Euler Teachers' MO 2007-20 62p

geometry problems from Euler Teachers' Math Olympiad with aops links in the names

collected inside aops here
it started in 2007, it has two rounds,
first one is correspodence
2020  second round did not take place

2007-20

2007 Euler Teachers' Olympiad  I p4

In the tetrahedron $ABCD$, the edge $CD$ is perpendicular to the plane $ABC$. Point $N$ is the midpoint of the edge $AB$, point $M$ is the midpoint of the edge $BD$, point $K$ divides $DC$ edge in ratio $DK: KC= 2: 1$ . Prove that line $CN$ is equidistant from lines $BK$ and $AM$.

2007 Euler Teachers' Olympiad  I p12

Given an isosceles triangle with base $a$, lateral side $b$ and apex angle of $12^o$. Prove that $b<5a$

2007 Euler Teachers' Olympiad  II p5

Points $A_1$ and $B_1$ lie on the sides $BC$ and $AC$ of the triangle $ABC$, respectively, segments $AA_1$ and $BB_1$ intersect at $K$ (see picture). Find the area of the quadrangle $A_1CB_1K$, if it is known that the area of triangle $A_1BK$ is $5$, the area of the triangle $AB_1K$ is $8$, and the area of the triangle $ABK$ is $10$.

2008 Euler Teachers' Olympiad I p6

A line divides the triangle into two new shapes with equal perimeters and areas. Prove that the center of the circle inscribed in the triangle lies on this segment.

2008 Euler Teachers' Olympiad I p7

All faces of the tetrahedron are triangles, the lengths of the sides of which are $a, b$ and $c$. Find the volume of the tetrahedron.

2008 Euler Teachers' Olympiad I p10
Prove the inequality $a^2+b^2+c^2\le 9R^2$ if $a,b,c$ are the lengths of the sides of the triangle $ABC$, and $R$ is the radius of the circle circumscribed around this triangle.

2008 Euler Teachers' Olympiad I p12

The midpoint of each side of the base of the quadrangular pyramid is connected with a segment to the intersection point of the medians of the opposite side face. Prove that:

a) these segments intersect and the intersection point are divided in a ratio of $3: 2$, counting from the side of the foundation;
b) the midpoints of these segments are vertices of parallelogram.
Find the ratio of the area of this parallelogram to the area the base of the pyramid.

2008 Euler Teachers' Olympiad  II p5

Find the volume of the solid obtained by rotating the cube with an edge equal to $a$, around a line containing the diagonal of one of the faces of the cube.

2009 Euler Teachers' Olympiad I p4

Lines parallel to the sides of the triangle and touching the inscribed in it circle, cut off from him three smaller triangles. Prove that the sum areas of cut off triangles not less than a third of the area of the original the triangle.

2009 Euler Teachers' Olympiad I p5
There is an tetrahedron which has both an inscribed and a circumscribed sphere.  $3$ edges of the tetrahedron are pairwise perpendicular. Find the radius of the inscribed sphere if the radius of the circumscribed sphere is $3\sqrt3$.

2009 Euler Teachers' Olympiad I p11
Prove that in any triangle the length of the bisector does not exceed the length median drawn from the same vertex.

2009 Euler Teachers' Olympiad II p4

The radii of the excircles of a triangle are equal, respectively $2, 3$ and $6$ cm. Find the radius of the circle inscribed in this triangle.

2009 Euler Teachers' Olympiad  II p5

A  plane divides the medians of the faces $ABC, ACD$, and $ABD$ of a tetrahedron $ABCD$ coming from the vertex $A$ into ratios $1: 2$, $1: 1$ and $1: 2$ (counting from point $A$). Find the ratio of the volumes of the parts, on which this plane divides the given tetrahedron .

2010 Euler Teachers' Olympiad I p5

Point $K$ lies on side $BC$ of the parallelogram $ABCD$, and point $M$ lies on side $AD$
Segments $CM$ and $DK$ intersect at point $L$, and segments $AK$ and $BM$ intersect at point $N$.
Find the largest value of the ratio of area of the quadrangles $KLMN$ and $ABCD$.

2010 Euler Teachers' Olympiad I p6

Through the vertex $D$ of the rectangular parallelepiped $ABCDA_1B_1C_1D_1$ held a parallelepiped
section by a plane intersecting the edges $AA_1,BB_1$, and $CC_1$ at points $K, M$, and $N$,
respectively. Find ratio of the volume of the pyramid with the vertex at point $P$ and the base $DKMN$
to the volume of the given parallelepiped if it is known that the point $P$ divides the edge $DD_1$ with
respect to $DP: PD_1= m:n$.

