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 |
|---|---|---|---|---|---|---|---|---|---|
13.18. (CSSR, 74). Prove that if the sides of the inscribed hexagon $A B C D E F$ satisfy the equations
$$
A B=B C, C D=D E, E F=F A
$$
then the area of triangle $A C E$ does not exceed the area of triangle $B D F$. | 13.18. If point $O$ is the center of the circumscribed circle around the given hexagon $A B C D E F$ with radius $R$ (Fig. 86) and
$$
\alpha=\angle C A E, \beta=\angle A E C, \gamma=\angle A C E
$$
then from the equalities of the sides indicated in the condition, we have
$$
\begin{gathered}
\angle A O B=\angle B O C... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,041 |
13.19. (Australia, 82). Prove that if the extensions of the angle bisectors of angles $A, B$, and $C$ of triangle $ABC$ intersect the circumcircle of the triangle at points $A_{1}$, $B_{1}$, and $C_{1}$ respectively, then the inequality
$$
A A_{1} + B B_{1} + C C_{1} > A B + B C + A C
$$
holds. | 13.19. We will prove that \( A A_{1} > (A B + A C) / 2 \). Indeed, by Ptolemy's theorem (Theorem 69), we have
\[
A A_{1} \cdot B C = A B \cdot A_{1} C + A C \cdot A_{1} B
\]
(Fig. 87) and, considering the equality of the inscribed angles \( B A A_{1} \) and \( C A A_{1} \)
. Prove that if $A D, B E$ and $C F$ are the angle bisectors of triangle $A B C$, then the area of triangle $D E F$ does not exceed one quarter of the area of triangle $A B C$. | 13.20. Notations
$$
a=BC, b=AC, c=AB, S=S_{ABC}, S_{0}=S_{DEF}
$$
Then, by the property of the angle bisector of a triangle (Fig. 88), we have
$$
\frac{AF}{b}=\frac{BF}{a}=\frac{AF+BF}{b+a}=\frac{c}{a+b}
$$
from which
$$
AF=\frac{bc}{a+b}
$$
. Prove that if 10 points are placed in a circle of diameter 5, then the distance between some two of them is less than 2. | 13.22. Let's highlight in this circle with center $O$ a concentric circle with diameter 2. Then, if some two of the given points lie inside this circle, the distance between them is less than 2, and the statement of the problem is valid. Otherwise, in the remaining annulus of the given circle, at least 9 points are loc... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,045 |
13.23. (Jury, Poland, 82). Points $A_{1}, A_{2}, \ldots, A_{t}$ on a circle with center $O$ and radius 1 are located such that
$$
\overrightarrow{O A}_{1}+\overrightarrow{O A}_{2}+\ldots+\overrightarrow{O A}_{n}=0
$$
Prove that for any point $B$ the inequality
$$
B A_{1}+B A_{2}+\ldots+B A_{n} \geqslant n
$$
holds. | 13.23. Denote
$$
\overrightarrow{O A}_{i}=a_{i}, \quad \overrightarrow{O B}=b \quad(i=1, \ldots, n)
$$
then we have
$$
\left|a_{i}\right|=1, \quad \overrightarrow{B A}_{i}=\overrightarrow{O A}_{i}-\overrightarrow{O B}=a_{i}-b
$$
H
$$
\begin{aligned}
& \sum_{i=1}^{n} B A_{i}=\sum_{i=1}^{n}\left|a_{i}-b\right|=\sum_... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 29,046 |
13.24*. (VNR, 81). Prove that for any points $A$, $B, C, D, E$ on a plane, the inequality
$A B+C D+D E+E C \leqslant A C+A D+A E+B C+B D+B E$.
