Navvye/Olympiad-Problems-CSV huggingface dataset:
1970 IMO Longlists
Let the numbers $1, 2, \ldots , n^2$ be written in the cells of an $n \times n$ square board so that the entries in each column are arranged increasingly. What are the smallest and greatest possible sums of the numbers in the $k^{th}$ row? ($k$ a positive integer, $1 \leq  k \leq n$.)
1970 IMO Longlists
Given $100$ coplanar points, no three collinear, prove that at most $70\%$ of the triangles formed by the points have all angles acute.
1970 IMO Longlists
For which digits $a$ do exist integers $n \geq 4$ such that each digit of $\frac{n(n+1)}{2}$ equals $a \ ?$
1969 IMO Longlists
$(BEL 1)$ 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 Longlists
$(BEL 2) (a)$ 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 $ABC.$
1969 IMO Longlists
$(BEL 3)$ Construct the circle that is tangent to three given circles.
1969 IMO Longlists
$(BEL 4)$ 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 $O.$
1969 IMO Longlists
$(BEL 5)$ 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.
1969 IMO Longlists
$(BEL 6)$ Evaluate $\left(\cos\frac{\pi}{4} + i \sin\frac{\pi}{4}\right)^{10}$ in two different ways and prove that $\dbinom{10}{1}-\dbinom{10}{3}+\frac{1}{2}\dbinom{10}{5}=2^4$
1969 IMO Longlists
$(BUL 1)$ Prove that the equation $\sqrt{x^3 + y^3 + z^3}=1969$ has no integral solutions.
1969 IMO Longlists
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.
1969 IMO Longlists
$(BUL 3)$ 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
$(BUL 4)$ Let $M$ be the point inside the right-angled triangle $ABC (\angle C = 90^{\circ})$ such that $\angle MAB = \angle MBC = \angle MCA =\phi.$ Let $\Psi$ be the acute angle between the medians of $AC$ and $BC.$  Prove that $\frac{\sin(\phi+\Psi)}{\sin(\phi-\Psi)}= 5.$
1969 IMO Longlists
$(BUL 5)$ 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 $r.$
1969 IMO Longlists
$(CZS 1)$ Given a unit cube, find the locus of the centroids of all tetrahedra whose vertices lie on the sides of the cube.
1969 IMO Longlists
$(CZS 2)$ Let $p$ be a prime odd number. Is it possible to find $p-1$ natural numbers $n + 1, n + 2, . . . , n + p -1$ such that the sum of the squares of these numbers is divisible by the sum of these numbers?
1969 IMO Longlists
$(CZS 3)$ 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 $|x| \le \frac{1}{2}(a +\sqrt{a^2 + 4b})$ Conversely, if $x$ satisfies the above inequality, 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 $x^2+px+ q = 0.$
1969 IMO Longlists
$(CZS 4)$ Let $K_1,\cdots , K_n$ be nonnegative integers. Prove that $K_1!K_2!\cdots K_n! \ge \left[\frac{K}{n}\right]!^n$, where $K = K_1 + \cdots + K_n$
1969 IMO Longlists
$(CZS 5)$ 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 $s = \frac{a + b + c +d}{2}$ and $P$ be the area of the quadrilateral. Prove that $P^2 = (s - a)(s - b)(s - c)(s - d) - abcd \cos^2\frac{\alpha +\gamma}{2}$
1969 IMO Longlists
$(CZS 6)$ Let $d$ and $p$ be two real numbers. Find the first term of an arithmetic progression $a_1, a_2, a_3, \cdots$ with difference $d$ such that $a_1a_2a_3a_4 = p.$ Find the number of solutions in terms of $d$ and $p.$
1969 IMO Longlists
$(FRA 1)$ 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 $a \neq b,$ 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 $\frac{\omega}{2}$ numbers smaller than $\omega$
1969 IMO Longlists
$(FRA 2)$ 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 $0_m, 1_m, . . . , (n-1)_m$ 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 $x_m = 0 \neq x_1$ and that the sequence is periodic starting only from some number $k > 1.$
1969 IMO Longlists
$(FRA 3)$ 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 $T = 2S- B  -2I + 2.$
1969 IMO Longlists
$(FRA 4)$ A right-angled triangle $OAB$ has its right angle at the point $B.$ An arbitrary circle with center on the line $OB$ is tangent to the line $OA.$ Let $AT$ be the tangent to the circle different from $OA$ ($T$ is the point of tangency). Prove that the median from $B$ of the triangle $OAB$ intersects $AT$ at a point $M$ such that $MB = MT.$
1969 IMO Longlists
$(FRA 5)$ Let $\alpha(n)$ be the number of pairs $(x, y)$ of integers such that $x+y = n, 0 \le y \le x$, and let $\beta(n)$ be the number of triples $(x, y, z)$ such that$ x + y + z = n$ and $0 \le z \le y \le x.$ Find a simple relation between $\alpha(n)$ and the integer part of the number $\frac{n+2}{2}$ and the relation among $\beta(n), \beta(n -3)$ and $\alpha(n).$ Then evaluate $\beta(n)$ as a function of the residue of $n$ modulo $6$. What can be said about $\beta(n)$ and $1+\frac{n(n+6)}{12}$? And what about $\frac{(n+3)^2}{6}$?

