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Chisinau City MO 1949-56, 1973-79 (Moldova) 63p

geometry problems from Chisinau City Mathematical Olympiads (Moldova) with aops links in the names



1949-56, 1973-79 


The sides of the triangle $ABC$ satisfy the relation $c^2 = a^2 + b^2$. Show that angle $C$ is right.

Show that in a right-angled triangle the bisector of the right angle divides into equal parts the angle between the altitude and the median, drawn from the same vertex.

Inside the angle $ABC$ of $60^o$, point $O$ is selected, which is located at distances from the sides of the angle $a$ and $b$, respectively. Determine the distance from the top of the angle to this point.

Show that a line passing through the feet of two altitudes of an acute-angled triangle cuts off a similar triangle.

Show that the straight lines passing through the feet of the altitudes of an acute-angled triangle form a triangle in which the altitudes of the original triangle are angle bisectors.

Formulate a criterion for the conguence of triangles by two medians and an altitude.

The areas of two right-angled triangles have ratio equal to the squares of their hypotenuses. Show that these triangles are similar.

Prove the inequality $2\sqrt{(p-b)(p-c)}\le a$, where $a, b, c$ are the lengths of the sides, and $p$ is the semiperimeter of some triangle..

Let $M$ be an arbitrary point of a circle circumscribed around an acute-angled triangle $ABC$. Prove that the product of the distances from the point $M$ to the sides $AC$ and $BC$ is equal to the product of the distances from $M$ to the side $AB$ and to the tangent to the circumscribed circle at point $C$.

Through the point of intersection of the diagonals of the trapezoid, a straight line is drawn parallel to its bases. Determine the length of the segment of this straight line, enclosed between the lateral sides of the trapezoid, if the lengths of the bases of the trapezoid are equal to $a$ and $b$.

Find the locus of the points that are the midpoints of the chords of the secant to the given circle and passing through a given point.

Determine the locus of points that are the midpoints of segments of equal length, the ends of which lie on the sides of a given right angle.

Construct a triangle, the base of which lies on the given line, and the feet of the altitudes, drawn on the sides, coincide with the given points.

Construct a triangle by its altitude , median and angle bisector originating from one vertex.

Prove that the bisectors of the angles of a rectangle, extended to their mutual intersection, form a square.

A trapezoid and an isosceles triangle are inscribed in a circle. The larger base of the trapezoid is the diameter of the circle, and the sides of the triangle are parallel to the sides of the trapezoid. Show that the trapezoid and the triangle have equal areas.

On the radius $OA$ of a certain circle, as on the diameter, a circle is built. A ray is drawn from the center $O$, intersecting the larger and smaller circles at points $B$ and $C$, respectively. Show that the lengths of arcs $AB$ and $AC$ are equal.

Determine the locus of points, for each of which the difference between the squares of the distances to two given points is a constant value.

Determine the locus of points, from which the tangent segments to two given circles are equal.

Determine the locus of points, for whom the ratio of the distances to two given points has a constant value.

On the plane $n$ points are chosen so that exactly $m$ of them lie on one straight line and no three points not included in these $m$ points lie on one straight line. What is the number of all lines, each of which contains at least two of these points?

Show that triangle $ABC$ is right-angled if its angles satisfy the ratio $\cos^2A + \cos ^2B +\ cos ^2C=1$.

Show that the sum of the distances from any point of a regular tetrahedron to its faces is equal to the height of this tetrahedron.

Find the locus of the projections of a given point on all planes containing another point $B$.

On two intersecting lines $\ell_1$ and $\ell_2$, segments $AB$ and $CD$ of a given length are selected, respectively. Prove that the volume of the tetrahedron $ABCD$ does not depend on the position of the segments $AB$ and $CD$ on the lines $\ell_1$ and $\ell_2$.


Each point in space is colored in one of four different colors. Prove that there is a segment $1$ cm long with endpoints of the same color.

A finite number of chords is drawn in a circle $1$ cm in diameter so that any diameter of the circle intersects at most $N$ of these chords. Prove that the sum of the lengths of all chords is less than $3.15 \cdot  N$ cm.

If $A$ and $B$ are points of the plane, then by $A * B$ we denote a point symmetric to $A$ with respect to $B$. Is it possible, by applying the operation $*$ several times, to obtain from the three vertices of a given square its fourth vertex?

Inside the triangle $ABC$, point $O$ was chosen so that the triangles $AOB, BOC, COA$ turned out to be similar. Prove that triangle $ABC$ is equilateral.

The sides of the triangle $ABC$ lie on the sides of the angle $MAN$. Construct a triangle $ABC$ if the point $O$ of the intersection of its medians is given.

Through point $P$, which lies on one of the sides of the triangle $ABC$, draw a line dividing its area in half.

 Altitude $AH$ and median $AM$ of the triangle $ABC$ satisfy the relation: $\angle ABM = \angle CBH$. Prove that triangle $ABC$ is isosceles or right-angled.