2010 Euler Teachers' Olympiad I p9

The side edges $PA, PB, PC$ of the pyramid $PABC$ are equal to $2, 2$, and $3$, respectively the base
of the ABC is a regular triangle. It is known that the area of the lateral faces of the pyramid are equal to
each other. Find the volume of the pyramid $PABC$ .

2010 Euler Teachers' Olympiad I p11

Prove that in every quadrilateral the segments connecting the midpoints its opposite sides, as well as the

segment connecting the midpoints of the diagonals pass through one point and divide it in half.

2010 Euler Teachers' Olympiad II p5

The two triangular pyramids $MABC$ and $NABC$ have a common base and do not have other common points. All the vertices of both pyramids lie on the same sphere. Find the lengths of the edges $MA$ and $MB$, if it is known that they are equal, and the lengths of all other edges of both pyramids are $\sqrt3$.

2011 Euler Teachers' Olympiad I p4
Two disjoint circles are arranged so that one of their common internal tangents are perpendicular to one of their common external tangents. Find the area of the triangle formed by these tangents and the third common tangent of given circles if their radii are $r_1$ and $r_2$.

2011 Euler Teachers' Olympiad I p6
A sphere touches all the edges of a tetrahedron, whose two opposite edges are equal to $a$ and $b$, and all other edges are equal to each other. Find the radius of this sphere.

2011 Euler Teachers' Olympiad I p10
A square $ABDE$ is constructed on the hypotenuse $AB$ of a right triangle $ABC$ in the half-plane to which triangle $ABC$ does not belong. Find the distance from vertex $C$ to the center of the square, if it is known that $BC=a$ and $AC= b$

2011 Euler Teachers' Olympiad I p12
Prove the inequality $P> 4R$, where$P$ is the perimeter and $R$ is the radius of the circumscribed circumference of an acute-angled triangle.

2011 Euler Teachers' Olympiad II p3
The base of the $PABCD$ pyramid is the trapezoid $ABCD$, whose base $AD$ is twice the base BC. Line $MN$ os the midline of the triangle $ABP$ parallel to side $AB$. Find the ratio of the volumes of solids on which the plane $DMN$ divides the given pyramid.

2012 Euler Teachers' Olympiad I p6
Prove that the sum of the squares of the distances from the vertices of a regular heptagon to an arbitrary straight line passing through its center does not depend on the position of this line.

2012 Euler Teachers' Olympiad I p7
A circle and a parabola have exactly two common points, one of which is the point of tangency of this parabola and this circle. Is it true in general, that the second common point of circle and parabola, is also their point of tangency?

2012 Euler Teachers' Olympiad I p8
Find the volume of the pyramid $PABCD$ at the base of which is quadrilateral $ABCD$ with sides $5, 5, 10$ and $10$, the smaller diagonal of which is equal to $4\sqrt5$, if it is known that all the side faces of the pyramid are inclined to the plane bases at an angle of $45^o$.

2012 Euler Teachers' Olympiad II p3

Construct a line passing through a vertex of a convex quadrilateral dividing it into two equal parts (areas).

2012 Euler Teachers' Olympiad II p4

a) Prove that any section of a regular tetrahedron by a plane passing through the midpoints of its two crossing edges, divides the tetrahedron into two equal parts.

b) Prove that the statement formulated in part (a) is true for any (not necessarily regular) tetrahedron .

2012 Euler Teachers' Olympiad II p7

A quadrilateral is inscribed in a rectangle so that on each side the rectangle lies one vertex of the quadrilateral . Prove that the perimeter of the quadrilateral at least twice the diagonal of the rectangle.

2013 Euler Teachers' Olympiad I p4
Let $A, B$ and $C$ be the vertices of an equilateral triangle, $P$ be an arbitrary point in the plane of this triangle. Prove that $AP\le BP+CP$.

2013 Euler Teachers' Olympiad I p5
In a regular quadrangular pyramid, the cosine of the angle between opposite side faces is $8/9$ . Find the section of this pyramid with a plane, passing through the edge of the base, the area of which is the largest.

