geometry problems from Savin Competition for grades 6-8 (7-9 after 2016) in Russia, by Kvant magazine, with aops links in the names
collected inside aops:
1991- 2020
A plane point with coordinates $(a,b)$ can be connected by segments with points $(a - b, a)$ and $(a,b - a)$. Is it possible to connect with broken lines, consisting of such segments the points
a) $(19, 90)$ and $(1990,3383) $ ?
b) $(234,1001)$ and $(661, 7007)$ ?
The altitude of a triangle, drawn on its larger side, is not more than the sum of the lengths of the perpendiculars drawn from an arbitrary point on this side to the other two sides of this triangle. Prove it.
Triangle $BMN$ is inscribed in rhombus $ABCD$ made of two equilateral triangles, as shown in the figure. Prove that if the value of at least one of the angles of the triangle $BMN$ is $60^o$, then it is equilateral.
An equilateral triangle $ABC$ is located on the plane. Find all points $M$ of this plane for which triangles $ABM$ and $ACM$ are isosceles.
The angle $A$ of triangle $ABC$ is $60$, and side $AC$ is one and a half times longer than side $AB$. Cut the triangle into three pieces that can be folded into a regular hexagon.
Convex quadrilaterals $ABCD$ and $A'B'C'D'$ are such that segment $AB$ is parallel to segment $CD$, segment $B'C'$ is parallel to segment $D'A '$, and the lengths of the corresponding sides are equal: $AB = A'B'$, $BC = B ' C '$, $CD = C'D'$ and $DA = D'A '$. Prove that $ABCD$ is a parallelogram.
From nine identical square cards, first fold a square, and then a pyramid so that any two cards that have two common vertices in the first location will keep one of them common in the second location.
Three straight lines were drawn through point $P$ of an equilateral triangle, passing through its vertices. These straight lines have divided the triangle into $6$ triangles, as shown in the figure. The sum of the areas of the red triangles turned out to be equal to the sum of the areas of the blue triangles. Prove that point $P$ belongs to at least one of the medians of an equilateral triangle.
$50$ gangsters simultaneously shoot at each other, and each shoots at the nearest gangster (if there are several of them, then at one of them) and kills him. Find the smallest possible number of people killed.
Find the smallest values of the lengths of the sides of the rectangle of the picture, if the lengths of the sides of all the squares into which it is divided are integers.
On legs $AC$ and $BC$ of right-angled triangle $ABC$, points $M$ and $N$ are taken, respectively, so that $AM = BC$ and $MC = BN$. Prove that the angle between $AN$ and $BM$ is $45^o$. 
A square was placed on the strip, the side of which is equal to the width of the strip, and so that its border crossed the border of the strip at four points. Prove that two lines crossing these points meet at an angle of $45^o$.
Equilateral triangles $ABC$ and $PQR$ are located so that vertex $C$ lies on side $PQ$, and vertex $R$ lies on side $AB$. Prove that quadrilateral $ABQP$ is a trapezoid.
A cube is rolled over the surface of the table, turning it over the edges. Is it possible to turn it over $12$ times so that it turns over once through each edge and, as a result, ends up in the same place?
A point is marked inside the triangle. Prove that the length of the largest of the segments connecting it to the vertices is at least twice the length of the smallest of the segments connecting it to the midpoints of the sides of the triangle.
Three nodes $A, B$ and $C$ are marked on the checkered paper. The angle $ABC$ is $45^o$, and segments $AB$ and $BC$ have no nodes except their ends. Prove that triangle $ABC$ is right-angled.
In triangle $ABC$ points $M$ are taken on the side $AB$ and $N$ on the side$BC$ . Prove that if the angle $BNM$ is less than the angle $ANM$, then the angle $BNM$ is greater than the angle $AMN$.
The unfolded cube consists of six conguent squares. Is it possible to make a sweep of a parallelepiped out of five conguent rectangles?
Using a compass and a straight edge, construct a triangle if the point of intersection of its medians, the center of the circumscribed circle and the point of its intersection with one of the bisectors are known.
The angle bisector, median and altitude, drawn respectively from the vertices $A, B$ and $C$ of triangle $ABC$, intersect at point $O$. Prove that if $AB$ is the largest side of the triangle, then $BO> AO$; and if the smallest, then $BO <AO$.
Points $M$ and $N$ lie on the lateral side $BC$ of an isosceles triangle $ABC$, with $MN = AN$. The angles $BAM$ and $NAC$ are equal. Find the angle $MAC$.
