problem stringlengths 1 13.6k | solution stringlengths 0 18.5k ⌀ | answer stringlengths 0 575 ⌀ | problem_type stringclasses 8
values | question_type stringclasses 4
values | problem_is_valid stringclasses 1
value | solution_is_valid stringclasses 1
value | source stringclasses 8
values | synthetic bool 1
class | __index_level_0__ int64 0 742k |
|---|---|---|---|---|---|---|---|---|---|
3. The factory's production over four years has increased by 4 times. By what percentage did the production on average increase each year compared to the previous year? | 3. About \(41 \%\). Hint. If \(x\) is the required percentage, then (see problem 3 of 3): \(\left(1+\frac{x}{100}\right)^{4}=4\) and so on. | 41 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 43,556 |
4 (!). Solve the systems of equations:
1) \(\left\{\begin{array}{l}x^{2}-2 x=0 \\ x^{3}+y=6\end{array}\right.\)
2) \(\left\{\begin{array}{l}y^{2}-4 y+3=0, \\ 2 x+y=9 .\end{array}\right.\) | 4. 1) 0 and 6 or \(2 \cdot\) and -2 . Hint. Solve the combined system of equations:
\[
\left\{\begin{array} { l }
{ x = 0 } \\
{ x ^ { 3 } + y = 6 , }
\end{array} \text { or } \left\{\begin{array}{l}
x=2, \\
x^{3}+y=6
\end{array}\right.\right.
\]
2) 4 and 1 or 3 and 3 . Hint. Solve the combined system of equations:
... | (0,6)or(2,-2) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 43,557 |
6. It is known that the graph of the function \(y=x^{2}+p x+q\) passes through the points \(A(1 ; 1)\) and \(B(3 ; 1)\). Does this graph pass through the point \(C(4 ; 5)\) ? | 6. Solution. Since the graph passes through the point \(A(1 ; 1)\), then \(1=1^{2}+p \cdot 1+q\). Since the graph passes through the point 86
\(B(3 ; 1)\), then \(1=3^{2}+p \cdot 3+q\). We will find \(p\) and \(q\) by solving the system of equations:
\[
\left\{\begin{array}{l}
p+q=0, \\
3 p+q=-8,
\end{array} \quad p=-... | 不经过点C | Algebra | math-word-problem | Yes | Yes | olympiads | false | 43,559 |
8(!). Does y have a maximum or minimum value if:
1) \(x^{2}-6 x+2 y=0\)
2) \(3 x^{2}+12 x-2 y-4=0\)
3) \(y=\frac{2 x}{1+x^{2}}\);
4) \(y=\frac{2 x-1}{x^{2}+2 x+1}\). | 8. 1) Solution. The discriminant of the quadratic equation in \(x\) must be non-negative, therefore, 9 \(-2 y \geqslant 0, y \leqslant 4.5\). The maximum value of \(y\) is 4.5, and there is no minimum value.
2) The maximum value does not exist, and the minimum value of \(y\) is -8.
3) \(-1 \leqslant y \leqslant 1\).
4... | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 43,561 |
9. Show that one of the equations
\[
c x^{2}+m x-a=0 \text { or } a x^{2}+n x+c=0
\]
necessarily has at least one root. | 9. It is solved. The discriminant of the first equation is equal to \(m^{2}+\) \(+4 a c\), and the second one is \(n^{2}-4 a c\). Since the sum of these discriminants is equal to \(m^{2}+n^{2} \geqslant 0\), at least one of them is non-negative, and therefore the corresponding equation has at least one root. | proof | Algebra | proof | Yes | Yes | olympiads | false | 43,562 |
10. It is required to fence a rectangular area adjacent to a wall. The fence should have a length of \(60 \mathrm{~m}\). What should be the length and width of this area so that its area is the largest? | 10. Solution. First method. Let the width of the site be \(x \mu\), then its length will be ( \(60-2 x\) ) \(m\), and the area will be \(y=x(60-2 x)\) square meters. Considering the obtained equation as a quadratic equation in terms of \(x\), we get: \(2 x^{2}-60 x+y=0\). The discriminant of this equation is non-negati... | 450 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 43,563 |
11. Prove that the product of two consecutive natural numbers cannot be equal to \(25 k+1\) for \(k=\{0 ; 1 ; \ldots\}\). | 11. Solution. First method. Let the first number be \(n\), then the second number is \(n+1(n \in N)\). Suppose that the equality \(n(n+1)=25 k+1\) holds. Then the discriminant of the quadratic equation in \(n\)
\[
n^{2}+n-(25 k+1)=0
\]
is \(1+4 \cdot(25 k+1)=\ldots=5 \cdot(20 k+1)\).
The number \(5(20 k+1)\) is divi... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 43,564 |
12. The price of a diamond is proportional to the square of its mass. If a diamond is broken into two parts, in which case will the total price of the two parts be the lowest? | 12. Solution. Let the price of a diamond be calculated by the formula \(y=a m^{2}\), where \(m\) is its mass. Let the mass of the first piece be \(\frac{m}{2}+x\). Then the mass of the second piece will be \(m-\frac{m}{2}-x=\frac{m}{2}-x\). The total cost of the two pieces will be:
\[
a\left(\frac{m}{2}+x\right)^{2}+a... | 0 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 43,565 |
14. Solve the equations in the most rational way:
1) \(1969 x^{2}-1974 x+5=0\)
2) \((a+b-2 c) x^{2}+(b+c-2 a) x+(c+a-2 b)=0\). | 14. 1) \(\mathrm{P}\) solution. It is not hard to notice that the number 1 is a root of the given equation. Since the product of the roots of this equation is \(\frac{5}{1969}\), and one of the roots is 1, the second root is \(\frac{5}{1969}\). Clearly, there can be no other roots.
2) \(\mathrm{P}\) solution. It is imp... | Algebra | math-word-problem | Yes | Yes | olympiads | false | 43,567 | |
15. If two rectangles have equal perimeters and equal areas, then the lengths of their sides are respectively equal. Prove it. | 15. Instruction. Using Vieta's theorem, show that in both cases the lengths of the sides are the roots of the same quadratic equation. | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,568 |
16(!). Determine \(p\) so that the sum of the absolute values of the roots of the equation \(z^{2}+p z-6=0\) is equal to 5. | 16. Solution. Let \(x\) and \(y\) be the roots of the given equation. Since the product of the roots is -6, they have different signs. Let \(x>0\) and \(y<0\). Then, using the conditions of the problem and Vieta's theorem, we can form the following system of two equations with two unknowns:
\[
\left\{\begin{array}{l}
... | 1or-1 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 43,569 |
17. Factorize:
1) \(a^{2}+2 b^{2}-2 c^{2}+3 a b+a c\)
2) \(a^{2}-2 b^{2}-2 c^{2}-a b+5 b c-a c\). | 17. 1) \((a+b-c)(a+2b+2c)\)
2) \((a-2b+c)(a+b-2c)\).