Find the number of triples $(x, y, z)$ with the property $x+ y+ z \le n, 0 \le z \le y \le x$ as a function of the residue of $n$ modulo $6.$What can be said about the relation between this number and the number $\frac{(n+6)(2n^2+9n+12)}{72}$?
1969 IMO Longlists
$(FRA 6)$ Consider the integer $d = \frac{a^b-1}{c}$, where $a, b$, and $c$ are positive integers and $c \le a.$ Prove that the set $G$ of integers that are between $1$ and $d$ and relatively prime to $d$ (the number of such integers is denoted by $\phi(d)$) can be partitioned into $n$ subsets, each of which consists of $b$ elements. What can be said about the rational number $\frac{\phi(d)}{b}?$
1969 IMO Longlists
$(GBR 1)$ The polynomial $P(x) = a_0x^k + a_1x^{k-1} + \cdots + a_k$, where $a_0,\cdots, a_k$ are integers, is said to be divisible by an integer $m$ if $P(x)$ is a multiple of $m$ for every integral value of $x$. Show that if $P(x)$ is divisible by $m$, then $a_0 \cdot k!$ is a multiple of $m$. Also prove that if $a, k,m$ are positive integers such that $ak!$ is a multiple of $m$, then a polynomial $P(x)$ with leading term $ax^k$can be found that is divisible by $m.$
1969 IMO Longlists
$(GBR 2)$ Let $a, b, x, y$ be positive integers such that $a$ and $b$ have no common divisor greater than $1$. Prove that the largest number not expressible in the form $ax + by$ is $ab - a - b$. If $N(k)$ is the largest number not expressible in the form $ax + by$ in only $k$ ways, find $N(k).$
1969 IMO Longlists
$(GBR 3)$ A smooth solid consists of a right circular cylinder of height $h$ and base-radius $r$, surmounted by a hemisphere of radius $r$ and center $O.$ The solid stands on a horizontal table. One end of a string is attached to a point on the base. The string is stretched (initially being kept in the vertical plane) over the highest point of the solid and held down at the point $P$ on the hemisphere such that $OP$ makes an angle $\alpha$ with the horizontal. Show that if $\alpha$ is small enough, the string will slacken if slightly displaced and no longer remain in a vertical plane. If then pulled tight through $P$, show that it will cross the common circular section of the hemisphere and cylinder at a point $Q$ such that $\angle SOQ = \phi$, $S$ being where it initially crossed this section, and $\sin \phi = \frac{r \tan \alpha}{h}$.
1969 IMO Longlists
$(GBR 4)$ The segment $AB$ perpendicularly bisects $CD$ at $X$. Show that, subject to restrictions, there is a right circular cone whose axis passes through $X$ and on whose surface lie the points $A,B,C,D.$ What are the restrictions?
1969 IMO Longlists
$(GBR 5)$ Let us define $u_0 = 0, u_1 = 1$ and for $n\ge 0, u_{n+2} = au_{n+1}+bu_n, a$ and $b$ being positive integers. Express $u_n$ as a polynomial in $a$ and $b.$ Prove the result. Given that $b$ is prime, prove that $b$ divides $a(u_b -1).$
1969 IMO Longlists
$(GDR 1)$ Find all real numbers $\lambda$ such that the equation $\sin^4 x - \cos^4 x = \lambda(\tan^4 x - \cot^4 x)$