## § 14. Geometric Problems on Extremum
(see Appendix G: definitions 35, 37; theorems $1,6,64,75$ ) | 13.24. Note that if points $A, B, C, D, E$ lie on the same line, the inequality is satisfied. Indeed, let point $E$ lie between points $C$ and $D$, then we have $(A C + A D) +$
$$
\begin{aligned}
& + (B C + B D + A E + B E) \geqslant \\
& \quad \geqslant C D + C D + A B = \\
& \quad = (C E + E D) + C D + A B
\end{alig... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 29,047 |
14.1. (Czechoslovakia, 80). The length of the largest side of an isosceles trapezoid is 13, and the perimeter is 28.
a) Find the sides of the trapezoid if its area is 27.
b) Can the area of such a trapezoid be 27,001? | 14.1. Let $AD$ be the larger base and $BH$ the height of the given trapezoid $ABCD$ (Fig. 92). Then
$$
AD=13
$$
(otherwise, $AB=CD=13, AD+BC=28-2 \cdot 13=2$ and $S_{ABCD}=$ $=BH \cdot(AD+BC) / 2 \leqslant 13 \cdot(2 / 2)=13<27)$,
$$
\begin{gathered}
AB=x, \\
BC=28-13-2 x=15-2 x, \quad AH=\frac{13-(15-2 x)}{2}=x-1_{... | AB=BC=CD=5 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,048 |
14.2. (Jury, Belgium, 82). Among all triangles of a given perimeter, find the one for which the radius of the inscribed circle is maximized. | 14.2. Let $a, b, c$ be the sides of a triangle with a given semiperimeter $p$, $S$ be its area, and $r$ be the radius of the inscribed circle. Then, by the theorem of means, we have
$$
\begin{aligned}
(r p)^{2}=S^{2}=p(p-a)(p-b)(p-c) & \leqslant \\
& \leqslant p\left(\frac{(p-a)+(p-b)+(p-c)}{3}\right)^{3}=\frac{p^{4}}... | r\leqslant\frac{p}{\sqrt{27}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,049 |
14.3. (Curie, USA, 77). Through a point located at a distance $k<1$ from the center of a circle of radius 1, a pair of perpendicular chords is drawn. Find the greatest and the least value of the sum of their lengths. | 14.3. Let through point $A$, located at a distance $k$ from the center $O$ of the circle, perpendicular chords $K L$ and $M N$ be drawn. Drop perpendiculars $O B$ and $O C$ to the chords $K L$ and $M N$ respectively and denote $\angle A O B=\alpha$ (Fig. 93). Then
$$
\begin{aligned}
& K L=2 B L=2 \sqrt{1-k^{2} \cos ^{... | 2\sqrt{4-2k^{2}}2(1+\sqrt{1-k^{2}}) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,050 |
14.4. (SFRY, 73). Prove that the sum of the distances between the midpoints of opposite sides of a quadrilateral is equal to its semiperimeter if and only if the quadrilateral is a parallelogram. | 14.4. Let $K, L, M, N$ be the midpoints of the sides $AB, BC, CD, DA$ of quadrilateral $ABCD$ (Fig. 94). Then we have
$$
\overrightarrow{KM}=\frac{1}{2}(\overrightarrow{KA}+\overrightarrow{AD}+\overrightarrow{DM})+\frac{1}{2}(\overrightarrow{KB}+\overrightarrow{BC}+\overrightarrow{CM})=\frac{1}{2}(\overrightarrow{AD}+... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,051 |
14.5. (England, 82). Prove that if for a point $O$ lying inside a quadrilateral $A B C D$ of area $S$, the equality
$$
2 S=O A^{2}+O B^{2}+O C^{2}+O D^{2}
$$
holds, then the quadrilateral $A B C D$ is a square, and the point $O$ is its center. | 14.5. Since the following relations hold (Fig. 95)
$$
\begin{aligned}
& O A^{2}+O B^{2}+O C^{2}+O D^{2}= \\
& =\frac{1}{2}\left(O A^{2}+O B^{2}\right)+\frac{1}{2}\left(O B^{2}+O C^{2}\right)+\frac{1}{2}\left(O C^{2}+O D^{2}\right)+\frac{1}{2}\left(O D^{2}+O A^{2}\right) \geqslant \\
& \quad \geqslant O A \cdot O B+O B... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,052 |
14.6. (New York, 80). Let $A_{i} H_{i} (i=1,2,3)$ be the altitudes of triangle $A_{1} A_{2} A_{3}$, the area of which is $S$. Prove that triangle $A_{1} A_{2} A_{3}$ is equilateral if and only if
$$
S=\frac{1}{6} \sum_{i=1}^{3} A_{i} A_{i+1} \cdot A_{i} H_{i} \quad\left(A_{4}=A_{1}\right)
$$ | 14.6. Let
$$
\begin{array}{lll}