In a square with side $a$, segments $AN$, $BK$, $CL$ and $DM$ are drawn so that the area of the green quadrilateral is equal to the sum of the areas of the brown triangles. Prove the equality $AM + BN + CK + DL = 2a$.
The convex quadrilateral $ABCD$ is such that the intersection point of the bisectors of the angles $DAC$ and $DBC$ lies on the side $CD$. Prove that the intersection point of the bisectors of angles $ADB$ and $ACB$ lies on side $AB$.
Several circles are located on the plane. There are at least five circles, any three of them have a common point. Prove that all these circles pass through one common point.
Two circles are given on the plane. Through the centers of each of the circles, straight lines are drawn, tangent to the second circle. Points $A_1$ and $A_2$ are the intersection points with the first circle of a straight line passing through its center, and $B_1$ and $B_2$ are similar points for the second circle. Prove that lines $A_1 B_1$ and $A_2 B_2$ are parallel.
Think of a triangle $ABC$ for which there are two points $M$ such that $MA + BC = MB + AC = MC + AB.$
Points $M$ and $K$ are taken on the hypotenuse of an isosceles right-angled triangle $ABC$ so that point $K$ lies between points $A$ and $M$, and the angle $MCK$ is $45^o$. Prove the equality $MK^2 = AK^2 + BM^2$.
On the sides of the convex quadrilateral $ABCD$, points $M, N, P$ and $Q$ are taken, respectively, dividing the sides of the quadrilateral in a ratio of $1: 2$, as shown in the figure. Prove that if the quadrilateral $MNPQ$ is a parallelogram, then $ABCD$ is also a parallelogram.
All vertices of the polyline $ABCDE$ lie on a circle, as shown in the figure. The angles at vertices $B, C$ and $D$ are equal to $45^o$. Prove the equality $AB^2 + CD^2 = BC^2 + DE^2$.
A flea jumps along a circle divided into several arcs. Before each of her jumps, she calculates the length of the arc on which she is, and then jumps so as to move clockwise to the arc of the calculated length. In particular, if a flea hits the border of two arcs, then it jumps further clockwise along the boundary points, thereby visiting all arcs. Prove that the flea will visit all arcs anyway.
A square inscribed in a circle with a radius of $5$ is divided into several identical squares. After removing two small squares from the square, the rest of it was placed in a circle with a radius of $4$. Is it possible to remove one more square so that the remainder fits in a circle with a radius of $3$?
$AP$ and $BQ$ are the bisectors of the acute angles of the right-angled triangle $ABC$. Segments $CM$ and $CN$ are medians of triangles $APC$ and $CBQ$, respectively. Prove that the sum of the angles at the vertices $M$ and $N$ of the five-pointed star $MNPCQ$ is equal to the sum of the angles at the other three vertices.
The angles of some $19$-gon are divisible by $10^o$. Prove that this $19$-gon has at least one pair of parallel sides.
The square is cut by straight lines, parallel to the sides of the square, into rectangles, which are colored in black and white in a checkerboard pattern, the sum of the areas of the black rectangles is equal to the sum of the areas of the white rectangles. Prove that the rectangles can be moved so that all the black rectangles make up one rectangle.
On the plane, two equilateral triangles $ABC$ and $ACO$ are given and a circle is drawn with center at point $O$, passing through points $A$ and $C$. For any point $M$ of this circle, prove the equality $MA^2 + MC^2 = MB^2$.
Point $A$ begins to move from a vertice of the square with side $a$ along its perimeter at a constant speed. At the same time, point $B$ begins to move from the same vertice with four times greater speed along the perimeter. Point $M$ is the midpoint of segment $AB$. What path will point $M$ travel to the moment when point $A$ again reaches the original vertex?
An equilateral triangle contains five pairwise disjoint circles of radius $1$. Prove that six pairwise disjoint circles of radius $1$ can be placed in this triangle.
In an equilateral triangle $ABC$ with side $a$, points $M, N, P$ and $Q$ are located as shown in the figure, with $MA + AN = PC + CQ = a$. Prove that the angle between $MQ$ and $PN$ is $60^o$.
In triangle $ABC$ through point $P$, three segments are drawn parallel to the sides of the triangle, as shown in the figure. Prove that the areas of the triangles $KLM$ and $PQR$ are equal.
The circle and quadrilateral are located as shown in the figure. The sums of the lengths of opposite circular arcs lying inside the quadrilateral are equal. Prove that a circle can be circumscribed around this quadrilateral.