2013 Euler Teachers' Olympiad I p9
Two circles of radii $4$ and $8$ touch at point $A$. Through point $A$, a line that intersects the larger circle at point $B$, and the smaller circle at point $C$. Find $AB$ if it is known that $BC = 6\sqrt2$.

2013 Euler Teachers' Olympiad I p12
Construct a triangle $ABC$ given its two sides $AC$ and $AB$, if it is known that $\angle BAC = 2 \angle ABC$ .


2013 Euler Teachers' Olympiad II p3

Prove that there is a triangular pyramid with face areas equal to $3, 4, 5$ and $10$.

2014 Euler Teachers' Olympiad I p2
The side of the square is a. The midpoints of the sides of the square are connected by segments with its opposite vertices (see figure). Find the area of the resulting octagon.

2014 Euler Teachers' Olympiad I p6
A sphere of radius $R$ is inscribed into a tetrahedron. Tangents are drawn to this ball planes parallel to the faces of the tetrahedron. These planes are cut off from the original tetrahedron $4$ smaller tetrahedra, each of which also contains a sphere. The radii of these balls are $R_1,R_2,R_3$ and $R_4$. Prove that $R_1+R_2+R_3+R_4=2R$.

2014 Euler Teachers' Olympiad I p9
Construct a straight line through vertex $A$ of triangle $ABC$ so that the sum the distance from this line to points $B$ and $C$ is the greatest.

2014 Euler Teachers' Olympiad II p5
Points$K, L$ and $M$ are selected on the sides $AB, BC$ and $AC$ of triangle $ABC$, respectively so that the segments $AL, BM$ and $CK$ intersect at one point and $\overrightarrow{AL}+\overrightarrow{BM} + \overrightarrow{CK}= \overrightarrow{0}$ . Prove that these points are the midpoints of the sides of the given triangle.

2015 Euler Teachers' Olympiad I p2
A quadrilateral is said to be equidiagonal if its diagonals are equal. Prove that if the segment connecting the midpoints of opposite sides convex quadrilateral $ABCD$, divides it into two equidiagonal quadrilateral, then quadrilateral $ABCD$ is also equidiagonal.

2015 Euler Teachers' Olympiad I p7
At the base of the pyramid of volume $V$ lies a parallelogram. Side rib lengths of this pyramid are different and different from the lengths of the ribs of its base. Find the volume a triangular pyramid made up of the side faces of this pyramid.

2015 Euler Teachers' Olympiad II p2

In the $ABCD$ tetrahedron: $AB = 8, BC = 10, AC = 12, BD = 15$. It is known that four segments connecting the vertices of the tetrahedron with the centers of the circles inscribed in opposite faces intersect at one point. Find the lengths of edges $DA$ and $DC$

2016 Euler Teachers' Olympiad I p5
At the base of the pyramid with apex $S$ lies the square $ABCD$. It is known that $AS = 7, BS= DS = 5$. What values can the side length of the square $ABCD$ take? Also find the length of the side edge $CS$ of such a pyramid.

2016 Euler Teachers' Olympiad I p9 
In triangle $ABC$, bisectors $AD$ and $BE$ are drawn. It turned out that $DE$ is the bisector of triangle $ADC$. Find the angle $\angle BAC$

2016 Euler Teachers' Olympiad I p11
A triangle $ABC$ with sides $BC=a,AC=b, AB=c$ is inscribed in a circle. The tangent to this circle at point $C$ is perpendicular to the straight line $AB$. Prove that $(b^2-a^2)^2=c^2(a^2+b^2)$.

2016 Euler Teachers' Olympiad II p4
At the base of the quadrangular pyramid $PABCD$ lies the parallelogram $ABCD$. On the lateral edges $PC$ and $PD$, points $M$ and $N$ are selected, respectively, so that points $A, B, M$ and $N$ lie in the same plane and the volume of the $PABMN$ pyramid is $4.5$ times less than the volume pyramids $PABCD$ (ratio of volumes $=2:9$). Find the ratio $PM: MC$.

2017 Euler Teachers' Olympiad I  p6
Find the smallest area of the common part of a regular octagon and its image when rotated by some angle wrt to the center of the original octagon.