$(a)$ has no solution,

$(b)$ has exactly one solution,

$(c)$ has exactly two solutions,

$(d)$ has more than two solutions (in the interval $(0, \frac{\pi}{4}).$
1969 IMO Longlists
$(GDR 2)^{IMO1}$ 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 $n.$
1969 IMO Longlists
$(GDR 3)$ 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 $n \le 6.$
1969 IMO Longlists
$(GDR 4)$ Find the maximal number of regions into which a sphere can be partitioned by $n$ circles.
1969 IMO Longlists
$(GDR 5)$ Given a ring $G$ in the plane bounded by two concentric circles with radii $R$ and $\frac{R}{2}$, prove that we can cover this region with $8$ disks of radius $\frac{2R}{5}$. (A region is covered if each of its points is inside or on the border of some disk.)
1969 IMO Longlists
$(HUN 1)$ Let $a$ and $b$ be arbitrary integers. Prove that if $k$ is an integer not divisible by $3$, then $(a + b)^{2k}+ a^{2k} +b^{2k}$ is divisible by $a^2 +ab+ b^2$
1969 IMO Longlists
$(HUN 2)$ Prove that $1+\frac{1}{2^3}+\frac{1}{3^3}+\cdots+\frac{1}{n^3}<\frac{5}{4}$
1969 IMO Longlists
$(HUN 3)$ In the plane $4000$ points are given such that each line passes through at most $2$ of these points. Prove that there exist $1000$ disjoint quadrilaterals in the plane with vertices at these points.
1969 IMO Longlists
$(HUN 4)$IMO2 If $a_1, a_2, . . . , a_n$ are real constants, and if $y = \cos(a_1 + x) +2\cos(a_2+x)+ \cdots+ n \cos(a_n + x)$ has two zeros $x_1$ and $x_2$ whose difference is not a multiple of $\pi$, prove that $y = 0.$
1969 IMO Longlists
$(HUN 5)$ Let $r$ and $m (r \le m)$ be natural numbers and $Ak =\frac{2k-1}{2m}\pi$. Evaluate $\frac{1}{m^2}\displaystyle\sum_{k=1}^{m}\displaystyle\sum_{l=1}^{m}\sin(rA_k)\sin(rA_l)\cos(rA_k-rA_l)$
1969 IMO Longlists
$(HUN 6)$ 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.
1969 IMO Longlists
$(MON 1)$ 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.
1969 IMO Longlists
$(MON 2)$ Given reals $x_0, x_1, \alpha, \beta$, find an expression for the solution of the system $$x_{n+2} -\alpha x_{n+1} -\beta x_n = 0, \qquad  n= 0, 1, 2, \ldots$$
1969 IMO Longlists
$(MON 3)$ Let $A_k (1 \le k \le h)$ be $n-$element sets such that each two of them have a nonempty intersection. Let $A$ be the union of all the sets $A_k,$ and let $B$ be a subset of $A$ such that for each $k (1\le k \le h)$ the intersection of $A_k$ and $B$ consists of exactly two different elements $a_k$ and $b_k$. Find all subsets $X$ of the set $A$ with $r$ elements satisfying the condition that for at least one index $k,$ both elements $a_k$ and $b_k$ belong to $X$.
1969 IMO Longlists
$(MON 4)$ Let $p$ and $q$ be two prime numbers greater than $3.$ Prove that if their difference is $2^n$, then for any two integers $m$ and $n,$ the number $S = p^{2m+1} + q^{2m+1}$ is divisible by $3.$
1969 IMO Longlists
$(MON 5)$ Find the radius of the circle circumscribed about the isosceles triangle whose sides are the solutions of the equation $x^2 - ax + b = 0$.
1969 IMO Longlists
Given $n>4$ points in the plane, no three collinear. Prove that there are at least $\frac{(n-3)(n-4)}{2}$ convex quadrilaterals with vertices amongst the $n$ points.
1969 IMO Longlists
$(NET 1)$ The vertices of an $(n + 1)-$gon are placed on the edges of a regular $n-$gon so that the perimeter of the $n-$gon is divided into equal parts. How does one choose these $n + 1$ points in order to obtain the $(n + 1)-$gon with