Hint. Each expression can be considered as a quadratic trinomial, for example, in terms of \(a\) and find its roots. | (-2b+)(+b-2c) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 43,570 |
18*. Simplify the fractions:
1) \(\frac{\left(x^{2}-x-5\right)\left(x^{2}-x-2\right)+2}{\left(x^{2}-x-5\right)\left(x^{2}-x-1\right)+4}\);
2) \(\frac{x+5-5 \sqrt{x-1}}{x-1-3 \sqrt{x-1}}\). | 18. 1) \(\frac{x^{2}-x-4}{x^{2}-x-3}\). Hint. Use the following substitution: \(y=x^{2}-x-5\).
2) \(\frac{\sqrt{x-1}-2}{\sqrt{x-1}} \cdot\) Hint. Use the substitution: \(\sqrt{x-1}=y\) | \frac{x^{2}-x-4}{x^{2}-x-3} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 43,571 |
23*. Plot the graphs of the equations:
1) \(x=y^{2}\)
2) \(x=(y-2)^{2}\)
3) \(x=-y^{2}+1\)
4) \(x=(y+1)^{2}-2\). | 23. 1) \(\mathrm{P}\) is a solution. If we consider the variable \(y\) as independent and the variable \(x\) as dependent, the desired graph will be a parabola depicted in figure 35. By swapping the \(O x\) and \(O y\) axes, we obtain the desired graph of the equation in the coordinate system familiar to us (fig. 36).
... | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 43,575 |
1. Prove that the median of a triangle is less than half the sum of the sides enclosing it. | 1. Extend the median $C M_{3}$ beyond point $M_{3}$ and construct point $C_{1}$ such that $C M_{3}=M_{3} C$. Consider the triangle $C C_{1} A$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,577 |
4. The bisector of angle $C$ of triangle $ABC$ meets its side $AB$ at point $L_{3}$. Prove that if $AC > BC$, then $AL_{3} > BL_{3}$. Prove the converse theorem. | 4. Notice that $A<180^{\circ}-B$. Reflect the smaller side $B C$ with respect to the bisector $C L_{3}$. If $B_{1}$ is the reflected vertex, then consider the triangle $A B_{1} L_{3}$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,580 |
5. Based on the base $AC$ of the isosceles triangle $ABC$, two arbitrary points $E$ and $F$ are taken, through which two lines are drawn, forming angles with the base $AC$ equal to the angles at the base of the triangle, and intersecting at point $D$ (inside the triangle), meeting its sides $BC$ and $BA$ at points $L$ ... | 5. Consider the three isosceles triangles that have formed. From the equality $A B=B C$, it follows that $B K+K A=B L+L C$, $B K+K D+D F=B L+L D+D E$ and $B K+K D=B L+L D$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,581 |
6. Inside triangle $ABC$, a point $M$ is taken. Prove that angle $ABC$ is less than angle $AMC$.
untranslated text remains the same as requested. | 6. Draw the line $B M$ and use the exterior angle theorem of a triangle twice.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。
6. Draw the line $B M$ and use the exterior angle theorem of a triangle twice. | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,582 |
7. Prove that if the bisector of a triangle divides its perimeter in half, then the triangle is isosceles.
Prove that if the bisector of a triangle divides its perimeter in half, then the triangle is isosceles. | 7. F i r s t m e t h o d. Prove by contradiction using axial symmetry and the theorem about the ratio of sides of a triangle.
S e c o n d m e t h o d. Let $C A + A L_{3} = C B + B L_{3}$, where $C L_{3}$ is the angle bisector of angle $C$ in triangle $A B C$. On the extension of $C A$, mark a segment $A M$ equal to $A... | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,583 |
9. From points $A$ and $B$, located on one side of an acute angle $O$, perpendiculars $A A_{1}$ and $B B_{1}$ are dropped to the other side of the angle. Prove that if $O A < O B$, then $\angle O A A_{1} \geqslant \angle O B B_{1}$. | 9. Note that $\angle O B B_{1}+\angle A_{1} A B \leqslant 180^{\circ}$, and $\angle O A A_{1}+\angle A_{1} A B=$ $=180^{\circ}$. Therefore, $\angle O A A_{1} \geqslant \angle O B B_{1}$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,585 |
10. Prove that the greater height of a triangle corresponds to the smaller side and vice versa. | 10. Let $A H_{1}>B H_{2}$ and assume that $B C>A C$. Combine the right triangles $A C H_{1}$ and $B C H_{2}$ by their equal acute angles and consider the quadrilateral with two adjacent right angles. Use the result from problem № 9. | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,586 |
11. From the vertex of the right angle $C$ of the right triangle $ABC$, a height $CH_{3}$ is dropped to the hypotenuse. Prove that $\angle A C H_{3} \geqslant$ $\geqslant \angle A B C, \quad \angle B C H_{3} \geqslant \angle B A C$.
18 | 11. Extend the leg $A C$ beyond vertex $C$ and construct point $C_{1}$ such that $A C_{1}=A B$. Drop a perpendicular $C_{1} D_{1}$ from point $C_{1}$ to $A B$. Notice that $\angle A C_{1} D_{1}=\angle A B C, \angle A C_{1} D_{1} \leqslant \angle A C H_{3}$ (see problem № 9 ). | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,587 |
14. Prove that if two angles and the side adjacent to one of them of one triangle are respectively equal to the corresponding elements of another triangle, then the triangles are congruent. | 14. F i r s t m e t h o d. Assume the opposite. Superimpose one triangle on the other in a proper manner (Fig. 1) and use the theorem about the exterior angle of a triangle.
Fig. 1
S e c o ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,590 |
15. Prove that two triangles are equal to each other if two sides and the angle opposite the larger of them in one triangle are respectively equal to the corresponding elements of the other triangle. | 15. Let $A C=A_{1} C_{1}, B C=B_{1} C_{1}, B C>A C, \angle A=\angle A_{1}$. Suppose that $A B>A_{1} B_{1}$. Construct a point $B_{2}$ on $A B$ such that $A B_{2}=$ $=A_{1} B_{1}$. From the equality of triangles $A B_{2} C$ and $A_{1} B_{1} C_{1}$, it follows that $C B=C B_{2}$, hence angle $B$ is acute, while angle $A ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,591 |
16. Prove that if the medians $C M_{3}$ and $C^{\prime} M_{3}^{\prime}$ of triangles $A B C$ and $A^{\prime} B^{\prime} C^{\prime}$ are equal, and the corresponding parts of angles $C$ and $C^{\prime}$, into which these medians divide them, are also equal, then the triangles are equal. | 16. First method. Assume the opposite. Superimpose one triangle on the other and, by establishing the equality of the resulting triangles, discover a contradiction with the theorem about the exterior angle of a triangle.
