geometry problems from Adygea Teachers' Geometry Olympiad with aops links in the names
[ОЛИМПИАДА ПО ГЕОМЕТРИИ
для учителей математики общеобразовательных организаций]
Find the area of the $MNRK$ trapezoid with the lateral side $RK = 3$ if the distances from the vertices $M$ and $N$ to the line $RK$ are $5$ and $7$, respectively.
2017 Adygea Teachers' Geometry Olympiad p2
It turned out for some triangle with sides $a, b$ and $c$, that a circle of radius $r = \frac{a+b+c}{2}$ touches side $c$ and extensions of sides $a$ and $b$. Prove that a circle of radius $ \frac{a+c-b}{2}$ is tangent to $a$ and the extensions of $b$ and $c$.
2017 Adygea Teachers' Geometry Olympiad p3
Jack has a quadrilateral that consists of four sticks. It turned out that Jack can form three different triangles from those sticks. Prove that he can form fourths triangle that is different from the others.
2017 Adygea Teachers' Geometry Olympiad p4
A regular tetrahedron $SABC$ of volume $V$ is given. The midpoints $D$ and $E$ are taken on $SA$ and $SB$ respectively and the point $F$ is taken on the edge $SC$ such that $SF: FC = 1: 3$. Find the volume of the pentahedron $FDEABC$.
2017 Adygea Teachers' Geometry Olympiad p2
It turned out for some triangle with sides $a, b$ and $c$, that a circle of radius $r = \frac{a+b+c}{2}$ touches side $c$ and extensions of sides $a$ and $b$. Prove that a circle of radius $ \frac{a+c-b}{2}$ is tangent to $a$ and the extensions of $b$ and $c$.
2017 Adygea Teachers' Geometry Olympiad p3
Jack has a quadrilateral that consists of four sticks. It turned out that Jack can form three different triangles from those sticks. Prove that he can form fourths triangle that is different from the others.
A regular tetrahedron $SABC$ of volume $V$ is given. The midpoints $D$ and $E$ are taken on $SA$ and $SB$ respectively and the point $F$ is taken on the edge $SC$ such that $SF: FC = 1: 3$. Find the volume of the pentahedron $FDEABC$.
Can the distances from a certain point on the plane to the vertices of a certain square be equal to $1, 4, 7$, and $8$ ?
It is known that in a right triangle:
a) The height drawn from the top of the right angle is the geometric mean of the projections of the legs on the hypotenuse;
b) the leg is the geometric mean of the hypotenuse and the projection of this leg to the hypotenuse.
Are the converse statements true? Formulate them and justify the answer.
Given a cube $ABCDA_1B_1C_1D_1$ with edge $5$. On the edge $BB_1$ of the cube , point $K$ such thath $BK=4$.
a) Construct a cube section with the plane $a$ passing through the points $K$ and $C_1$ parallel to the diagonal $BD_1$.
b) Find the angle between the plane $a$ and the plane $BB_1C_1$.
source: remshagu.ru/Koncurs_dly_uchiteley/Geometry_teachers/
a) The height drawn from the top of the right angle is the geometric mean of the projections of the legs on the hypotenuse;
b) the leg is the geometric mean of the hypotenuse and the projection of this leg to the hypotenuse.
Are the converse statements true? Formulate them and justify the answer.
Is it possible to formulate the criterion of a right triangle based on these statements? If possible, then how? If not, why?
Two circles intersect at points $A$ and $B$. Through point $B$, a straight line intersects the circles at points $C$ and $D$, and then tangents to the circles are drawn through points $C$ and $D$. Prove that the points $A, D, C$ and $P$ - the intersection point of the tangents - lie on the same circle.
Given a cube $ABCDA_1B_1C_1D_1$ with edge $5$. On the edge $BB_1$ of the cube , point $K$ such thath $BK=4$.
a) Construct a cube section with the plane $a$ passing through the points $K$ and $C_1$ parallel to the diagonal $BD_1$.
b) Find the angle between the plane $a$ and the plane $BB_1C_1$.
