possible value for the number of students in this school.
1966 IMO Shortlist, 35
Let $ax^{3}+bx^{2}+cx+d$ be a polynomial with integer coefficients $a,$ $b,$ $c,$ $d$ such that $ad$ is an odd number and $bc$ is an even number. Prove that (at least) one root of the polynomial is irrational.
1967 IMO Shortlist, 2
Find all real solutions of the system of equations:
\[\sum^n_{k=1} x^i_k = a^i\] for $i = 1,2, \ldots, n.$
1992 IMO Longlists, 40
The colonizers of a spherical planet have decided to build $N$ towns, each having area $1/1000$ of the total area of the planet. They also decided that any two points belonging to different towns will have different latitude and different longitude. What is the maximal value of $N$?
1967 IMO Longlists, 2
Prove that
\[\frac{1}{3}n^2 + \frac{1}{2}n + \frac{1}{6} \geq (n!)^{\frac{2}{n}},\]
and let $n \geq 1$ be an integer. Prove that this inequality is only possible in the case $n = 1.$