Second method. Extend the medians \( C M_{3} \) and \( C^{\prime} M^{\prime}_{3} \) beyond points ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,592 |
17. On the external bisector of angle $C$ of triangle $A B C$, a point $M$ is taken. Prove that $A C + C B < A M + M B$. | 17. Reflect point $B$ with respect to the bisector. Connect the obtained point $B^{\prime}$ on the extension of side $A C$ with point $M$ and consider triangle $A M B^{\prime}$, in which $A B^{\prime}=A C+C B, M B^{\prime}=M B$. Similarly, consider the case when point $M$ lies on the other bisector. | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,593 |
21. Prove that if the segment $G M_{3}$ of the median $C M_{3}$ of triangle $A B C$ ($G$ - centroid of the triangle) is equal to $\frac{1}{2} A B$, then $C G=2 G M_{3}$. | 21. Extend the median $\mathrm{CM}_{3}$ beyond point $M_{3}$ and construct point $C^{\prime}$ such that $G M_{3}=M_{3} C^{\prime}$. From vertices $C, B$ and point $C^{\prime}$, drop perpendiculars $C C_{0}, B B_{0}, C^{\prime} C^{\prime}{ }_{0}$ to the median $A M_{1}$ and prove that $C C_{0}=B B_{0}, B B_{0}=C^{\prime... | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,597 |
22. Prove that the midline of a triangle is not greater than half of the side it does not intersect. | 22. On the midline $M_{1} M_{2}$
Fig. 2 drop perpendiculars $A A_{1}, B B_{1}$, and $C C_{1}$. Prove that $A_{1} B_{1}=2 M_{1} M_{2}$, considering three pairs of equal right triangles. Not... | M_{1}M_{2}\leqslant\frac{1}{2}AB | Geometry | proof | Yes | Yes | olympiads | false | 43,598 |
23. The hypotenuses of two right triangles are equal. Prove that each leg of one triangle cannot be greater than each of the legs of the other triangle. | 23. Attach both triangles $A B C$ and $A_{1} B_{1} C_{1}$ to each other so that their hypotenuses coincide. Notice that the segment $C C_{1}$ intersects side $A B$. From this, the assumption that $A C > A_{1} C_{1}$, $B C > B_{1} C_{1}$, leads to a contradiction. | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,599 |
24. Given two non-intersecting lines and their center of symmetry $O$. A right angle with its vertex at point $O$ rotates around its vertex, with one side intersecting line $a$ at point $A$, and the other side intersecting line $b$ at point $B$. Prove that the distance from point $O$ to the line $A B$ remains constant. | 24. Extend the ray $O A$ to intersect the line $b$ and, using the properties of central symmetry, establish the equality of two right triangles. The distance from point $O$ to the line $A B$ is equal to the distance from this point to the given line $b$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,600 |
25. Through a point $M$, located on the base $A B$ of an isosceles triangle $A B C$, a secant is drawn, intersecting its lateral sides at points $P$ and $Q$ such that $M P=M Q$. Prove that this secant cuts off equal segments from the points $A$ and $B$ on the lateral sides (here the sides are considered as lines). | 25. Drop perpendiculars $P P_{1}$ and $Q Q_{1}$ from points $P$ and $Q$ to the base $A B$ of the triangle and consider pairs of equal triangles: $M P P_{1}, M Q Q_{1}$ and $P P_{1} A, Q Q_{1}$ B. | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,601 |
27. Prove that a line passing through the midpoints of two sides of a triangle is perpendicular to the line drawn perpendicular to the third side through its midpoint.
## § 2. Quadrilaterals | 27. Drop perpendiculars from the vertices of the triangle onto the line and consider the quadrilateral with two right angles and two equal sides, for which the perpendicular bisector of the given triangle is the axis of symmetry. | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,603 |
29. Angles $B$ and $D$ of quadrilateral $A B C D$ are equal, and diagonal $A C$ is bisected by the other diagonal. Prove that the opposite sides of the quadrilateral are equal. | 29. Prove by contradiction, reflecting triangle $A B C$ about the point of intersection of the diagonals of the quadrilateral. | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,605 |
31. Prove that if two opposite sides of a quadrilateral are equal, as well as two angles adjacent to the third side, then the other two angles of the quadrilateral are equal. | 31. Connect the midpoint of the third side to the vertices located on the opposite side, and consider the three triangles formed. | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,607 |
35. The distances from vertices $A$ and $B$ of quadrilateral $ABCD$ to side $CD$ are equal. Prove that if $AC + CB = AD + DB$, then $AD = BC$ and $AC = BD$.