2017 Euler Teachers' Olympiad I  p8
Find the length of a side $AB$ of a triangle $ABC$ if it is known that $AC = 4,BC = 6$ and angle $A$ is twice the angle $B$

2017 Euler Teachers' Olympiad I  p11
Let $B$ and $C$ be points of tangency with a circle of radius $R$ tangent to it, drawn from point $A$. Let also $r$ be the radius of the circle inscribed in triangle $ABC$, and $r_a$ is the radius of
its excircle tangent to side $BC$ (equal to $a$). Prove that $4R^2=r^2+r_a^2+\frac{a^2}{2}$

2017 Euler Teachers' Olympiad II  p6
a) Prove that any trihedral angle with vertex $P$ has a ray $PO$ making equal angles to the edges of this triangular corner.
b)  Let each of the rays $AK, BL, CM$ and $DN$ in the tetrahedron $ABCD$ make equal corners with the edges of "their" triangular corner. Formulate and prove at least one necessary and sufficient condition for rays $AK, BL, CM$ and $DN$ intersect at one point. What planimetry theorem is analogous to this statement?

2018 Euler Teachers' Olympiad I p5 
Consider a tetrahedron $ABCD$ whose altitude $DH$ passes through the intersection point of altitudes of triangle $ABC$.
a) Prove that the opposite edges of this tetrahedron are perpendicular.
b) Prove that the lines on which the altitudes of this tetrahedron lie pass through one point.
c) Find out whether the converse is true: “If the lines on which lie the altitudes of the tetrahedron pass through one point, then the bases of the altitudes of this the tetrahedron are the points of intersection of the heights of the corresponding faces. ''

2018 Euler Teachers' Olympiad I p12
The regular heptagon $ABCDEFG$ is inscribed in the unit circle. Prove that the squares of the lengths of the segments $AB$, $AC$ and $AD$ are the roots of the equation $x^3-7x^2+14x-7=0$

2018 Euler Teachers' Olympiad II p5
Given a tetrahedron with lengths of edges equal to $1$. Prove that there exists on the surface of this tetrahedron a set of segments with total length less than $1+\sqrt3$, such that any two vertices of the tetrahedron are connected by a polyline consisting of these segments.

2018 Euler Teachers' Olympiad II p7
Let $a, b, c$ be the lengths of the sides of the triangle, $r$ be the radius of the inscribed circle to this triangle. Prove that if $r=\frac{a+b-c}{2}$, then this triangle is right-angled."

2019 Euler Teachers' Olympiad I p1
In triangle $ABC$, the lengths of two sides are known: $AB = 1$ and $AC = 3$. Find all values that can take:
a) the length $h$ of the altitude dropped from the vertex $A$,
b) length $m$ of the median drawn from the vertex $A$,
c) the length $l$ of the bisector drawn from the vertex $A$.
Justify your answers.

2019 Euler Teachers' Olympiad I p9
The sides of the triangle $ABC$ are known: $AB = 5, BC = 6, AC = 7$. The circle passing through points $A$ and $C$ intersects lines $BA$ and $BC$ at points $E$ and $F$, respectively, different from the vertices of the triangle. Segment $EF$ touches the inscribed circle in triangle $ABC$. Find the length of the line segment$EF$.

2019 Euler Teachers' Olympiad I p10
Prove that there are no equilateral triangles on the plane, all the coordinates of the vertices are integers.

2019 Euler Teachers' Olympiad II p3
In the rectangular parallelepiped $ABCDA_1B_1C_1D_1$, the lengths of its three edges: $AB= 5$,$AD= 12$,$AA_1 = 8$. The point $M$ on the edge $AA_1$ is such that $AM = 5$. Find the volume of the pyramid $MB_1C_1D$.

2019 Euler Teachers' Olympiad II p7
Construct a square $ABCD$ at its vertex $A$ and two points $M$ and $N$, lying on lines $BC$ and $CD$ respectively.

2020 Euler Teachers' Olympiad I p10
Squares $ABCD, DCGH, BEFG$ and $ELKM$ are positioned as shown on the picture. Find the area of triangle $DGK$ if you know that the area of square $ABCD$ is $20$

2020 Euler Teachers' Olympiad I p12
Find the conditions under which the image of an angle in its orthogonal projection onto a plane is less than the original angle.

2020 Euler Teachers' Olympiad II p3

The diagonals of a convex quadrilateral are perpendicular. Prove that the projections of their intersection points on all four sides quadrilaterals lie on the same circle.

2020 Euler Teachers' Olympiad II p5

All faces of a tetrahedron are triangles, the lengths of the sides of each of which are are equal to $a, b$ and $c$. Find the radius of the ball circumscribed around the given tetrahedron.

source: https://www.euler-foundation.org/?page_id=432