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European Math Tournament 2018-22 VII-VIII (Belarus) (EMT) 55p

geometry problems from European Math Tournament from Belarus with aops links in the names

it started in 2018, 
collected inside aops here

Seniors stands for 7-8 grades

2018 - 2022

Seniors Team Contests

2018 European Math Tournament Seniors Team p5
Four points $A, B, C, D$ are such that $D$ and $C$ lie on one side relative to the line $AB$ and the equality $AB = AD + BC$ holds. The bisectors of the angles $ABC$ and $BAD$ intersect at point $E$. Prove that $CE = DE$.

There is a convex quadrilateral $ABCD$ on the plane such that $\angle B = \angle C, \angle D = 90^o$ and $AB = 2CD$. Prove that the angle bisector of $\angle ACB$ is perpendicular to the line $CD$.

Point $P$ is marked on the $CD$ side of trapezoid $ABCD$ ($AD\parallel  BC$) . It turned out that $AP$ and $BP$ are the bisectors of angles $A$ and $B$, respectively. Let $BC = m$ and $AD = n$. Find $AB$.

Point $P$ is marked inside the square $ABCD$ so that $AP = AB$ and $\angle CPD = 90^o$. Prove that $DP=2CP$.

In the triangle $ABC$, point$ D$ is the midpoint of $AB$, and point $E$, divides the side $BC$ in the ratio $2:1$, counting from point $B$. It turned out that $\angle BAE = \angle CDA$. Find the angle $\angle BAC$.


Seniors Math Battles

2018 European Math Tournament Seniors Math Battle 1.2
In the quadrangle $ABCD$, the sides $AB$ and $CD$ are parallel, and the diagonals $AC$ and $BD$ are perpendicular. Prove that $AB + CD \le AD + BC$.

2018 European Math Tournament Seniors Math Battle 2.2
A triangle $ABC$ is given, in which $\angle A = 30^o, \angle B = 40^o$. On side $AB$, point $D$ is taken such that $\angle BDC = 80^o$. Prove that $AD = BC$.

2018 European Math Tournament Seniors Math Battle 3.2
A triangle $ABC$ is given in which $\angle B = 120^o$. On sides $AB$ and $BC$, points $E$ and $F$, respectively, are taken such that $AE = EF = FC$. Let $M$ be the midpoint of $AC$. Find $\angle  EMF$.

2018 European Math Tournament Seniors Math Battle 4.2
Take the paper square $ABCD$. Let $M$ be the midpoint of $AD$. We bend the square so that the fold passes through the point $M$, and the midpoint of $AB$ falls on the diagonal $BD$. Bend the sheet, the segment of the fold is denoted as $ME$. Prove that the segment $ME$ is equal to the side of the square $ABCD$.

2018 European Math Tournament Seniors Math Battle 5.2
$ABCD$ and $AEDF$ are rectangles, the straight line $DE$ divides the $BC$ side in half. Prove that triangle $ABE$ is isosceles.
In a triangle $ABC$, the side $AC$ is shorter than the side $AB$, and one of the corners, into which the median $AF$ divides the angle $A$ is twice the other (i. e. either $\angle FAB = 2\angle FAC$ or $\angle F AC = 2\angle FAB$). The point $D$ on the line $AF$ is such that $BD \perp AB$. Prove that $AD = 2AC$.

A square $ABCD$ is given. Point $P$ is selected inside $ABCD$ such that $AP = AB$ and $\angle CPD = 90^o$. Prove that $DP = 2CP$.

In a convex quadrilateral $ABCD, \angle B = 50^o, \angle A = 80^o$. There is a point $K$ marked on the extension of the side $AB$ beyond the point $A$ such that $AK = CD$. It turns out that $KD \perp BC$. Find out the value of the angle between $AC$ and bisector of the angle $BCD$.

There is a point $C$ marked on a segment $AB$. Distinct points $X$ and $Y$ are such points that the triangles $ABX$ and $ACY$ are equilateral. Denote the midpoints of $YC$ by $K$ and the midpoint of $BX$ by $M$ respecively. It turns out that the triangle $AMK$ is a right triangle. Find the quotient $AC/BC$.

You are given an acute-angled triangle $ABC$. Let us denote by $\ell$ the perpendicular bisector of $AC$. A straight line parallel to $BC$ and passing through point $A$ intersects $\ell$ at point $X$, and a line parallel to $AB$ and passing through point $C$ intersects $\ell$ at point $Y$. It turned out that $AX$ divides $BY$ in half. Prove that the altitude of triangle $ABC$ from vertex $B$, divides $AY$ in half.