## § 3. Circle | 35. Reflect point $B$ with respect to $DC$ and connect the obtained point $B^{\prime}$ with point $A$. If $AB^{\prime}$ intersects $CD$ at point $M$, then $AC + CB > AB^{\prime}$. For all points $P$ located on side $CD$, $AP + PB > AM + MB$. Therefore, segments $CM$ and $DM$ cannot be unequal, otherwise it would contra... | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,611 |
36. Prove that the geometric locus of points of tangency of pairs of circles touching a given line at two given points on it is a circle. | 36. Draw a common tangent $t$ to two touching circles at their point of contact $T$. If $t$ meets a given line at point $M$, then establish that the segment $T M$ maintains a constant length. | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,612 |
37. Prove that the angle at which a diameter of a circle is seen from a point on the circle is not greater than $90^{\circ}$. | 37. Let $A B$ be the diameter of circle $O$, and $C$ be a point on the circle. Consider two isosceles triangles $A O C$ and $B O C$ and note that the sum of the angles in a triangle does not exceed $180^{\circ}$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,613 |
38. A quadrilateral $A B C D$ is circumscribed around a circle $O$. Prove that $\angle B O C+\angle D O A=\angle A O B+\angle C O D$. | 38. Drop perpendiculars from the center of the circle to the sides of the quadrilateral. Consider four pairs of equal angles and note that their sum is $360^{\circ}$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,614 |
40. On the extensions of the chord $A B$ of a given circle, equal segments $M A$ and $N B$ are laid out. From points $M$ and $N$ on opposite sides of $M N$, tangents $M T_{1}$ and $N T_{2}$ are drawn to the circle. Prove that the segment $T_{1} T_{2}$ bisects the segment $M N$. | 40. Construct a circle concentric with the given one and passing through points $M$ and $N$, extend the tangents until they intersect with the constructed circle, and for the resulting quadrilateral, construct the axis of symmetry. Establish the perpendicularity of the chord $T_{1} T_{2}$ to this axis. Use problem № 27... | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,616 |
41. In a circle, two intersecting equal chords are drawn. Prove that the common point of these chords divides them into respectively equal parts. | 41. Connect the ends of equal chords $A B$ and $C D$, intersecting at point $M$, and establish the equality of segments $A C$ and $B D$, triangles $A B C$ and $D B C$, and angles $C B M$ and $M C B$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,617 |
42. A quadrilateral is circumscribed around a circle, one of its diagonals is bisected by the other. Prove that the quadrilateral is a kite. | 42. Let the diagonals $A C$ and $B D$ intersect at point $O$, such that $B O = O D$. Reflect triangle $A B C$ about the midpoint $M$ of side $A C$ to form triangle $A C B'$. Assume that $A O > A M$. Quadrilateral $A C D B'$ has the property that the distances from its vertices $D$ and $B'$ to side $A C$ are equal. Furt... | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,618 |
43. Given two concentric circles with center $O$ and points $A$ and $B$, lying on the outer circle. From these points, tangents are drawn to the inner circle such that one tangent meets the other at point $M$, which is not on the perpendicular from point $O$ to the chord $A B$. Tangents are drawn from points $A$ and $B... | 43. Let the lines $A M$ and $B M$ intersect the circumcircle at points $C$ and $D$. The tangents at points $C$ and $D$ intersect at point $Q$, which is symmetric to $P$ with respect to the axis of symmetry of the quadrilateral $A B C D$. Considering that the diagonals $A C$ and $B D$ bisect the segment $P Q$ (see probl... | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,619 |
46. From point $S$ to circle $O$, tangents $S A$ and $S B$ are drawn ($A, B$ - points of tangency), which are intersected by a third tangent at points $M$ and $M_{1}$. Prove that angles $A O M$ and $S O M_{1}$ are either equal or their sum is $180^{\circ}$. | 46. Let a circle lie outside the triangle $S M M_{1}$ and touch the side $M M_{1}$ at point $T$. Consider that $\angle A O M=\angle M O T, \angle T O M_{1}=$ $=\angle M_{1} O B, \angle A O S=\angle S O B$. Hence, $2 \angle A O M+\angle T O S=2 \angle B O M_{1}-$ $-\angle T O S$, or $\angle A O M=\angle B O M_{1}-\angle... | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,622 |
47. Prove that the altitudes of an acute triangle intersect at one point. | 47. Let the altitudes $A H_{1}$ and $C_{3}$ intersect at point $H$. Reflect the line $H_{1} H_{3}$ with respect to $A H_{1}$ and $C H_{3}$. Prove that point $A$ is equidistant from the reflected lines, which implies that these lines intersect at some point $P$. The vertices $A$ and $C$ lie on the same axis of symmetry ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,623 |
48. Prove that if the sums of the opposite sides of a quadrilateral are equal, then its angle bisectors intersect at one point. | 48. According to the condition $A B+C D=A D+B C$. From this, $A B-B C=$ $=A D-D C$, therefore if $A B>B C$, then $A D>D C$. On the sides $A B$ and $A D$, we lay off segments $A M$ and $A N$ so that $B C=B M, D C=$ $=D N$. The bisector of angle $A$ intersects side $B C$ (or $C D$) at point $K$, which is established base... | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,624 |
49. Angles $A$ and $C$ of the convex quadrilateral $A B C D$ are equal, and moreover, $A B+C D=B C+A D$. Prove that the quadrilateral is a kite. | 49. Inscribe a circle in the given quadrilateral and establish the equality of sides $A B$ and $B C$, using the property of tangents drawn from a single point to a circle. | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,625 |
50. Prove that the segment of the common internal tangent, enclosed between the external tangents drawn to two equal circles, is not greater than the distance between the centers of the circles. | 50. Connect the centers $O_{1}$ and $O_{2}$ with points $P$ and $Q$, where the internal tangent intersects the external ones. The point of intersection of the diagonals of the resulting quadrilateral $O_{1} P O_{2} Q$ is its center of symmetry. Therefore, the opposite angles of the quadrilateral are equal, but $\angle ... | PQ\leqslantO_{1}O_{2} | Geometry | proof | Yes | Yes | olympiads | false | 43,626 |
53. On the median $C M_{3}$ of triangle $A B C$, an arbitrary point $P$ is taken. The lines $A P$ and $B P$ intersect the sides $B C$ and $A C$ at points $A_{1}$ and $B_{1}$, respectively. Prove that if the segments $A A_{1}$ and $B B_{1}$ are equal, then the triangle is isosceles. | 53. At points $A, B, C$ of the plane of triangle $ABC$ (Fig. 4), erect perpendiculars and lay off segments $A A_{0}, B B_{0}$, and $C C_{0}$ such that line $B_{0} C_{0}$ passes through point $A_{1}$, and line $A_{0} C_{0}$ passes through point $B_{1}$. It will turn out that $A A_{0}=B B_{0}$, since segments $A A_{1}$ a... | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,629 |
54. Using stereometry, prove that the medians of a triangle intersect at one point. | 54. To the plane of the triangle at points $A, B, C$, erect perpendiculars and lay off equal segments $A A_{0}, B B_{0}, C C_{0}$ (Fig. 5), of which two segments ($A A_{0}$ and $B B_{0}$) are directed to one side of the plane, and the third to the other side of the plane. Draw planes $A_{0} B C_{0}$ and $A B_{0} C_{0}$... | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,630 |