$(a)$ maximal area;

$(b)$ minimal area?
1969 IMO Longlists
$C$ is a point on the semicircle diameter $AB$, between $A$ and $B$. $D$ is the foot of the perpendicular from $C$ to $AB$. The circle $K_1$ is the incircle of $ABC$, the circle $K_2$ touches $CD,DA$ and the semicircle, the circle $K_3$ touches $CD,DB$ and the semicircle. Prove that $K_1,K_2$ and $K_3$ have another common tangent apart from $AB$.
1969 IMO Longlists
$(NET 3)$ Let $x_1, x_2, x_3, x_4,$ and $x_5$ be positive integers satisfying

$$x_1 +x_2 +x_3 +x_4 +x_5 = 1000,$$

$$x_1 -x_2 +x_3 -x_4 +x_5 > 0,$$

$$x_1 +x_2 -x_3 +x_4 -x_5 > 0,$$

$$-x_1 +x_2 +x_3 -x_4 +x_5 > 0,$$

$$x_1 -x_2 +x_3 +x_4 -x_5 > 0,$$

$$-x_1 +x_2 -x_3 +x_4 +x_5 > 0$$

$(a)$ Find the maximum of $(x_1 + x_3)^{x_2+x_4}$

$(b)$ In how many different ways can we choose $x_1, . . . , x_5$ to obtain the desired maximum?
1969 IMO Longlists
$(NET 4)$ A boy has a set of trains and pieces of railroad track. Each piece is a quarter of circle, and by concatenating these pieces, the boy obtained a closed railway. The railway does not intersect itself. In passing through this railway, the train sometimes goes in the clockwise direction, and sometimes in the opposite direction. Prove that the train passes an even number of times through the pieces in the clockwise direction and an even number of times in the counterclockwise direction. Also, prove that the number of pieces is divisible by $4.$
1969 IMO Longlists
$(NET 5)$ The bisectors of the exterior angles of a pentagon $B_1B_2B_3B_4B_5$ form another pentagon $A_1A_2A_3A_4A_5.$ Construct $B_1B_2B_3B_4B_5$ from the given pentagon $A_1A_2A_3A_4A_5.$
1969 IMO Longlists
$(NET 6)$ A curve determined by $y =\sqrt{x^2 - 10x+ 52}, 0\le x \le 100,$ is constructed in a rectangular grid. Determine the number of squares cut by the curve.
1969 IMO Longlists
Prove that a regular polygon with an odd number of edges cannot be partitioned into four pieces with equal areas by two lines that pass through the center of polygon.
1969 IMO Longlists
$(POL 2)$ Given two segments $AB$ and $CD$ not in the same plane, find the locus of points $M$ such that $MA^2 +MB^2 = MC^2 +MD^2.$
1969 IMO Longlists
$(POL 3)$ Given a polynomial $f(x)$ with integer coefficients whose value is divisible by $3$ for three integers $k, k + 1,$ and $k + 2$. Prove that $f(m)$ is divisible by $3$ for all integers $m.$
1969 IMO Longlists
For each of $k=1,2,3,4,5$ find necessary and sufficient conditions on $a>0$ such that there exists a tetrahedron with $k$ edges length $a$ and the remainder length $1$.
1969 IMO Longlists
Let $a$ and $b$ be two natural numbers that have an equal number $n$ of digits in their decimal expansions. The first $m$ digits (from left to right) of the numbers $a$ and $b$ are equal. Prove that if $m >\frac{n}{2},$ then $a^{\frac{1}{n}} -b^{\frac{1}{n}} <\frac{1}{n}$
1969 IMO Longlists
Given triangle $ ABC $ with points $ M $ and $ N $ are in the sides $ AB $  and $ AC $ respectively.