a_{1}=A_{2} A_{3}, & a_{2}=A_{1} A_{3}, & a_{3}=A_{1} A_{2}, \\
h_{1}=A_{1} H_{1}, & h_{2}=A_{2} H_{2}, & h_{3}=A_{3} H_{3},
\end{array}
$$
then the following equalities hold
$$
\begin{aligned}
a_{3} h_{1}+a_{1} h_{2}+a_{2} h_{3} & = \\
& =a_{1} h_{1} \frac{a_{3}}{a_{1}}+a_{2} h_{2} \... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,053 |
14.7. (PFTB, 68). Find the ratios between the sides of a triangle satisfying the equality
$$
\frac{a \cos \alpha+b \cos \beta+c \cos \gamma}{a \sin \beta+b \sin \gamma+c \sin \alpha}=\frac{p}{9 R}
$$
where $a, b, c$ are the sides of the triangle, $\alpha, \beta, \gamma$ are the measures of the opposite angles, $P$ is... | 14.7. Let $h_{a}, h_{b}, h_{c}$ be the altitudes of the triangle satisfying the given condition, and $S$ be its area, then (Fig. 96)
$$
a \sin \beta=h_{c}, \quad b \sin \gamma=h_{a}, \quad c \sin \alpha=h_{b}
$$
and the original equality is equivalent to the condition
$$
P\left(h_{a}+h_{b}+h_{c}\right)=9 R(a \cos \a... | =b=,\quadh_{}=h_{b}=h_{} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,054 |
14.8. (New York, 79). Prove that the vertices of the smallest area regular $n$-gon ($n>3$) inscribed in a given regular $n$-gon coincide with the midpoints of the sides of the latter. | 14.8. Let the regular $n$-gon $B_{1} \ldots B_{n}$ with area $S_{B}$ be inscribed in the regular $n$-gon $A_{1} \ldots A_{n}$ with area $S_{A}$. Then, if they do not coincide, on each side $A_{i} A_{i+1}$ for $i=1, \ldots, n\left(A_{n+1}=\right.$ $=A_{1}$ ) there lies exactly one vertex, for definiteness $B_{i}$. Indee... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,055 |
14.9. (SFRY, 74). For arbitrary $n \geqslant 3$ points $A_{1}$, $A_{2}, \ldots, A_{n}$ on a plane, no three of which lie on the same line, let $\alpha$ denote the smallest of the angles $\angle A_{i} A_{j} A_{k}$ formed by triples $A_{i}, A_{j}, A_{k}$ of distinct points. For each value of $n$, find the greatest value ... | 14.9. We will prove that the greatest value of $\alpha$ is $180^{\circ} / n$. Indeed, let some arrangement of points on a plane correspond to the value $\alpha$. Consider a line, say $A_{1}^{\prime} A_{2}^{\prime}$, such that all points are located in one half-plane relative to this line, and choose a point $A_{3}^{\pr... | \alpha=\frac{180}{n} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,056 |
14.10. (MS, 83). Find the square of the smallest size in which 5 circles of radius 1 each can be arranged so that no two of them have common interior points. | 14.10. Let \(ABCD\) be a square with center \(O\) and side \(a\), containing 5 non-overlapping circles of radius 1. Then the centers of the circles lie within the inner square \(A_1B_1C_1D_1\) with center \(O\) and side \(a-2\) (where \(A_1B_1 \parallel AB\); see Fig. 99). The lines connecting the midpoints of opposite... | 2\sqrt{2}+2 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,057 |
14.11. (Jury, GDR, 82). Prove that if the radius of the circle inscribed in a triangle is half the radius of the circle circumscribed around it, then the triangle is equilateral. | 14.11. Let points $A_{\mathbf{1}}, B_{1}, C_{1}$ be the midpoints of sides $B C, A C, A B$ of a given triangle $A B C$, for which the radii $R$ and $r$ of the circumscribed and inscribed circles, respectively, satisfy the equation $R=2$ r. Then the radius $\rho$ of the circle circumscribed around triangle $A_{1} B_{1} ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,058 |
14.12. (USA, 79). On a plane, an angle is drawn, and a point $O$ is marked inside it. Draw a segment $B C$ through point $O$ with endpoints on the sides of the angle, such that the value
$$
\frac{1}{B O}+\frac{1}{C O}
$$
is maximized. | 14.12. We will prove that if segments $B_{1} C_{1}$ and $B_{2} C_{2}$ are drawn through point $O$, with endpoints $B_{1}$ and $B_{2}$ lying on side $A B$ of the given angle