In convex quadrilateral $ABCD$, the sides $DA$ and $BC$ have been extended by their lengths at points $A$ and $C$. Obtained points $P$ and $Q$. It turned out that the diagonal $BD$ intersects the segment $PQ$ in its midpoint $K$. Let $M$ be the midpoint of $BD$. Prove that $AKCM$ is a parallelogram.

Points $X$ and $Y$ were marked inside the rectangle so that the distances from point $X$ to the sides of the rectangle are related as $1: 2: 3: 4$ (in some order), and from the point $Y$ - as $9: 10: 11: 12$ (in some order, possibly in another). Can the distances from the midpoint of segment XY to the sides of the rectangle to be treated as $5: 6: 7: 8$?

Given a square $ABCD$ with side $AB = 1$. Points $M$ are marked on sides $AB$ and $AD$ and $N$, respectively, so that $\angle ABN + \angle MCN + \angle MDA = 90^o$. Prove that the perimeter triangle $AMN$ is less than $2$.

On the side $BC$ of a right-angled triangle $ABC$ with a right angle $B$ there was such point $K$ such that $2 (AB + BK) = KC$ and $\angle AKB = 60^o$. Find the angles of triangle $ABC$.

You are given a convex quadrilateral $ABCD$. Its sides $AB$ and $CD$ have been extended beyond points $B$ and $D$ to their lengths, receiving points $P$ and $R$. Sides $BC$ and $DA$ extended beyond points $C$ and $A$ by doubled lengths, obtaining points $Q$ and $S$. Lines $PQ$ and $RS$ intersect line $AC$ at points $X$ and $Y$. Find the ratio XY: AC

Two rays are drawn in square $ABCD$ from vertex $A$, dividing the angle by three equal parts. One ray intersects the diagonal $BD$ at point $N$, the second intersects the extension of $BC$ at point $M$. Through point $B$, draw a line perpendicular to $BD$ and intersecting line $MN$ at point $K$. Find $\angle KAC$.

In an isosceles triangle $ABC$ ($AB = AC$), the angle A is obtuse. Point $M$ on the extension of side $AC$ beyond point $C$ is such that $AC = CM$. Perpendicular bisector of the segment $AM$ intersects line $AB$ at point $P$. It turned out that lines $PM$ and $BC$ are perpendicular. Prove that $APM$ is an equilateral triangle.

A square is inscribed in the triangle so that on one side of the triangle there are two the vertices of the square, on the other two - one each. It turned out that the center of the square and the point the intersections of the medians of the triangle coincided. Find the angles of the triangle.

You are given an isosceles triangle $PQR$ with an apex angle $Q$ equal to $108^o$. Point $O$ is located inside the triangle $PQR$ so that$ \angle ORP = 30^o$ and $\angle OPR = 24^o$. Find the value of angle $\angle QOR$.

You are given a trapezoid $ABCD$ ($AD \parallel BC$). Points $E$ and $K$ are selected on sides $AB$ and $AD$ respectively. It is known that $CK$ is perpendicular to $AD$, $\angle  DCE = \angle ABC$ and $AB + 2AK =AD + BC$. Prove that $BE = BC$,

In triangle $ABC$, $\angle A = 2 \angle C$. On the side $AB$, a point $M$ is chosen such that $AM = AC$. It turned out that $CM = AB$. Find the angles of triangle$ ABC$.

In square $ABCD$, point $M$ is the midpoint of $BC$. Point $L$ on the side of $CD$ is such that $AM$ is bisector of $\angle BAL$. Point $S$ on $AD$ is such that $\angle LMS = 45^o$. Prove that the segment $MS$ is divided in half by $AL$.

In a right-angled triangle $ABC$, points $M$ and $N$ are taken on the legs $AB$ and $BC$, respectively, so that $AM=CB$ and $CN=MB$. Find the acute angle between line segments $AN$ and $CM$.

The two squares in the figure have a common side $AB$. A point $K$ is taken on the diagonal of one of them, the distance from which to the vertex $C$ of the other square is equal to its diagonal. Find the angle $\angle ACK$.


Point $D$ is marked on side $BC$ of triangle $ABC$ so that $CD = 2BD$. It turned out that $\angle  ADC = 60^o$ and $\angle ABC = 45^o$. Find $\angle BAC$.

The figure shows three squares. Find the marked angle.