If  $ \dfrac{BM}{MA} +\dfrac{CN}{NA} = 1 $ , then prove that the centroid of  $ ABC $ lies on $ MN $ .
1969 IMO Longlists
$(SWE 1)$ Six points $P_1, . . . , P_6$ are given in $3-$dimensional space such that no four of them lie in the same plane. Each of the line segments $P_jP_k$ is colored black or white. Prove that there exists one triangle $P_jP_kP_l$ whose edges are of the same color.
1969 IMO Longlists
$(SWE 2)$ For each $\lambda (0 < \lambda < 1$ and $\lambda = \frac{1}{n}$ for all $n = 1, 2, 3, \cdots)$, construct a continuous function $f$ such that there do not exist $x, y$ with $0 < \lambda < y = x + \lambda \le 1$ for which $f(x) = f(y).$
1969 IMO Longlists
$(SWE 3)$ 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.
1969 IMO Longlists
$(SWE 4)$ Let $a_0, a_1, a_2, \cdots$ be determined with $a_0 = 0, a_{n+1} = 2a_n + 2^n$. Prove that if $n$ is power of $2$, then so is $a_n$
1969 IMO Longlists
Which natural numbers can be expressed as the difference of squares of two integers?
1969 IMO Longlists
$(SWE 6)$ Prove that there are infinitely many positive integers that cannot be expressed as the sum of squares of three positive integers.
1969 IMO Longlists
$(USS 1)$ Prove that for a natural number $n > 2, (n!)! > n[(n - 1)!]^{n!}.$
1969 IMO Longlists
$(USS 2)$ Prove that for $a > b^2,$ the identity ${\sqrt{a-b\sqrt{a+b\sqrt{a-b\sqrt{a+\cdots}}}}=\sqrt{a-\frac{3}{4}b^2}-\frac{1}{2}b}$
1969 IMO Longlists
$(USS 3)$ $(a)$ Prove that if $0 \le a_0 \le a_1 \le a_2,$ then $(a_0 + a_1x - a_2x^2)^2 \le (a_0 + a_1 + a_2)^2\left(1 +\frac{1}{2}x+\frac{1}{3}x^2+\frac{1}{2}x^3+x^4\right)$

$(b)$ Formulate and prove the analogous result for polynomials of third degree.
1969 IMO Longlists
Given real numbers $x_1,x_2,y_1,y_2,z_1,z_2$ satisfying $x_1>0,x_2>0,x_1y_1>z_1^2$, and $x_2y_2>z_2^2$, prove that: $$ {8\over(x_1+x_2)(y_1+y_2)-(z_1+z_2)^2}\le{1\over x_1y_1-z_1^2}+{1\over x_2y_2-z_2^2}. $$ Give necessary and sufficient conditions for equality.
1969 IMO Longlists
$(USS 5)$ Given $5$ points in the plane, no three of which are collinear, prove that we can choose $4$ points among them that form a convex quadrilateral.
1969 IMO Longlists
$(YUG 1)$ Suppose that positive real numbers $x_1, x_2, x_3$ satisfy

$x_1x_2x_3 > 1, x_1 + x_2 + x_3 <\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}$

Prove that:

$(a)$ None of $x_1, x_2, x_3$ equals $1$.

