IMO 1969 - Longlist Problems

(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 $P_1$.

(BEL 2)
(a) Find the equations of regular hyperbolas passing through the points $A(\alpha,0)$, $B(\beta,0)$, and $C(0,\gamma)$.
(b) Prove that all such hyperbolas pass through the orthocenter $H$ of the triangle $ABC$.
(c) Find the locus of the centers of these hyperbolas.
(d) Check whether this locus coincides with the nine-point circle of the triangle $ABC$.

(BEL 3) Construct the circle that is tangent to three given circles.

(BEL 4) Let $O$ be a point on a nondegenerate conic. A right angle with vertex $O$ intersects the conic at points $A$ and $B$. Prove that the line $AB$ passes through a fixed point located on the normal to the conic through the point $O$.

(BEL 5) Let $G$ be the centroid of the triangle $OAB$.
(a) Prove that all conics passing through the points $O,A,B,G$ are hyperbolas.
(b) Find the locus of the centers of these hyperbolas.

(BEL 6) Evaluate $(\cos(\pi/4)+i\sin(\pi/4))^{10}$ in two different ways and prove that
$$\binom{10}1-\binom{10}3+\frac12\binom{10}5=2^4.$$

(BUL 1) Prove that the equation $\sqrt{x^3+y^3+z^3}=1969$ has no integral solutions.

(BUL 2) Find all functions $f$ defined for all $x$ that satisfy the condition
$$xf(y)+yf(x)=(x+y)f(x)f(y),$$
for all $x$ and $y$. Prove that exactly two of them are continuous.

(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 $\pi$ cm², 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.

(BUL 4) Let $M$ be the point inside the right-angled triangle $ABC$ $(\angle C=90^\circ)$ such that
$$\angle MAB=\angle MBC=\angle MCA=\varphi.$$
Let $\psi$ be the acute angle between the medians of $AC$ and $BC$. Prove that $\frac{\sin(\varphi+\psi)}{\sin(\varphi-\psi)}=5$.

(BUL 5) Let $Z$ be a set of points in the plane. Suppose that there exists a pair of points that cannot be joined by a polygonal line not passing through any point of $Z$. Let us call such a pair of points unjoinable. Prove that for each real $r>0$ there exists an unjoinable pair of points separated by distance $r$.

(CZS 1) Given a unit cube, find the locus of the centroids of all tetrahedra whose vertices lie on the sides of the cube.

(CZS 2) Let $p$ be a prime odd number. Is it possible to find $p-1$ natural numbers $n+1,n+2,\ldots,n+p-1$ such that the sum of the squares of these numbers is divisible by the sum of these numbers?

(CZS 3) Let $a$ and $b$ be two positive real numbers. If $x$ is a real solution of the equation $x^2+px+q=0$ with real coefficients $p$ and $q$ such that $|p|\le a$, $|q|\le b$, prove that
$$|x|\le\frac12\left(a+\sqrt{a^2+4b}\right).\tag{1}$$
Conversely, if $x$ satisfies (1), prove that there exist real numbers $p$ and $q$ with $|p|\le a$, $|q|\le b$ such that $x$ is one of the roots of the equation $x^2+px+q=0$.

(CZS 4) Let $K_1,\ldots,K_n$ be nonnegative integers. Prove that
$$K_1!K_2!\cdots K_n!\ge [K/n]!^n,$$
where $K=K_1+\cdots+K_n$.

(CZS 5) A convex quadrilateral $ABCD$ with sides $AB=a$, $BC=b$, $CD=c$, $DA=d$ and angles $\alpha=\angle DAB$, $\beta=\angle ABC$, $\gamma=\angle BCD$, and $\delta=\angle CDA$ is given. Let $s=(a+b+c+d)/2$ and $P$ be the area of the quadrilateral. Prove that
$$P^2=(s-a)(s-b)(s-c)(s-d)-abcd\cos^2\frac{\alpha+\gamma}{2}.$$

(CZS 6) Let $d$ and $p$ be two real numbers. Find the first term of an arithmetic progression $a_1,a_2,a_3,\ldots$ with difference $d$ such that $a_1a_2a_3a_4=p$. Find the number of solutions in terms of $d$ and $p$.

(FRA 1) Let $a$ and $b$ be two nonnegative integers. Denote by $H(a,b)$ the set of numbers $n$ of the form $n=pa+qb$, where $p$ and $q$ are positive integers. Determine $H(a)=H(a,a)$. Prove that if $a\ne b$, it is enough to know all the sets $H(a,b)$ for coprime numbers $a,b$ in order to know all the sets $H(a,b)$. Prove that in the case of coprime numbers $a$ and $b$, $H(a,b)$ contains all numbers greater than or equal to $\omega=(a-1)(b-1)$ and also $\omega/2$ numbers smaller than $\omega$.

(FRA 2) Let $n$ be an integer that is not divisible by any square greater than $1$. Denote by $x_m$ the last digit of the number $x^m$ in the number system with base $n$. For which integers $x$ is it possible for $x_m$ to be $0$? Prove that the sequence $x_m$ is periodic with period $t$ independent of $x$. For which $x$ do we have $x_t=1$. Prove that if $m$ and $x$ are relatively prime, then $0_m,1_m,\ldots,(n-1)_m$ are different numbers. Find the minimal period $t$ in terms of $n$. If $n$ does not meet the given condition, prove that it is possible to have $x_m=0\ne x_1$ and that the sequence is periodic starting only from some number $k\gt 1$.

(FRA 3) A polygon (not necessarily convex) with vertices in the lattice points of a rectangular grid is given. The area of the polygon is $S$. If $I$ is the number of lattice points that are strictly in the interior of the polygon and $B$ the number of lattice points on the border of the polygon, find the number $T=2S-B-2I+2$.