Diagonals of a convex quadrilateral $ABCD$ intersect at point $O$. It is known that $AB = BC = CD$, and the angle AOD is equal to the half-sum of the angles $BAD$ and $CDA$. Prove that $ABCD$ is a rhombus.
Is there a quadrilateral, any vertex of which can be moved to another location so that the new quadrilateral is congruent to the original one?
On the side $AD$ of the parallelogram $ABCD$, point $M$ is taken, and on the sides $AB$ and $CD$, points $P$ and $Q$, respectively, such that the segments $PM$ and $QM$ are parallel to the diagonals of the parallelogram $BD$ and $AC$, respectively. Prove that the areas of triangles $PBM$ and $QCM$ are equal.
In the star pentagon $ACEBD$, some intersection points of the sides bisect the sides, namely: $AQ = QC$, $BR = RD$, $CR = RE$, and $DS = SA$. Prove that points $T$ and $P$ divide the segment $BE$ into three equal parts.
Point $Z$ is given on side $AB$ of triangle $ABC$. Construct points $X$ and $Y$ on sides $BC$ and $AC$ so that the areas of triangles $AYZ$, $BXZ$ and $CXY$ are the same.
Points $A_1$, $B_1$ and $C_1$ are taken on the sides of triangle $ABC$, as shown in the figure. The angles marked equally turned out to be equal to each other: the angle $AC_1B_1$ is equal to the angle $B_1A_1C$, the angle $BA_1C_1$ is equal to the angle $C_1B_1A$, and the angle $CB_1A_1$ is equal to the angle $A_1C_1B$. Prove that points $A_1$, $B_1$ and $C_1$ are the midpoints of the sides of triangle $ABC$.
Point $P$ is given on the side $AB$ of parallelogram $ABCD$. Construct an inscribed parallelogram with apex at point $P$, the sides of which are cut off from parallelogram $ABCD$ by four triangles of equal area.
Prove that for any point located inside the triangle, there is at least one of its vertices, which is no more than the length of its average side (not smallest, not largest), divided by $\sqrt2$.
$6$ conguent parallelograms of a unit area were cut out of paper, with which it was possible to paste over the entire surface of a cube with an edge $1$. Can these parallelograms not be rectangles?
The sum of the angles $BAD$ and $CAD$, as well as the sum of the angles $CDA$ and $BDA$, is $90^o$. Points M and N are the midpoints of sides $BC$ and $AD$ of quadrilateral $ABCD$. Prove the perpendicularity of lines MN and $AD$.
The curved triangle $OAB$ is a quarter circle. A ray emanates from vertex $A$, which, after reflecting successively from radius $OB$ at point $K$, from arc $AB$ at point $L$ and from radius $OA$ at point $M$, reaches vertex $B$. Prove that if $L$ is the midpoint of arc $AB$, then the angles $AKL$ , $KLM$ and $LMB$ are equal to $45^o$.
A convex quadrilateral is such that the sum of the distances from any point of its interior to all its sides is the same for all points. Is it true that this quadrilateral is a parallelogram?
Can two bisectors of the exteranl angles of a triangle intersect on its circumcircle?
Two convex quadrangles $ABCD$ and $KLMN$ are located so that they give an octagon at the intersection. All eight altitudes in the triangles bordering this octagon, drawn on its sides, are equal. Prove that quadrilaterals have equal perimeters and equal areas.
Inside the convex quadrilateral $ABCD$, there is a point $O$ such that the feet of the perpendiculars $P, Q, R$, and $S$ split the sides of the quadrilateral into parts that satisfy the inequalities $DP\ge PA$, $AQ\ge QB$, $BR\ge RC$, and $CS\ge SD$. Prove that point $O$ is the center of a circle circumscribed around the quadrilateral $ABCD$.
$ABCD$ and $CEFK$ are parallelograms. The segments $BE$ and $DK$ are parallel, and the lengths of these segments are equal to $a$ and $b$. Find the length of segment $AF$.
The angle $A$ of rhombus $ABCD$ is $60^o$. The line passing through point $C$ intersects lines $AB$ and $AD$ at points $M$ and $N$. Prove that the angle between lines $MD$ and $NB$ is $60^o$.
The convex hexagon $ABCFGH$ is such that $CF = FG = HA$, and the values of its angles $A, C$ and $G$ are respectively equal to the values of its angles $B, F$ and $H$. Prove that a circle can be circumscribed around this hexagon.