$(b)$ Exactly one of these numbers is less than $1.$
1969 IMO Longlists
$(YUG 2)$ A park has the shape of a convex pentagon of area $50000\sqrt{3} m^2$. A man standing at an interior point $O$ of the park notices that he stands at a distance of at most $200 m$ from each vertex of the pentagon. Prove that he stands at a distance of at least $100 m$ from each side of the pentagon.
1969 IMO Longlists
$(YUG 3)$ Let four points $A_i (i = 1, 2, 3, 4)$ in the plane determine four triangles. In each of these triangles we choose the smallest angle. The sum of these angles is denoted by $S.$ What is the exact placement of the points $A_i$ if $S = 180^{\circ}$?
1967 IMO Longlists
Prove that all numbers of the sequence $$ \frac{107811}{3}, \quad \frac{110778111}{3}, \frac{111077781111}{3}, \quad \ldots $$ are exact cubes.
1967 IMO Longlists
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 $n = 1.$
1967 IMO Longlists
Prove the trigonometric inequality $\cos x < 1 - \frac{x^2}{2} + \frac{x^4}{16},$ when $x \in \left(0, \frac{\pi}{2} \right).$
1967 IMO Longlists
Suppose, medians $m_a$ and $m_b$ of a triangle are orthogonal. Prove that:

(a) The medians of the triangle correspond to the sides of a right-angled triangle.

(b) If $a,b,c$ are the side-lengths of the triangle, then, the following inequality holds:$$5(a^2+b^2-c^2)\geq 8ab$$
1967 IMO Longlists
Solve the system of equations:



$

\begin{matrix} 

x^2 + x - 1 = y \\ 

y^2 + y - 1 = z \\ 

z^2 + z - 1 = x.

\end{matrix}

$
1967 IMO Longlists
Solve the system of equations:



$

\begin{matrix}

|x+y| + |1-x| = 6 \\ |x+y+1| + |1-y| = 4.

\end{matrix}

$
1967 IMO Longlists
Find all real solutions of the system of equations:

$$\sum^n_{k=1} x^i_k = a^i$$ for $i = 1,2, \ldots, n.$
1967 IMO Longlists
The parallelogram $ABCD$ has $AB=a,AD=1,$ $\angle BAD=A$, and the triangle $ABD$ has all angles acute. Prove that circles radius $1$ and center $A,B,C,D$ cover the parallelogram if and only

$$a\le\cos A+\sqrt3\sin A.$$
1967 IMO Longlists
Circle $k$ and its diameter $AB$ are given. Find the locus of the centers of circles inscribed in the triangles having one vertex on $AB$ and two other vertices on $k.$
1967 IMO Longlists
The square $ABCD$ has to be decomposed into $n$ triangles (which are not overlapping) and which have all angles acute. Find the smallest integer $n$ for which there exist a solution of that problem and for such $n$ construct at least one decomposition. Answer whether it is possible to ask moreover that (at least) one of these triangles has the perimeter less than an arbitrarily given positive number.
1967 IMO Longlists
Let $n$ be a positive integer. Find the maximal number of non-congruent triangles whose sides lengths are integers $\leq n.$
1967 IMO Longlists
Given a segment $AB$ of the length 1, define the set $M$ of points in the

following way: it contains two points $A,B,$ and also all points obtained from $A,B$ by iterating the following rule: With every pair of points $X,Y$ the set $M$ contains also the point $Z$ of the segment $XY$ for which $YZ = 3XZ.$
1967 IMO Longlists
Find whether among all quadrilaterals, whose interiors lie inside a semi-circle of radius $r$, there exist one (or more) with maximum area. If so, determine their shape and area.
1967 IMO Longlists
Which fractions $ \dfrac{p}{q},$ where $p,q$ are positive integers $< 100$, is closest to $\sqrt{2} ?$ Find all digits after the point in decimal representation of that fraction which coincide with digits in decimal representation of $\sqrt{2}$ (without using any table).
1967 IMO Longlists
Suppose $\tan \alpha = \dfrac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. Prove that the number $\tan \beta$ for which $\tan {2 \beta} = \tan {3 \alpha}$ is rational only when $p^2 + q^2$ is the square of an integer.
1967 IMO Longlists
Prove the following statement: If $r_1$ and $r_2$ are real numbers whose quotient is irrational, then any real number $x$ can be approximated arbitrarily well by the numbers of the form $\ z_{k_1,k_2} = k_1r_1 + k_2r_2$ integers, i.e. for every number $x$ and every positive real number $p$ two integers $k_1$ and $k_2$ can be found so that $|x - (k_1r_1 + k_2r_2)| < p$ holds.