55. On the height $C H_{3}$ of triangle $A B C$ (point $H_{3}$ lies between vertices $A$ and $B$), a point $P$ is given. Lines $A P$ and $B P$ intersect sides $B C$ and $A C$ at points $A_{1}$ and $B_{1}$, respectively. Prove that $H_{3} C$ is the bisector of angle $A_{1} H_{3} B_{1}$. Prove the converse theorem. | 55. The pair of lines $H_{3} A_{1}, H_{3} B_{1}$ divides harmonically the pair of lines $H_{3} C$ and $H_{3} A$ (property of a complete quadrilateral). But by the condition, the lines $\mathrm{CH}_{3}$ and $A B$ are perpendicular, and therefore, the lines $C_{3}$ and $A B$ are axes of symmetry for the pair of lines $H_... | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,631 |
57. In the plane of a triangle, a point $M$ is given, which is reflected sequentially relative to all vertices of the triangle once and then a second time. Prove that after the last reflection, the reflected point coincides with point $M$. | 57. Use the fact that the sum of two reflections with respect to two points is a parallel translation, and establish that the sum of the specified reflections with respect to the vertices of a triangle is an identity transformation. | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,633 |
58. A line passing through the center of a parallelogram intersects its sides at points $P$ and $Q$, which are connected to the vertices of the parallelogram. Prove that the points of intersection of segments $A P$, $B P$, $C Q$, $D Q$ with the diagonals of the parallelogram are the vertices of a new parallelogram. | 58. Consider the parallelograms $A P C Q$ and $Q D P B$, which have the center of the given parallelogram as their center, and show that in the resulting quadrilateral, the diagonals bisect each other. | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,634 |
59. Prove that the points symmetric to point $M$ with respect to the midpoints of the sides of a quadrilateral are the vertices of a parallelogram. | 59. Use the property of the midline of a triangle to show that the opposite sides of the resulting quadrilateral are parallel and equal to the corresponding diagonals of the given quadrilateral. | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,635 |
60. Points $A_{1}, B_{1}, C_{1}$ are constructed symmetric to point $M$ with respect to the midpoints of the sides of triangle $A B C$. Prove that: 1) triangles $A B C$ and $A_{1} B_{1} C_{1}$ are equal; 2) the lines $A A_{1}, B B_{1}, C C_{1}$ intersect at one point. | 60. Use the property of the midline of a triangle to prove that the corresponding sides of triangles $A B C$ and $A_{1} B_{1} C_{1}$ are equal. Then show that $A B_{1} A_{1} B$ and $B_{1} C B C_{1}$ are parallelograms with a common center of symmetry, through which the diagonals $A A_{1}, B B_{1}$ and $C C_{1}$ pass. | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,636 |
61. In the plane of quadrilateral $A B C D$, a point $M$ is given. Points $M_{1}$, $M_{2}$, and $M_{3}$ are constructed as follows: $M_{1}$ is symmetric to $M$ with respect to the midpoint of side $A B$; $M_{2}$ is symmetric to $M_{1}$ with respect to the midpoint of side $B C$; $M_{3}$ is symmetric to $M_{2}$ with res... | 61. Establish the equality and parallelism of segments $M_{3} D$, $C M_{2}$, $M_{1} B$ and $A M$. Therefore, the quadrilateral $M_{3} D M A$ is a parallelogram and point $M$ is symmetric to point $M_{3}$ with respect to the midpoint of side $A D$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,637 |
62. Two opposite sides of one quadrilateral are respectively equal and parallel to two opposite sides of another quadrilateral. Prove that the midline of the first quadrilateral, which bisects its two other opposite sides, is parallel and equal to the corresponding midline of the other quadrilateral. (Both quadrilatera... | 62. Let for quadrilaterals $A B C D$ and $A_{1} B_{1} C_{1} D_{1}$ it holds: $\overrightarrow{A B}=\overrightarrow{A_{1} B_{1}}, \overrightarrow{D C}=\overrightarrow{D_{1} C_{1}}$, where $M$ and $N$ are the midpoints of $A D$ and $B C$, and $M_{1}, N_{1}$ are the midpoints of $A_{1} D_{1}$ and $B_{1} C_{1}$. Translate ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,638 |
64. The line $m$ does not intersect the sides of triangle $ABC$. Through the vertices of the triangle, parallel lines are drawn, intersecting line $m$ at points $A_{1}, B_{1}, C_{1}$ respectively, such that $A A_{1} + B B_{1} = C C_{1}$. Prove that the lines $m$, possessing the given property, belong to a pencil. | 64. Through the midpoint $M_{3}$ of side $A B$, draw a line parallel to $C C_{1}$ and intersecting line $m$ at point $N_{3}$. According to the condition $C C_{1}=2 M_{3} N_{3}$, and therefore $C M_{3}$ intersects line $m$ at point $D$ such that $C M_{3}=M_{3} D$. Thus, point $D$ is fixed and lines $m$ belong to a penci... | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,640 |
65. In the plane of a parallelogram, a point $P$ is given, through which two lines parallel to the sides of the parallelogram are drawn, intersecting them at points $A$ and $C, B$ and $D$ respectively. Prove that the center $M$ of the parallelogram, point $P$, and point $S$ of intersection of the midlines of quadrilate... | 65. Use the fact that in quadrilateral $A B C D$, point $S$ and the midpoints $K$ and $L$ of diagonals $A C$ and $B D$ lie on the same line, and $K S=S L$. Prove that $P L M K$ is a parallelogram, and establish that $M P$ and $L K$, as diagonals of this parallelogram, intersect at point $S$ and are bisected by it. | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,641 |
66. The midline of a quadrilateral divides it into two quadrilaterals. Prove that the midpoints of the diagonals of these two quadrilaterals are vertices of a parallelogram or lie on the same line, representing a degenerate parallelogram. | 66. Use the fact that the midline and the corresponding median of a triangle, intersecting, are bisected. Apply this property twice. | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,642 |
70. The segment connecting the midpoints $M_{1}$ and $M_{3}$ of two opposite sides of a convex quadrilateral intersects its diagonals at points $P$ and $Q$. Prove that if $M_{1} P=M_{3} Q$, then the quadrilateral is a trapezoid or a parallelogram. | 70. Let point $M_{1}$ be the midpoint of $A B$, and point $M_{3}$ be the midpoint of $D C$ (Fig. 6). Assume the opposite and draw lines $B B_{1}$ and $C C_{1}$ through vertices
Fig. 6 $B$ ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,646 |
71. A secant is drawn through the centroid of triangle $ABC$. Lines drawn through the vertices of the triangle parallel to each other intersect the secant at points $A_{1}, B_{1}, C_{1}$ respectively. Prove that the sum of the two segments, located on one side of the secant, from the obtained three segments $A A_{1}, B... | 71. If points $A$ and $B$ are on the same side of the secant, then draw a line through the midpoint $M_{3}$ of side $AB$ parallel to $AA_{1}$. Use the property of the midline of a trapezoid and the property of the point of intersection of the medians of a triangle. | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,647 |
72. On segments $A B$ and $B C$ (point $B$ lies between $A$ and $C$) of the same straight line, equilateral triangles $A M B$ and $B N C$ are constructed on the same side of the line. Prove that the midpoints of segments $M C$, $A N$, and point $B$ are the vertices of an equilateral triangle. Determine the validity of ... | 72. Establish the equality of triangles $A B N$ and $M B C$ and use the rotation of one of these triangles around point $B$ by an angle of $60^{\circ}$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,648 |
73. Side $AC$ of triangle $ABC$ is rotated by an angle of $+90^{\circ}$ around point $A$, and side $BC$ is rotated by an angle of $-90^{\circ}$ around point $B$. Prove that if $AC_{1}$ and $BC_{2}$ are the new positions of the rotated sides, then the midpoint of segment $C_{1} C_{2}$ does not depend on the position of ... | 73. The sum of two rotations about two centers by $90^{\circ}$ in the same direction is a central symmetry.