$O$ and $I$ are respectively, the centers of the circumcircle and the incircle of triangle $ABC$. M is the midpoint of the arc $AC$ of circumcircle that does not contain point $B$. Prove that the value of the angle $ABC$ is $60^o$ if and only if $MI = MO$.
A square $ABCD$ is inscribed in a circle. The points $P$ and $Q$ taken on the circumference are such that the angle $PAQ$ is $45^o$. Line $AP$ intersects the the square at point $M$, and line $AQ$ at point $N$. Prove that lines $MN$ and $PQ$ are parallel.
Five circles of unit diameter were cut out of a $5 \times 5$ cardboard square. Prove that you can always cut two $1 \times 2$ rectangles out of the remaining cardboard.
The lengths of all sides of the decagon are equal to $1$. Nine of its sides touch a circle. Prove that the tenth side also touches this circle.
Prove that a convex quadrilateral $ABCD$ has at least two parallel sides if and only if the product of the areas of triangles $ABD$ and $BCD$ is equal to the product of the areas of triangles $ABC$ and $ACD$.
The perpendicular bisector of one of the sides of the triangle divides it into two parts, the areas of which differ by $3$ times. The perpendicular bisector of the other side divides it into two parts, the areas of which do not differ by $3$ times. How many times are the areas of the parts into which the perpendicular bisector of the third side divides the triangle differ?
Through the midpoints of the diagonals of the convex quadrilateral $ABCD$, straight lines are drawn parallel to the diagonals. These lines intersect at point $M$. Prove that the area of triangle $ABM$ is equal to the area of triangle $CMD$, and the area of triangle $BMC$ is equal to the area of triangle $AMD$.
A circle is given, inside which points $A$ and $C$ are marked, lying on the same diameter. Using a compass and a ruler without divisions, construct on the circle a point $B$ for which the value of the angle $ABC$ is the greatest possible.
Using only a compass, divide the given square into two parts of equal area.
There is a rectangular table with $m$ rows and $n$ columns. Rows can vary in height and columns can vary in width. We can indicate several cells of the table, and the area of each of them will be written in these cells. What is the smallest number of cells we can indicate so that after filling them in, we can find out the areas of all other cells in the table?
Three points $A, B$ and $C$ are given on the plane. It is allowed to select any two of them and rotate the segment connecting them wrt to its midpoint. After this operation was done several times, point $A$ coincided with the original position of point $B$, and point $B$ did not coincide with the original position of point $C$. Prove that point $C$ also did not coincide with the original position of point $A$.
Using a ruler and a compass with a fixed ''angle'', construct an acute-angled triangle, one angle of which is twice the size of the other.
A square with side $1$ is given on the plane. It is allowed to select any two of its vertices and move one of them to any distance in an arbitrary direction (within the plane), and move also the other vertices to the same distance in the opposite direction. After several such operations, we got a square congruent to the original one. Prove that the intersection area of the squares is greater than $0.8$.
Prove that if there are three right-angled triangles: with legs $a, b$ and hypotenuse $c$, with legs $x, y$ and hypotenuse $z$, with legs $a + x$, $b + y$, and hypotenuse $c + z$, then these triangles are similar.
A square field with a side of $180$ m was sown with rye. On any plot of $40$ m $\times$ $100$ m, the sides of which are parallel to the sides of the field, at least $91\%$ of its area is sown. What is the smallest percentage of field area that rye could have been sown?
The flat closed ten-link broken line $ABCDEFGHKLA$ has the same midpoints of pairs of links $AB$ and $FG$, $BC$ and $GH$, $CD$ and $HK$, $DE$ and $KL$. Prove that the midpoints of the links $EF$ and $LA$ also coincide.
What is the largest number of checkerboard cells that can be marked so that there is not a single obtuse triangle with vertices in the centers of the marked cells?
Several straight lines are drawn on the plane, which, crossing each other, give several five-pointed stars that do not overlap one another. For example, in the figure, nine straight lines form three stars. Could there be more such stars than straight lines?
A ray of light, directed from the top of the square, reflected $2005$ times from its walls according to the law "the angle of incidence is equal to the angle of reflection", ends its path at another vertex of the square. What is the smallest distance the ray could travel if the side of the square is $1$?
Triangles $ABC$, $ADE$, $BFD$, $CEG$, $BGH$ are equilateral and equally oriented. Prove that triangle $EFH$ is also equilateral.
A convex quadrilateral $ABCD$ is such that no matter how you cut it into three triangles, there is always a triangle of area $1$ among them. Prove that $ABCD$ is a parallelogram of area $2$.