Fig. 101
$B A C... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,059 |
15.1. (PRB, 66). Prove that for any tetrahedron, a triangle can be constructed whose sides are the 3 edges of the tetrahedron emanating from one of its vertices. | 15.1. Let $AB$ be the largest edge of the tetrahedron $ABCD$. Then we have (Fig. 105)
$$
\begin{aligned}
&(AC + AD - AB) + (BC + BD - BA) = \\
&(AC + CB - AB) + (AD + DB - AB) > 0
\end{aligned}
$$
therefore, at least one of the inequalities $AC + AD > AB$ or $BC + BD > BA$ holds, ensuring that a triangle can be forme... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,061 |
15.2. (Czechoslovakia, 67). Prove that if the edges of the tetrahedron \(ABCD\) satisfy the equations
\[
AB^2 + CD^2 = AC^2 + BD^2 = AD^2 + BC^2
\]
then at least one of its faces is an acute triangle. | 15.2. For the tetrahedron \(ABCD\) given in the problem, by the cosine rule we have (see Fig. 105)
\[ 2 AB \cdot AC \cos \angle BAC = AB^2 + AC^2 - BC^2 = AB^2 + AD^2 - BD^2 = 2 AB \cdot AD \cos \angle BAD, \]
from which it follows that
\[ \text{sign } \cos \angle BAC = \text{sign} \cos \angle BAD, \]
. Is it true that if the areas of four faces of one tetrahedron are equal to the areas of the corresponding faces of another, then the volumes of these tetrahedra are equal? | 15.3. Let perpendiculars to the ends of a segment $A B$ of length $d$ be drawn, not perpendicular to each other and to the segment $A B$. On these lines, there are segments of length $a$ with midpoints $A$ and $B$ respectively. The ends of these segments are vertices of a tetrahedron, each face of which has an area
$$... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,063 |
15.4. (CSSR, 71). Prove that there exists a tetrahedron \(ABCD\) all of whose faces are similar right triangles with acute angles at vertices \(A\) and \(B\). Determine which of the edges of this tetrahedron is the largest and which is the smallest, and find the length of the smallest edge if the length of the largest ... | 15.4. Suppose that the tetrahedron \(ABCD\) described in the problem exists, and we will prove that the edge \(AB\) is the largest, and the edge \(CD\) is the smallest. From the problem's conditions, it follows that each of the angles \(ACB\), \(ADB\), and one of the angles \(ACD\) or \(ADC\), say angle \(ACD\), is a r... | (\frac{\sqrt{5}-1}{2})^{3/2} | Geometry | proof | Yes | Yes | olympiads | false | 29,064 |
15.5. (GDR, 74). Determine into how many parts a regular tetrahedron is divided by six planes, each of which passes through one edge and the midpoint of the opposite edge of the tetrahedron. Find the volume of each of the parts if the volume of the tetrahedron is 1. | 15.5. Since any of the planes drawn contains one of the segments connecting the midpoints of opposite edges of the tetrahedron, all planes pass through the point of intersection of these segments, which lies inside the tetrahedron (see Theorem 92). Therefore, the six drawn planes divide the entire space into polyhedral... | \frac{1}{24} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,065 |
15.6. (CSSR, 76). Tetrahedra $A B C D$ and $A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ are positioned such that the lines $A A^{\prime}, B B^{\prime}, C C^{\prime}$, and $D D^{\prime}$ are parallel, the faces $A B C$ and $A^{\prime} B^{\prime} C^{\prime}$ have no common points, and the vertices $D$ and $D^{\prime}$ l... | 15.6. For any position of point $D$ in the plane of triangle $A^{\prime} B^{\prime} C^{\prime}$, at least one side of this triangle intersects with the line passing through the opposite vertex and point $D$. Let, for definiteness, $O$ be the intersection point of side $A^{\prime} B^{\prime}$ with line $C^{\prime} D$, a... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,066 |