90
In this symmetry, point $C_{1}$ is mapped to point $C_{2}$. Therefore, the midpoint $M$ of segment $C_{1} C_{2}$ is the center of symmetry, which, together with points $A$ and $B$, forms an isosceles right t... | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,649 |
74. Side $AC$ of triangle $ABC$ is rotated around vertex $A$ by an angle of $+90^{\circ}$ and after the rotation, it occupies the position $AC_{1}$. Side $BC$ is rotated around vertex $B$ by an angle of $+90^{\circ}$ and after the rotation, it occupies the position $BC_{2}$. Prove that segment $C_{1}C_{2}$ has a consta... | 74. The sum of the two specified rotations is a parallel translation. Therefore, the vector $\overrightarrow{C_{1}} C_{2}$ characterizes this translation. If point $C$ coincides with $A$, then $C_{1}$ also coincides with $A$, and point $C_{2}$ together with $A$ and $B$ form a right isosceles triangle with the right ang... | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,650 |
76. Prove that two quadrilaterals are equal if the sides and the midline of one of them are respectively equal to the corresponding elements of the other. | 76. Consider parallelograms, the vertices of which coincide with the ends of the midline and the midpoints of the diagonals of the given quadrilaterals, and establish their equality. Then translate the midline parallel to itself in the direction of one of the sides by a segment equal to half of this side, and consider ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,652 |
77. Prove that the internal bisectors of a parallelogram form a rectangle, the diagonals of which are parallel to the sides of the parallelogram. Establish the validity of the theorem for the bisectors of the external angles. | 77. Since the bisectors of two consecutive angles of a parallelogram are perpendicular, the quadrilateral $M N P Q$ formed by them is a rectangle. Extend the bisector $A M$ to intersect $B C$ at point $E$, and the bisector $C P$ to intersect $A D$ at point $F$ and establish that $A M=M E=C P=P F$, i.e., $M E C P$ is a ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,653 |
78. Prove that: 1) if the midlines of a quadrilateral are equal, then its diagonals are perpendicular, and vice versa; 2) if the midlines of a quadrilateral are perpendicular, then its diagonals are equal, and vice versa. | 78. Consider a parallelogram, the vertices of which coincide with the ends of the midlines of a quadrilateral (in the first case, this parallelogram is a rectangle, and in the second case, it is a rhombus), and use the property of the midline of a triangle. | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,654 |
82. Prove that if the diagonals of a trapezoid are equal, then the trapezoid is isosceles. | 82. Using parallel translation, prove that the diagonals $A C$ and $B D$ of trapezoid $A B C D$ form equal angles with the base $A D$. Triangles $A B D$ and $A C D$ are equal, from which it follows that the lateral sides $A B$ and $C D$ of the given trapezoid are equal. | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,658 |
83. Angles $A$ and $C$ of quadrilateral $ABCD$ are equal, and diagonal $AC$ is bisected by the other diagonal. Prove that the quadrilateral is a kite or a parallelogram. | 83. If angle $A O B$ is a right angle, then $B D$ is the axis of symmetry and quadrilateral $A B C D$ is a kite. If this angle is acute, then by reflecting point $C$ across the diagonal $B D$, we obtain point $C^{\prime}$ such that a circle can be circumscribed around quadrilateral $A C^{\prime} B D$, and this quadrila... | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,659 |
84. A plane undergoes a rotation transformation sequentially about the centers $A, B, C, D$ in the same direction by $90^{\circ}$ (about each center). Prove that the sum of these rotations is the identity transformation if and only if the segments $A C$ and $B D$ are equal and perpendicular. | 84. We will prove that segments $A C$ and $B D$ are equal and perpendicular. The sum of rotations around vertices $A$ and $B$ is a symmetry. The sum of rotations around vertices $C$ and $D$ is also a symmetry. The sum of these two symmetries is the identity. Therefore, both centers of symmetry coincide. From this cente... | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,660 |
85. On the sides of a quadrilateral, semicircles are constructed with the sides as diameters, and two opposite semicircles are turned inward while the other two are turned outward. Prove that the midpoints of these semicircles are the vertices of a parallelogram. | 85. Let the midpoints of the semicircles be denoted as $O_{1}, O_{2}, O_{3}, O_{4}$. The sum of rotations around these centers by angles $+90^{\circ}, -90^{\circ}, +90^{\circ}$, $-90^{\circ}$ maps vertex $A$ of quadrilateral $A B C D$ to itself. Therefore, the sum of these four rotations is the identity. The sum of the... | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,661 |
86. On the sides of a quadrilateral, semicircles are constructed, located outside the quadrilateral, with the sides as their diameters. Prove that the midpoints of the arcs of these semicircles are the vertices of a quadrilateral with equal and perpendicular diagonals. Prove the validity of the theorem in the case when... | 86. Let the midpoints of the semicircles constructed on the sides $A B, B C, C D, D A$ of the quadrilateral $A B C D$ outside it be denoted by $O_{1}, O_{2}, O_{3}, O_{4}$, respectively. The rotation of vertex $A$ around these centers sequentially by $90^{\circ}$ in the same direction brings point $A$ back to itself. T... | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,662 |
88. Prove that if the sum of the midlines of a quadrilateral is equal to its semiperimeter, then the quadrilateral is a parallelogram. | 88. Establish that the distance between the midpoints of two opposite sides of a quadrilateral is not greater than half the sum of the other two opposite sides,
Figure 7 i.e., \( m_{1} \le... | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,664 |
89. Prove that angle $C$ of triangle $A B C$ is equal to $135^{\circ}$, if $A H_{3}=2 h_{3}, B H_{3}=3 h_{3}\left(H_{3}\right.$ - the base of the height $\left.h_{3}\right)$.