There is a game of "battleship". A three-deck ship is hidden on a $7 \times 7$ checkered field. In what is the smallest number of shots you can probably sink it?
Points $X, Y$ and $Z$ are taken on the sides $BC, CA$ and $AB$ of triangle $ABC$ so that the segments $AX$, $BY$ and $CZ$ divide the corresponding sides of triangle $XYZ$ in half. Are points $X, Y$, and $Z$ necessarily the midpoints of the sides of triangle $ABC$?
All angles of a $400$-gon are measured in integer numbers of degrees. Prove that it has at least three parallel sides.
Can three rays divided angle $BAC$ into two pairs of congruent angles, and the segment $BC$ into two pairs of segments of equal length, as shown in the figure (and the corresponding congruent angles equal length segments are shown in one color)?
In a right-angled isosceles triangle, the point $P$ of intersection of its leg with the bisector of the opposite angle is marked. Using only a compass of an arbitrary but fixed ''angle'', draw the center of the inscribed circle of this triangle.
A segment connecting the midpoints of two opposite sides of a quadrilateral cut its diagonal in half. Prove that the quadrilateral is a trapezoid or parallelogram.
Three stripes of equal width at the intersection form a star-shaped hexagon $ABCDEFGHIJKL$. Prove that the following segments intersect at one point:
a) $AG$, $CI$ and $EK$,
b) $AG$, $BH$ and $FL$,
c) $CI$, $BH$ and $DJ$,
d) $EK$, $DJ$ and $FL$.
(A strip is the part of a plane between two parallel lines; the width of a strip is the distance between the lines that bound it.)
Prove that if the midpoint of a segment connecting the midpoints of two opposite sides of a convex quadrilateral lies on the diagonal of the quadrilateral, then this diagonal passes through the midpoint of the other diagonal.
The Pythagoreans considered rectangles of size $36$ and $44$ to be perfect, because the lengths of the sides of each such rectangle are expressed in integers, and the area of the rectangle is numerically equal to its perimeter. Find all perfect (that is, having the same properties):
a) right ,
b) isosceles triangles.
A point lying inside an acute-angled triangle was reflected symmetrically with respect to each of its three sides. The resulting three turned out to be on the circumscribed circle of the original triangle. Then only three obtained points were left on the drawing, and everything else was erased. Restore the original triangle given these three points.
The point lying inside the parallelogram was reflected symmetrically with respect to its sides. The four obtained points turned out to be the vertices of a square. Is the original parallelogram a square?
Prove that if the bisectors of the angles at the vertices of the "irregular" five-pointed star intersect at one point, then the bisectors of the angles of the inner pentagon intersect at one point.
Prove that it is impossible to draw a straight line passing through the point of intersection of the medians of a triangle that divides its area in the ratio $2: 1$.
Triangle $A'B'C'$ is inscribed in equilateral triangle $ABC$ as shown in the figure. The angles $BA'C'$ and $C'B'A$ are equal. The angles $BC'A'$ and $A'B'C$ are also equal. Prove that line $BB'$ bisects angle $ABC$.
$49$ points are marked on the plane, which are the nodes of a $6\times 6$ checkered table. Prove that any closed polyline, the vertices of which are all these points, has two parallel links.
Is it possible to mark several cells on the surface of a $3 \times 3 \times 3$ Rubik's cube so that exactly $5$ cells are marked on each ring of $12$ cells encircling the cube?
The triangular park has three paths: blue, green and red. Each path divides the park in two parts of equal area. Together, the paths divide the park into $7$ parts: four "triangular" and three "quadrangular" (fig. ). Could it be that out of two parts adjacent to the same side of the park, "quadrangular" is always no less in area than "triangular"?
The diagonal $AC$ of the quadrilateral $ABCD$ is the bisector of the angle$BCD$ , where the angle $BCD$ is $120^o$ and the angle $BAD$ is $30^o$. Prove that the perimeter of triangle $BCD$ is equal to the length of the segment $AC$.
On the sides $BC$ and $CD$ of the rhombus $ABCD$ took points $P$ and $Q$, respectively, so that $BP = CQ$. Prove that the center of gravity (the point of intersection of the medians) of the triangle $APQ$ lies on the diagonal $BD$ of the rhombus.
Prove that any convex quadrilateral can be cut into four pieces that add up to a rectangle.
You are given an isosceles trapezoid with legs of length $a$ and diagonals of length $b$. Find the product of the lengths of its bases.
