Ukraine From Tasks to Tasks 2010-16 V-IX 22p

 geometry problems from and Ukrainian Contest named as From Tasks to Tasks (Від задачок до задач) for grades 5-9 ( by magazine ''In the World of Mathematics''' ) with aops links

Від задачок до задач

collected inside aops here 

all problems in Ukrainian with solutions here

                                                    2010 - 2016

                                                                      (it lasted only these years)

2010 From Tasks to Tasks, Ukraine p5

In a right triangle $ABC$ ($\angle C = 90^o$) it is known that $AC = 4$ cm, $BC = 3$ cm. The points $A_1, B_1$ and $C_1$ are such that $AA_1 \parallel BC$, $BB_1\parallel A_1C$, $CC_1\parallel A_1B_1$, $A_1B_1C_1= 90^o$, $A_1B_1= 1$ cm. Find $B_1C_1$.

2010 From Tasks to Tasks, Ukraine p9

On the sides $AB, BC, CD$ and $DA$ of the parallelogram $ABCD$ marked the points $M, N, K$ and $F$. respectively. Is it possible to determine, using only compass, whether the area of the quadrilateral $MNKF$ is equal to half the area of the parallelogram $ABCD$?

2010 From Tasks to Tasks, Ukraine p13

You can inscribe a circle in the pentagon $ABCDE$. It is also known that $\angle ABC = \angle BAE = \angle CDE = 90^o$. Find the measure of the angle $ADB$.

2011 From Tasks to Tasks, Ukraine p3

Let $O$ be the center of the circumcircle, and $AD$ be the angle bisector of the acute triangle $ABC$. The perpendicular drawn from point $D$ on the line $AO$ intersects the line $AC$ at the point $P$. Prove that $AP = AB$.

2011 From Tasks to Tasks, Ukraine p8

On the median $AD$ of the isosceles triangle $ABC$, point $E$ is marked. Point $F$ is the projection of point $E$ on the line $BC$, point $M$ lies on the segment $EF$, points $N$ and $P$ are projections of point $M$ on the lines $AC$ and $AB$, respectively. Prove that the bisectors of the angles $PMN$ and $PEN$ are parallel.

2011 From Tasks to Tasks, Ukraine p14

The lengths of the four sides of an cyclic octagon are $4$ cm, the lengths of the other four sides are $6$ cm. Find the area of the octagon.

2012 From Tasks to Tasks, Ukraine p2

The triangle $ABC$ is equilateral. Find the locus of the points $M$ such that the triangles $ABM$ and $ACM$ are both isosceles.

2012 From Tasks to Tasks, Ukraine p4

Let $ABCD$ be an isosceles trapezoid ($AD\parallel BC$), $\angle BAD = 80^o$, $\angle BDA = 60^o$. Point $P$ lies on $CD$ and $\angle PAD = 50^o$. Find $\angle PBC$

2012 From Tasks to Tasks, Ukraine p9

In the triangle $ABC$, the angle $A$ is equal to $60^o$, and the median $BD$ is equal to the altitude $CH$. Prove that this triangle is equilateral.

2012 From Tasks to Tasks, Ukraine p13

The sides of a triangle are consecutive natural numbers, and the radius of the inscribed circle is $4$. Find the radius of the circumscribed circle.

2013 From Tasks to Tasks, Ukraine p4

The trapezoid is composed of three conguent right isosceles triangles as shown in the figure. It is necessary to cut it into $4$ equal parts. How to do it?

2013 From Tasks to Tasks, Ukraine p9

The perpendicular bisectors of the sides $AB$ and $CD$ of the rhombus $ABCD$ are drawn. It turned out that they divided the diagonal $AC$ into three equal parts. Find the altitude of the rhombus if $AB = 1$.

2013 From Tasks to Tasks, Ukraine p13

In the quadrilateral $ABCD$ it is known that $ABC + DBC = 180^o$ and $ADC + BDC = 180^o$. Prove that the center of the circle circumscribed around the triangle $BCD$ lies on the diagonal $AC$.

2014 From Tasks to Tasks, Ukraine p4

In the triangle $ABC$ it is known that $AC = 21$ cm, $BC = 28$ cm and $\angle C = 90^o$. On the hypotenuse $AB$, we construct a square $ABMN$ with center $O$ such that the segment $CO$ intersects the hypotenuse $AB$ at the point $K$. Find the lengths of the segments $AK$ and $KB$.

2014 From Tasks to Tasks, Ukraine p9

On a circle with diameter $AB$ we marked an arbitrary point $C$, which does not coincide with $A$ and $B$. The tangent to the circle at point $A$ intersects the line $BC$ at point $D$. Prove that the tangent to the circle at point $C$ bisects the segment $AD$.

2014 From Tasks to Tasks, Ukraine p15

Construct a right triangle given the hypotenuse and the median drawn to the leg.

2015 From Tasks to Tasks, Ukraine p5

A coordinate system was constructed on the board, points $A (1,2)$ and B $(3, 1)$ were marked, and then the coordinate system was erased. Restore the coordinate system at the two marked points.

2015 From Tasks to Tasks, Ukraine p10

Can the sum of the lengths of the median, angle bisector and altitude of a triangle be equal to its perimeter if

a) these segments are drawn from three different vertices?

b) these segments are drawn from one vertex?

2015 From Tasks to Tasks, Ukraine p14

On the side $AB$ of the triangle $ABC$ mark the points $M$ and $N$, such that $BM = BC$ and $AN = AC$. Then on the sides $BC$ and $AC$ mark the points$P$ and $Q$, respectively, such that $BP = BN$ and $AQ = AM$. Prove that the points $C, Q, M, N$ and $P$ lie on the same circle.

2016 From Tasks to Tasks, Ukraine p3

In fig. the bisectors of the angles $\angle DAC$, $\angle EBD$, $\angle ACE$, $\angle BDA$ and $\angle CEB$ intersect at one point. Prove that the bisectors of the angles $\angle TPQ$, $\angle PQR$, $\angle QRS$, $\angle RST$ and $\angle STP$ also intersect at one point.

2016 From Tasks to Tasks, Ukraine p8

Let $ABCD$ be a convex quadrilateral. It is known that $S_{ABD} = 7$, $S_{BCD}= 5$ and $S_{ABC}= 3$. Inside the quadrilateral mark the point $X$ so that $ABCX$ is a parallelogram. Find $S_{ADX}$ and $S_{BDX}$.

2016 From Tasks to Tasks, Ukraine p13

Let $ABC$ be an isosceles acute triangle ($AB = BC$). On the side $BC$ we mark a point $P$, such that $\angle PAC = 45^o$, and $Q$ is the point of intersection of the perpendicular bisector of the segment $AP$ with the side $AB$. Prove that $PQ \perp BC$.

source: http://www.mechmat.univ.kiev.ua/