## § 7. Circle | 89. Complete triangle $ABC$ to parallelogram $ACBD$ and prove that its diagonal $CD$ is equal to and perpendicular to side $AC$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,665 |
91. In one of two intersecting circles, a chord $AB$ is drawn, and in the other, a parallel chord $CD$. Prove that the segment $AC(AD)$ is seen from one of the intersection points of the circles at the same angle or supplementary to $180^{\circ}$ as the segment $BD(BC)$ is seen from the other intersection point of the ... | 91. Draw the common chord $M N$ of the circles and extend $C M$ to intersect the extension of $A B$ at point $S$ (Fig. 7). Obviously, $\angle B N D=\angle B A M+\angle M C D=\angle B A M+\angle M S B=\angle A M C$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,667 |
94. At two fixed points $A$ and $B$ of a circle and at a variable point $M$ of the arc $A B$, tangents are drawn. Prove that the segment of the variable tangent, intercepted by the two fixed tangents, is seen from the center of the circle under a constant angle. | 94. Consider two cases corresponding to the two arcs $A B$. In one case, the constant angle is equal to $90^{\circ}-\frac{\angle C}{2}$, and in the other $90^{\circ}+\frac{\angle C}{2}$, where $C$ is the point of intersection of the tangents at points $A$ and $B$. Pay attention to the case when the tangents at points $... | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,670 |
95. The bisector of a triangle divides it into two triangles, in which circles are inscribed. Prove that if the radii of these circles are equal, then the given triangle is isosceles. | 95. Use the fact that tangents drawn from the same point to a circle are equal, and also that if the angles between tangents drawn to equal circles are equal, then the tangent segments are also equal. | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,671 |
96. On the extensions of the chord $A B$ of a given circle, points $C$ and $D$ are taken such that the segments $C T_{1}$ and $D T_{2}$ of the tangents, drawn on the same side of the chord $A B$, are equal. Prove that the segments $A C$ and $B D$ are equal, and the line $T_{1} T_{2}$ is parallel to the line $A B$. | 96. Establish the equality of triangles $C T_{1} T_{2}$ and $D T_{1} T_{2}$, which leads to the parallelism of $C D$ and $T_{1} T_{2}$. Construct the axis of symmetry of the resulting isosceles trapezoid. Note that the problem can be referred to absolute geometry. | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,672 |
97. Chord $AB$ of a certain circle is extended in both directions, and equal segments $AC$ and $BD$ are laid off on the extensions. Tangent rays $CT_{1}$ and $DT_{2}$ are drawn from points $C$ and $D$ to the circle in the same half-plane relative to $CD$. Prove that the line $T_{1} T_{2}$ is parallel to the line $AB$. | 97. Establish the equality of angles $C T_{1} T_{2}$ and $D T_{2} T_{1}$ and the equality of segments $C T_{1}$ and $D T_{2}$ (see problem № 96). | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,673 |
99. The difference between angles $A$ and $B$ of triangle $ABC$ is $90^{\circ}$. Prove that the distance from the foot of the altitude dropped onto side $AB$ to the midpoint of this side is equal to the radius of the circumcircle of the given triangle. | 99. Determine that the circumcircle of the given triangle touches the altitude $C H_{3}$. Discover that the quadrilateral $\mathrm{OCH}_{3} M_{3}$ is a rectangle. | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,675 |
101. The vertices of a quadrilateral are arranged as follows: one at the center of a given circle, the second outside this circle, the last two on the tangents drawn from the second point to the circle, at equal distances from the center and on opposite sides of the chord connecting the points of tangency. Prove that a... | 101. Connect the center of the circle with the points of tangency and consider two equal right triangles. | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,677 |
102. Tangents are drawn at the ends of the chord $A B$ of circle $O$, meeting at point $S$. Prove that the midpoint $M$ of the arc $A B$, located outside triangle $A B C$, is the excenter of this triangle. | 102. Verify that $A M$ and $B M$ are the external bisectors of angles $A$ and $B$ of triangle $A B M$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,678 |
103. Circles $O$ and $O_{1}$ intersect at points $A$ and $B$. Lines $O A$ and $O_{1} A$ meet circles $O_{1}$ and $O$ respectively at points $C$ and $D$. Prove that points $B, O, D, C$ and $O_{1}$ lie on the same circle. | 103. Establish that points $C$ and $D$ belong to the circumcircle of triangle $O O_{1} B$. Use the properties of a quadrilateral that can be inscribed in a circle. | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,679 |
106. A secant is drawn through the intersection point of two circles, meeting these circles again at points $A$ and $B$. Determine the direction of the secant such that the segment $A B$ becomes the largest. | 106. Drop perpendiculars from the centers of the circles to an arbitrary secant; through the midpoint of one of the chords, draw a line parallel to the line of centers, and consider the resulting right triangle. | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 43,682 |
108. Common tangents are drawn to two circles of equal radii. Prove that the segment of the common internal tangent, enclosed between the external tangents, is equal to the segment of the external tangent, enclosed between the points of tangency. | 108. Connect each of the intersection points of the internal tangents of the given circles with their external tangents to the centers of the circles, and establish that the resulting quadrilateral is a rectangle. Use the central symmetry of the given circles. | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,684 |
109. Two equal circles of radius $R$ intersect at points $A$ and $B$ at an angle $\varphi\left(\varphi=\angle O_{1} A O_{2}\right)$. In one of the circles, a chord $A M$ is drawn, and in the other, a chord $A N$ is drawn such that the angle $M A N$ is equal to $\frac{\varphi}{2}$ (or the supplementary angle). Prove tha... | 109. Since the angle $O_{1} A O_{2}$ is equal to $\varphi$, then $\angle B A C = \angle M A N = \frac{\varphi}{2}$ (Fig. 8). From this, $\angle M B = \angle N C$. By translating the circles parallel to the vector $\overrightarrow{O_{1} O_{2}}$, verify that point $B$ coincides with point $C$, and therefore point $M$ coi... | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,685 |
111. Two circles touch each other at point $T$. A secant meets the first circle at points $A$ and $B$, and the second circle at points $C$ and $D$. Prove that the bisectors of angles $A T B$ and $C T D$ are either perpendicular or coincide. | 111. Let the circles touch each other externally. Draw lines $T C$ and $T D$, intersecting the first circle at points $C_{1}$ and $D_{1}$. Establish the parallelism of chords $A B$ and $C_{1} D_{1}$ and the perpendicularity of the bisectors of angles $A T B$ and $D_{1} T C_{1}$. Conduct similar reasoning for the second... | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,687 |