A billiard ball is released from the top $A$ of the square table $ABCD$. It is reflected from the sides according to the law “the angle of incidence is equal to the angle of reflection”. It is known that no two blows in a row fell on opposite sides, with the first blow coming from the side of the $BC$. After $2010$ shots from the boards, the ball finally hit the top of the table. In which?
In a right-angled triangle, the inscribed circle touches one of the midlines. Prove that the lengths of the sides of this triangle are related to each other as $3: 4: 5$.
Two disjoint circles are given on the plane. Is there a point $A$ outside the circles that any straight line passing through $A$ will necessarily "touch" at least one of these circles (that is, intersect or is tangent to)?
The company produces sets of $7$ tennis balls in two types of packaging: a long cylinder with a thickness of $1$ ball and a flat cylinder with a height of $1$ ball, as shown in the picture. The balls touch each other and the walls so that they do not dangle. The empty space is filled with plastic chips.
a) In filling a package - the long or flat, - requires more chips?
b) How many grams of crumb does it take to fill a flat pack if it takes $90$ grams to fill a long one ?
To solve (b), you will need an interesting fact, which Archimedes already knew: the ball occupies exactly $2/3$ of the volume the cylinder in which it is inscribed (the ball touches the walls, bottom and cover of the cylinder).
The angle $C$ at the apex of an isosceles triangle $ABC$ is $120^o$. From the vertex $C$, two rays were drawn into the triangle, which, having reflected from the base $AB$ at points $M$ and $N$ (according to the law "the angle of incidence is equal to the angle of reflection"), hit the side $AC$ at point $P$ and the side $BC$ at the point $Q$, respectively. Segment $PQ$ meets $CM$ at point $X$, and $CN$ at point $Y$. Prove that if $AP = CQ$, then triangles $PMX$, $XCY$ and $QNY$ are congruent.
During the big changes, one of the guys painted the schedule on the board so that you get a butterfly. And the teacher looked at the board and said:
- Look, guys, have we got a great task: in the left wing butterflies red is three times more than blue. And how many times in the right wing is more red than blue?
Rectangle $ABCD$ is split by two straight lines, intersecting at point $X$, into four rectangles.
a) Prove that if the point $X$ lies on the diagonal $AC$, then the areas of the upper left and lower right rectangles are equal (they are shaded in the figure).
b) Let it be known that the areas of the upper left and lower right rectangles are equal. Does the point $X$ then necessarily lie on diagonal $AC$?
The straight line that passes through the midpoints of the diagonals of a convex quadrangle divides its area in half. Prove that at least one of these diagonals passes through the midpoint of the other diagonal.
A cylindrical mug of diameter $1$ and height $2$, filled to the top with water, gets an angle of $45^o$ (the angle between the axis of the circle and the vertical) and starts spilling water. What fraction of the water, that fills the whole mug, is spilled out?
$4$ points are marked on a sheet of paper. Petya measured pairwise distances between them and got the number $1,2, 3, 4, 5, 6$ in some order. Prove that these the points are collinear.
The figure shows crystal inscribed in a cube of unit volume. Find crystal volume. (No matter from which side of the cube you look at the crystal, it is arranged the same. Crystal vertices lying on the edges of cube, halve them.)
Eight equilateral triangles are located as shown in the figure. Find the sum of the two marked angles.
From the vertices of the convex quadrilateral, perpendiculars were drawn on its diagonal. It turned out that the length of any diagonal is not more than the sum of the lengths of the perpendiculars drawn on it. Find the angle between segments connecting the midpoints of the opposite sides of the quadrilateral.
A triangle $ABC$ is drawn on the board. Point $A$ is reflected wrt line $BC$, point $B$ is reflected wrt line $CA$, point $C$ is reflected wrt line $AB$. It turned out that the three new points are the vertices of an equilateral triangle. Is it true that the original triangle equilateral ?
We will call a number squared if there a square, having such an area, can be drawn on a checkered plane with vertices at the grid nodes (for example, numbers $1,2, 4$ and $5$ are squared, but $3$ are not). Prove that if numbers $a$ and $b$ are squared, then the number $ab$ is also squared.
A rectangle can be placed without overlays on $100$ circles of radius $2$. Is it necessary that such a rectangle can be placed without overlapping on $400$ circles of radius $1$?
Two triangles have the same perimeters, and also the same area. Are these triangles necessarily congruent?