Is there a moment in a day when three hands - hour, minute and second - of a clock running correctly form angles of $120^o$ in pairs?

Let $O$ be the center of the regular triangle $ABC$. Find the set of all points $M$ such that any line containing the point $M$ intersects one of the segments $AB, OC$.

a) Let $S$ and $P$ be the area and perimeter of some triangle. The straight lines on which its sides are located move to the outside by a distance $h$. What will be the area and perimeter of the triangle formed by the three obtained lines?
b) Let $V$ and $S$ be the volume and surface area of some tetrahedron. The planes on which its faces are located are moved to the outside by a distance $h$. What will be the volume and surface area of the tetrahedron formed by the three obtained planes?

We will say that a convex polygon $M$ has the property $(*)$ if the straight lines on which its sides lie, being moved outward by a distance of $1$ cm, form a polygon $M'$, similar to this one.
a) Prove that if a convex polygon has property $(*)$ , then a circle can be inscribed in it.
b) Find the fourth side of a quadrilateral satisfying condition $(*)$ if the lengths of its three consecutive sides are $9, 7$, and $3$ cm.

A closed line on a plane is such that any quadrangle inscribed in it has the sum of opposite angles equal to $180^o$. Prove that this line is a circle.
Construct a right-angled triangle along its two medians, starting from the acute angles.

A straight line $\ell$ and a point $A$ outside of it are given on the plane. Find the locus of the vertices $C$ of the equilateral triangle $ABC$, the vertex $B$ of which lies on the straight line $\ell$.

Let $M$ be the point of intersection of the diagonals, and $K$ be the point of intersection of the bisectors of the angles $B$ and $C$ of the convex quadrilateral $ABCD$. Prove that points $A, B, M, K$ lie on the same circle if the following relation holds: $|AB|=|BC|=|CD|$

Construct a square from four points, one on each side.

Prove that any centrally symmetric convex octagon has a diagonal passing through the center of symmetry that is not parallel to any of its sides.

Three squares are built on the sides of the triangle to the outside. What should be the angles of the triangle so that the six vertices of these squares, other than the vertices of the triangle, lie on the same circle?

The diagonals of some convex quadrilateral are mutually perpendicular and divide the quadrangle into $4$ triangles, the areas of which are expressed by prime numbers. Prove that a circle can be inscribed in this quadrilateral.

Five points are given on the plane. Prove that among all the triangles with vertices at these points there are no more than seven acute-angled ones.

The convex $1976$-gon is divided into $1975$ triangles. Prove that there is a straight line separating one of these triangles from the rest.

Let $O$ be the center of a circle inscribed in a convex quadrilateral $ABCD$ and $|AB|= a$, $|CD|=$c.
Prove that$$\frac{a}{c}=\frac{AO\cdot BO}{CO\cdot DO}.$$

A triangle with a parallelogram inside was placed in a square. Prove that the area of a parallelogram is not more than a quarter of a square.

Determine the angles of a triangle in which the median, bisector and altitude, drawn from one vertice, divide this angle into four equal parts.

In an isosceles triangle $BAC$ ($| AC | = | AB |$) , point $D$ is marked on the side $AC$. Determine the angles of the triangle $BDC$ if $\angle A = 40^o$ and $|BC|: |AD|= \sqrt3$.

Let $\beta$ be the length of the bisector of angle $B$, and $a', c'$ be the lengths of the segments into which this bisector divides the side $AC$ of the triangle $ABC$. Prove the relation $\beta^2 = ac-a'c'$ and derive from this the formula $\beta^2=ac-\frac{b^2ac}{(a+c)^2}$.

Prove that $n$ ($\ge 4$) points of the plane are vertices of a convex $n$-gon if and only if any $4$ of them are vertices of a convex quadrilateral.

Prove that the side $AB$ of a convex quadrilateral $ABCD$ is less than its diagonal $AC$ if $|AB|+|BC| \le |AC| +| CD|$.

Five points are selected on the plane so that no three of them lie on one straight line. Prove that some four of these five points are the vertices of a convex quadrilateral.

On the plane $n$ points are selected that do not belong to one straight line. Prove that the shortest closed path passing through all these points is a non-self-intersecting polygon.

Prove that the largest area of a triangle with sides $a, b, c$ satisfying the relation $a^2 +b^2 c^2 = 3m^2$, equals to $\frac{\sqrt3}{4}m^2$.

Find the largest possible number of intersection points of the diagonals of a convex $n$-gon.

Show that in a right-angled triangle the bisector of the right angle divides into equal parts the angle between the altitude and the median, drawn from the same vertex.

The inner angles of the pentagon inscribed in the circle are equal to each other. Prove that this pentagon is regular.

Is it possible to cut a square into five squares?

Prove that the bases of the altitudes and medians of an acute-angled triangle lie on the same circle.

Prove that if every line connecting any two points of some finite set of points of the plane contains at least one more point of this set, then all points of the set lie on one straight line.

Prove that a section of a cube by a plane cannot be a regular pentagon.

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