Inside the quadrangle, a point is taken and connected with the midpoint of all sides. Areas of the three out of four formed quadrangles are $S_1, S_2, S_3$. Find the area of the fourth quadrangle.
Inside the triangle $T$ there are three other triangles that do not have common points. Is it true that one can choose such a point inside $T$ and draw three rays from it so that the triangle breaks into three parts, in each of which there will be one triangle?
In a cube-shaped box with an edge equal to $5$, there are two balls. The radius of one of the balls is $2$. Find the radius of the other ball if one of the balls touches the base and two side faces of the cube, and the other ball touches the first ball, base and two other side faces of the cube.
From which two statements about the trapezoid follows the third:
1) the trapezoid is tangential,
2) the trapezoid is right,
3) its area is equal to the product of the bases?
In planimetry, criterions of congruence of triangles with two sides and a larger angle, with two sides and the median drawn to the third side are known. Is it true that two triangles are congruent if they have two sides equal and the height drawn to the third side?
2020 Adygea Teachers' Geometry Olympiad p2
The square $ABCD$ is inscribed in a circle. Points $E$ and $F$ are located on the sides of the square, and points $G$ and $H$ are located on the smaller arc $AB$ of the circle so that the $EFGH$ is a square. Find the area ratio of these squares.
2020 Adygea Teachers' Geometry Olympiad p3
Is it true that of the four heights of an arbitrary tetrahedron, three can be selected from which a triangle can be made?
2020 Adygea Teachers' Geometry Olympiad p4
A circle is inscribed in an angle with vertex $O$, touching its sides at points $M$ and $N$. On an arc $MN$ nearest to point $O$, an arbitrary point $P$ is selected. At point $P$, a tangent is drawn to the circle $P$, intersecting the sides of the corner at points $A$ and $B$. Prove that that the length of the segment $AB$ is the smallest when $P$ is its midpoint.
2020 Adygea Teachers' Geometry Olympiad p2
The square $ABCD$ is inscribed in a circle. Points $E$ and $F$ are located on the sides of the square, and points $G$ and $H$ are located on the smaller arc $AB$ of the circle so that the $EFGH$ is a square. Find the area ratio of these squares.
2020 Adygea Teachers' Geometry Olympiad p3
Is it true that of the four heights of an arbitrary tetrahedron, three can be selected from which a triangle can be made?
2020 Adygea Teachers' Geometry Olympiad p4
A circle is inscribed in an angle with vertex $O$, touching its sides at points $M$ and $N$. On an arc $MN$ nearest to point $O$, an arbitrary point $P$ is selected. At point $P$, a tangent is drawn to the circle $P$, intersecting the sides of the corner at points $A$ and $B$. Prove that that the length of the segment $AB$ is the smallest when $P$ is its midpoint.
a) Two circles of radii $6$ and $24$ are tangent externally. Line $\ell$ touches the first circle at point $A$, and the second at point $B$. Find $AB$.
b) The distance between the centers$ O_1$ and $O_2$ of circles of radii $6$ and $24$ is $36$. Line $\ell$ touches the first circle at point $A$, and the second at point $B$ and intersects $O_1O_2$. Find $AB$.
In triangle $ABC$, the incircle touches the side $AC$ at point $B_1$ and one excircle is touching the same side at point $B_2$. It is known that the segments $BB_1$ and $BB_2$ are equal. Is it true that $\vartriangle ABC$ is isosceles?
In a triangle, one excircle touches side $AB$ at point $C_1$ and the other touches side $BC$ at point $A_1$. Prove that on the straight line $A_1C_1$ the constructed excircles cut out equal segments.
Two identical balls of radius $\sqrt{15}$ and two identical balls of a smaller radius are located on a plane so that each ball touches the other three. Find the area of the surface $S$ of the ball with the smaller radius.
source: remshagu.ru/Koncurs_dly_uchiteley/Geometry_teachers/
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