Equilateral triangles $OAB$ and $OCD$ have only one common point, vertex $O$. Points $M$ and $L$ are midpoints of sides $AB$ and $CD$ of these triangles, and points $K$ and $N$ are midpoint of segments $AD$ and $BC$. Prove that segments $ML$ and $KN$ are perpendicular.
Three triangles are given: an acute, a right and an obtuse. It is known that two of them can be attached to each other so that it turns out a triangle equal to the third. Which of the three given triangles is it equal to?
The circle crosses each side of the quadrilateral, and the sides cut off equal arcs from the circle (see figure). Prove that a circle can be inscribed in this quadrangle.
Let $K, L$ be the midpoints of the sides $BC$ and $DA$ of the quadrilateral $ABCD$. Prove that tripled area of quadrilateral $ABKL$ is larger than the area of quadrilateral $DCKL$.
Three positive numbers $x, y, z$ are such that for any triangle with sides $a, b, c$ exists triangle with sides $ax, by, cz$. Find all such numbers $x, y, z$.
The inscribed circle of triangle $ABC$ touches sides $AB$ and $AC$ at points $M$ and $N$. Through the midpoint of side BC, a straight line $\ell_A$ is drawn, perpendicular to $MN$. Similarly are difines lines $\ell_B$ and $\ell_C$. Prove that the lines $\ell_A$, $\ell_B$ and $\ell_C$ intersect at one point.
In equilateral triangle $ABC$ on the sides $AB$ and $BC$ are marekd points $E$ and $F$, respectively, so that $BE = CF$. Let $O$ be the center of the triangle, and $P$ be the intersection of $AF$ and $CE$. Prove that $OP$ is the bisector of angle $EPA$.
Several points were marked in red on the plane, so that the distance between any two red dots does not exceed $1$. Prove that all red dots can be covered by a pattern in the form of square with side $1$.
On the plane there are two conguent triangles that have no common points (neither inside nor at the boundaries). Line $\Pi_1$ divides the area of each of them in half, and line $\Pi_2$ divides their perimeters in half. Can lines $\Pi_1$ and $\Pi_2$ be perpendicular?
The height of a trapezoid whose diagonals are perpendicular is $h$. What is the smallest possible value of the length of the midline of such a trapezoid?
Given a square $ABCD$. Through vertex $C$ was drawn line $m$ that has no other points in common with the square (see figure). Points $E$ and $F$ are the projections of points $B$ and $D$ respectively on line $m$. Segments $BF$ and $DE$ intersect at the point $K$. Prove that the line $AK$ is perpendicular to line $m$.
A squadron of $10$ ships is located on a $10\times 10$ square checkered field. Ships are $1\times 2$ not in common rectangles with sides along the grid lines.
a) Prove that you can fire $32$ shots so that sure to get into some ship.
b) Is it possible to get by with $30$ shots?
Find all $n$ for which exists a convex $n$-gon whose vertices are lattice points (e.g. having integer coordinates) and all sides of the polygon to have the same length, equal to $5$ units.
In trapezoid $ABCD$, point M is the midpoint of the lateral side $CD$. Prove that if the point $M$ lies on the bisector of angle $A$, then it also lies on the bisector of angle $B$.
On sides $AB$ and $AC$ of an equilateral triangle $ABC$ are taken points $E$ and $F$, respectively, so that $AE: BE = CF: AF = 2$. Let $P$ be the intersection point of $BF$ and $CE$. Prove that the angle $CPA$ is right.
An equilateral triangle is inscribed in a regular hexagon of area $96$ as shown in the figure. Find the area of this triangle.
In a convex quadrilateral $ABCD$, $\angle A=30^o$, perimeter of triangle $BCD$ is equal to the length of the diagonal $AC$. Find $\angle C$.
Two angles are marked on a grid of equilateral triangles. Prove that they are equal.
On an infinite grid of equilateral triangles (its fragment is shown in the figure), a segment with vertices at grid nodes is given, whose length is equal to $a$. Prove that you can draw another segment with vertices at the nodes, whose length is equal to:
a) $\sqrt3 a$,
b) $\sqrt{37}a$.
From a circle, you can cut a quadrilateral, whose two opposite sides are $a$ and $c$, and the other two are $b$ and $d$. Tolik Vtulkin claims that then from this circle, you can also cut a quadrilateral, in which two opposite sides are equal to $a$ and $b$, and the other two are $c$ and $d$. Is Tolik correct?