112. Prove that if two circles intersect, then the segment of their common tangent, bounded by the points of tangency, is seen from the points of intersection of the circles at angles whose sum is $180^{\circ}$. | 112. Let the circles intersect at points $A$ and $B$, and the common tangent touches them at points $T_{1}$ and $T_{2}$. Establish that $\angle A T_{1} T_{2}=\angle T_{1} B A, \angle A T_{2} T_{1}=\angle T_{2} B A$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,688 |
113. Two circles have an internal tangency at point $K$. At an arbitrary point $P$ on the inner circle, a tangent is drawn, intersecting the outer circle at points $A$ and $B$. Prove that the segments $A P$ and $B P$ are seen from point $K$ at equal angles. | 113. If $K A$ and $K B$ intersect the inner circle at points $A_{1}$ and $B_{1}$, then establish the parallelism of the lines $A B$ and $A_{1} B_{1}$. Note that point $T$ is the midpoint of the arc $A_{1} T B_{1}$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,689 |
114. Two circles intersect at points $A$ and $B$. A tangent to one of them at an arbitrary point $T$ intersects the other circle at points $C$ and $D$. Prove that the angles under which the segment $C T$ is seen from point $A$ and the segment $T D$ - from point $B$, are either equal or their sum is $180^{\circ}$. | 114. Let point $T$ lie between $C$ and $D$. If $A T$ and $B T$ intersect the circle again at points $P$ and $Q$, then $\angle A T C = \angle A B T = \angle A P Q$, and therefore $P Q \| C D$. Hence, $P C = Q D$ and $\angle C A P = \angle Q B D$. If point $T$ lies outside the segment $C D$, then $\angle A T C + \angle A... | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,690 |
115. Through the intersection point of two circles, an arbitrary secant is drawn, which intersects the circles for the second time at points $A$ and $B$. Prove that the angles between the tangents drawn to the circles at points $A$ and $B$ are constant. | 115. Connect points $A$ and $B_{1}$ with the second intersection point of the given circles, as well as connect both intersection points. Use the properties of inscribed angles and angles formed by a tangent and a chord. Consider the case when the intersection point of the circles lies between points $A$ and $B_{1}$, a... | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,691 |
117. Three pairwise intersecting circles of the same radius have a common point $M$. Prove that the second points $A, B, C$ of intersection of these circles are the vertices of a triangle for which point $M$ is the orthocenter, and the radius of the circumscribed circle is equal to the radius of each of the given circl... | 117. Consider two cases: point $M$ lies inside triangle $A B C$, and point $M$ lies outside this triangle. In the first case, we have: $\angle M A C = \angle M B C, \angle M A B = \angle M C B, \angle M C A = \angle M B A$. From this, it follows that $M A \perp B C, M B \perp A C$. Notice that side $A B$ is visible fro... | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,693 |
118. Prove that the angle at which two Simson lines intersect, corresponding to two points, does not depend on the points themselves, but depends on the chord of the circle they subtend (i.e., equal chords correspond to equal angles between the Simson lines). | 118. If perpendiculars $P P_{1}, P P_{2}$ and $Q Q_{1}, Q Q_{2}$ are dropped from points $P$ and $Q$ of the circumcircle of triangle $A B C$ (Fig. 9) to the sides $A B$ and $A C$, respectively, then the angle $\varphi$ between the lines $P_{1} P_{2}$ and $Q_{1} Q_{2}$ is equal to the absolute value of the difference $\... | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,694 |
121. Two equally oriented equilateral triangles $A B C$ and $A_{1} B_{1} C_{1}$ are inscribed in a circle. Prove that the lines $A A_{1}, B B_{1}, C C_{1}$, when intersecting, also form an equilateral triangle. Verify the validity of a similar theorem for two inscribed regular polygons of the same name and orientation. | 121. Establish that $\angle A A_{1} B = \angle A_{1} B B_{1} = \angle 60^{\circ}$. For the polygon, prove that it is equiangular and cyclic, and therefore regular. | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,697 |
124. Through the vertices $A$ and $B$ of triangle $ABC$, a circle is drawn, intersecting sides $AC$ and $BC$ at points $M$ and $N$. Through point $M$, a line parallel to side $BC$ is drawn, and through point $N$, a line parallel to side $AC$ is drawn. These parallels meet side $AB$ at points $P$ and $Q$. Prove that the... | 124. Prove that $\angle A=\angle P M N=\angle N Q B$, using the properties of angles of a cyclic quadrilateral inscribed in a circle. | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,700 |
125. From the base $H_{3}$ of the height $C H_{3}$ of triangle $A B C$, perpendiculars $H_{3} A_{1}$ and $H_{3} B_{1}$ are dropped to the other two sides. Prove that the four points $A, B, A_{1}, B_{1}$ lie on the same circle. | 125. Notice that $\angle B_{1} A_{1} C = \angle B_{1} H_{3} C$, considering that a circle can be circumscribed around quadrilateral $C A_{1} H_{3} B_{1}$. Then prove that $\angle A = \angle B_{1} H_{3} C$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,701 |
126. Three lines pass through a single point $S$ and form six angles of $60^{\circ}$. Prove that the projections of any point $P$ (different from $S$) on these lines are the vertices of an equilateral triangle. | 126. Prove that the circle constructed on the diameter $S P$ passes through the feet of the perpendiculars dropped from point $P$ to the given lines. Use the properties of inscribed angles. | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,702 |
127. Prove that the perpendicular bisector of side $A B$ of triangle $A B C$ and the perpendicular to the larger of the other two sides at point $M$, which divides the broken line $A C B$ in half, intersect at point $P$ on the circumcircle of the triangle. | 127. Assume the opposite. Let the described circle intersect the perpendicular bisector at a point $S$, different from $P$ (Fig. 10). Drop a perpendicular from point $S$ to $BC$ and establish that its foot $M_{1}$ is the point for which $AC + CM_{1} = BM_{1}$, which contradicts the condition of the problem. To prove th... | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,703 |
129. Through the intersection point of two circles, two arbitrary secants are drawn, meeting the circles again at points $C$ and $C_{1}, D$ and $D_{1}$. Prove that the angle between the chords $C D$ and $D_{1} C_{1}$ is constant. | 129. Through points $C$ and $C_{1}$, $D$ and $D_{1}$, draw diameters (Fig. 11). From point $A$, drop perpendiculars to these diameters and show that the angles between these pairs of diameters are equal or sum up to $180^{\circ}$. Relate this angle to the angle $O A O_{1}$ and the angle between the extensions of the ch... | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,705 |
130. Through the point $A$ of intersection of two circles, a secant is drawn, meeting the circles at points $C$ and $D$. Prove that the perpendicular bisector of segment $C D$ passes through a fixed point as the secant describes a pencil with center at point $A$. | 130. Reflect point $A$ with respect to the midpoint of segment $O_{1} O_{2}$ and drop a perpendicular from the resulting point $A^{\prime}$ to the secant. Prove that the segment of this secant, bounded by the points of intersection with the circles, is bisected by the perpendicular. Drop perpendiculars from points $O_{... | proof | Geometry | proof | Yes | Yes | olympiads | false | 43,706 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.