Seniors Math Abaka

2018 European Math Tournament Geo 1
Each point of the inner square is $1$ cm from the nearest point of the outer square. It is known that the area between the squares is $20$ cm$^2$. Find the area of the smaller square.

2018 European Math Tournament Geo 2
On the side $AC$ of the isosceles triangle $ABC$ ($AB = AC$), the points $P$ and $Q$ are selected ($P$ lies between $A$ and $Q$). It turned out that $PQ = BQ$ and $\angle CBQ = \angle PBA$. What can be the angle  $\angle PBC$?

2018 European Math Tournament Geo 3
On the side $CD$ of the rectangle $ABCD$, point $E$ is chosen. The point symmetric to $C$ wrt segment $BE$ lies on the midline of the rectangle parallel to side $AB$. Find the angle  $\angle BED$.

2018 European Math Tournament Geo 4
Calculate the angles of an isosceles triangle if its height drawn to the base is half the bisector of the angle at the base.

In a triangle, the angles of which have the ratio $1: 2: 4$, all the bisectors are drawn. What is the largest number of isosceles triangles that can be identified in the resulting drawing?

What is the largest number of sides a figure can have that is a common part a triangle and a convex quadrilateral? Draw an example.

original wording
Какое наибольшее число сторон может иметь фигура, являющаяся общей частью треугольника и выпуклого четырехугольника? Нарисуйте пример
Draw two straight lines through the squares of $3\times 3$ so that all the cells of this square are cut.

Victor cut a triangle made of cardboard into two triangles and sent both parts to Pete, who again folded a triangle out of them. It turned out that Pete’s triangle is not equal to Victor’s. Give an example of a triangle and its cutting, as well as how to fold a new triangle, not equal to the old one.

Vanya divided a plane on equilateral triangles, the cutting scheme is shown in the figure. Help him find $\angle ABC$.
At what time between $4:00$ and $5:00$ the angle between the hour and minute hands will be $21^o$? If this can happen several times, then bring the very first.

A square consists of one white square and four equal grey rectangles. The perimeter of each rectangle is equal to $40$ cm. What is the area of big square?
What is the minimal number of three-cell corners that can be placed on the $8 \times  8$ board so that one cannot add another such corner? Give an example of this amount.

In a triangle, one of the angles is twice the second and differs $20$ degrees from the third. What values (in degrees) can take the greatest angle of such a triangle?

As a result of measuring four sides and one of the diagonals of some of the quadrilateral, the numbers $15, 23, 36, 50$ and $72$ were obtained. What could be equal to length of the measured diagonal?

What is the smallest number of cells of a $5\times 5$ square that can be painted over so that any four-cell rectangle had at least one filled cell? Give an example

A checkered square with side $n$ was cut along the borders of cells by more than $n$ rectangles of different areas, while none of the parts are square. When what is the smallest $n$ that is possible? Find the answer and give an example cutting. 

The rope was bent five times, then what happened was bent in half. There after made a transverse cut (not coinciding with the fold lines). The rope fell apart into pieces, the lengths of two of which were equal to $4$ cm and $7$ cm. Find a possible rope length (all values).

$AH$ and $CP$ are the altitudes of an isosceles triangle $ABC$. What could be the value of the angle $B$ if it is known that $AB = BC$ and $AC = 2HP$?

Two parallel straight lines were drawn on a square sheet of paper, the sheet was bent along the one line, then along the other and pierced at one place (not falling on the fold line). Thereafter the sheet was unbent. How many different holes could there be on the sheet?

On one of the sides of the triangle, $60$ points are taken, and on the other one $50$ points (all different from the vertices of the triangle). Each of the vertices at the base was connected by segments with all points on the opposite side. How many triangles turned out among the parts on which it turned out is the original triangle broken?

Cut a $10\times 10$ square into $16$ $2\times  3$ rectangles and one $4$ square shape type letter $\Gamma$.

Vasya drew an $8\times 8$ square on a checkered sheet and cut out four corner cells from it. What is the longest length of a closed self-non-intersecting polyline going along grid lines of the resulting shape (including along the edge)?

Three squares were drawn on a sheet, adjacent to each other, with sides $1,2$ and $x$. It turned out that a straight line drawn both on figure, divides a $2\times 2$ square into $2$ parts of equal area. Find $x$.

In a convex quadrilateral $ABCD,$ the bisectors of the angles $A$ and $B$ intersect in the midpoint of side $CD$, and angle $C$ is $70^o$. Find angle $D$.

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