Solve the problem in cases when the original quadrilateral:
a) is inscribed in the given circle (the vertices of the quadrilateral lie on the border of the circle),
b) not necessary inscribed, but convex (the diagonals lie inside the quadrilateral),
c) may be nonconvex (one of the diagonals may lie outside the quadrilateral).
There are two convex polygons. The first polygon has twice the the number of acute angles than the second polygon's obtuse angles, and the second has three times more acute angles than first polygon's obtuse angles. It is also known that each of them has at least one acute angle and that these polygons (one or both) there are also right angles.
a) Give an example of how this can be.
b) How many right angles has each of the polygons? (It is required to find all cases and prove that there are no others)
Petya invented a creiterion of congruence of quadrangles. He claims that given quadrangles $ABCD$ and $A'B'C'D'$ (not necessarily convex), where the three sides of one are respectively equal to the three sides of the other ($AB=A'B'$, $BC=B'C'$, $CD=C'D'$) and diagonals of one are respectively equal todiagonals of the other ($AC=A'C'$, $BD= B'D'$), then the quadrangles are congruent. Is Petya wrong?
Given an arbitrary acute-angled triangle.
a) Show how to cut it into six triangles with the same radius of circumscribed circles.
b) Prove that there are several ways to do this.
Each vertex of the rectangle (not a square) reflected symmetrically across a diagonal not passing through this vertex. We got four points.
a) Prove that these four points form a rectangle.
b) Can they be the vertices of a rectangle congruent to the original one?
c) Can they be the vertices of a rectangle similar but not equal to the original one?
Cut the shape in figure into two congruent parts.
Using a ruler and drawn part of the grid, it is required to construct the bisector of angle $A$. What is the smallest number of line segments required to do this? (The ruler is only allowed to draw straight lines.)
Construct a quadrilateral of perimeter $18$ and area $16$, all sides of which have odd length.
There is a cardboard triangle $ABC$ on the table. They do with it several times the next operation. Choose its vertex, denote it by $X$, flip the triangle and put it so that the point $X$ hits the same spot as before, and the angle $X$ of the triangle coincided with its previous position. Let us denote this operation by $P_X$. Prove that after sequence of opereratons $P_A$ , $P_B$, $P_C$, $P_A$, $P_B$, $P_C$ the triangle will return to its original position.
The circle intersects the sides of the triangle at $6$ points (see figure).
a) Prove that if $a = b$ and $c = d$, then $e = f$.
b) Prove that if $b = c$ and $d = e$, then $f = a$.
Points $A_1,B_1,C_1$ are midpoints of sides of the equilateral triangle $ABC$. Also on its sides lie points $M$ and $K$ as shown in the figure. Prove that the areas of the two shaded quadrangles are equal.
Given a triangle $ABC$. Is it true that on its sides there are necessarily four points that are the vertices of an isosceles trapezoid, the bases of which are parallel to $AB$?
Given a segment $a$ and an equilateral triangle with side $b$. Construct (with a compass and ruler) inside this triangle an equilateral triangle with side a such that so that the vertices of the outer triangle lay on the extensions of the sides of the inner.
Baron Munchausen said that what no matter what triangle is given to him, he can cut it into two polygons, and then cut each of them into $7$ congruent polygons. Could the baron's words be true?
Given a segment $AB$ and a circle. Construct points $C$ and $D$ on the circle so that so that the quadrilateral $ABCD$ is an isosceles trapezoid with base $AD$.
Point $K$ is marked on side $AB$ of acute-angled triangle ABC. Midpoint of segment $CK$ is equidistant from points $A$ and $B$. Prove that $AK <BC$.
You are given $k$ sticks, the length of each is a natural number, while there are no two sticks of the same length. It is known that from any three sticks, you can make a triangle with a perimeter of no more than $1000$. What is the greatest possible value of $k$?
Baron Munchausen fenced off his property with a fence in the shape of a $n$-gon, and asserts that each internal angle of this $n$-gon is either less than $10^o$ or greater $350^o$. Could the baron be right? Decide the problem for
a) $n = 10$
b) $n = 11$
c) $n = 101$
Two squares are inscribed in a right-angled triangle, as shown in the figure. Prove that three marked points lie on one straight line.
On the sides of the triangle inward isosceles triangles with angle $\alpha$ at top. For what is the largest $\alpha$ the triangle is guaranteed to be covered?
They put a square on the strip, side which is equal to the width of the strip, moreover so that its border crossed the border stripes at four points (see figure). Prove that two lines passing crosswise through these points, intersect under angle $45^o$.
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