15.7. (NBR, 68). Inside the tetrahedron $A B C D$ is a point $O$ such that the lines $A O, B O, C O, D O$ intersect the faces $B C D, A C D, A B D, A B C$ of the tetrahedron at points $A_{1}, B_{1}$, $C_{1}, D_{1}$ respectively, and the ratios
$$
\frac{A O}{A_{1} O}, \frac{B O}{B_{1} O}, \frac{C O}{C_{1} O}, \frac{D O... | 15.7. Let \( V \) be the volume of the tetrahedron \( ABCD \), and \( k \) be the desired number. Then we have
\[
\frac{V}{V_{OBCD}} = \frac{AA_1}{OA_1} = \frac{AO}{A_1O} + \frac{OA_1}{OA_1} = k + 1
\]
\[
\frac{V}{V_{OACD}} = \frac{V}{V_{OABD}} = \frac{V}{V_{OABC}} = k + 1,
\]
from which
\[
k + 1 = \frac{4V}{V_{OBC... | 3 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,067 |
15.8. (MS, 66). On the edges $A B, A C, A D$ of a given tetrahedron $A B C D$, for each value $n \in \mathbf{N}$, points $K_{n}, L_{n}, M_{n}$ are chosen respectively such that
$$
A B=n A K_{n}, A C=(n+1) A L_{n}, A D=(n+2) A M_{n}
$$
Prove that all planes $K_{n} L_{n} M_{n}$ pass through the same line. | 15.8. We will prove that all lines \( K_{n} L_{n} \) for \( n \in \mathbb{N} \) pass through a fixed point \( O \), lying on the line passing through vertex \( A \) and parallel to line \( B C \). Indeed, if the line \( K_{n} L_{n} \) intersects line \( B C \) at point \( P \) (lying on the ray \( C B \); see Fig. 108)... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,068 |
15.9. (PNR, 79). Prove that four lines connecting the vertices of a tetrahedron with the centers of the circles inscribed in the opposite faces intersect at one point if and only if the three products of the lengths of opposite edges are equal to each other. | 15.9. For the two lines connecting points $B$ and $C$ with the centers of the circles inscribed in triangles $A C D$ and $A B D$ to intersect, it is necessary and sufficient that they lie in the same plane. This, in turn, is equivalent to the bisectors of angles $A B D$ and $A C D$ intersecting edge $A D$ at the same p... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,069 |
15.10. (NPR, 76). A plane intersects three edges of a tetrahedron emanating from one vertex. Prove that this plane divides the surface of the tetrahedron into parts proportional to the volumes of the corresponding parts of the tetrahedron if and only if it passes through the center of the sphere inscribed in the tetrah... | 15.10. Let $V, S$ and $r$ denote the volume, surface area of a tetrahedron, and the radius of the inscribed sphere, respectively. One of the parts into which a plane divides the tetrahedron is a pyramid with its base lying in this plane. Let $V_{1}, S_{i}$ and $r_{\mathbf{i}}$ denote the volume, the lateral surface are... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,070 |
15.11. (Czechoslovakia, 68). Prove that if a tetrahedron has two pairs of opposite mutually perpendicular edges, then the midpoints of all its edges lie on one sphere. | 15.11. Let the edges of the tetrahedron \(ABCD\) satisfy the conditions \(AC \perp BD\) and \(AD \perp BC\). We draw through each edge of the tetrahedron a plane parallel to the opposite edge. The resulting three pairs
. Prove that if a regular tetrahedron with edge length \( a \) is inscribed in a regular tetrahedron with edge length \( b \) such that each face of the latter contains exactly one vertex of the inscribed tetrahedron, then \( 3a \geqslant b \). | 15.12. Let a regular tetrahedron \( T_{1} \) be inscribed in a regular tetrahedron \( T_{2} \). Then the radius \( R_{1} \) of the sphere \( S \) circumscribed around the tetrahedron \( T_{1} \) is not less than the radius \( r_{2} \) of the sphere inscribed in the tetrahedron \( T_{2} \). Indeed, by drawing tangent pl... | 3a\geqslantb | Geometry | proof | Yes | Yes | olympiads | false | 29,072 |
15.13. (GDR, 83). In the tetrahedron $A B C D$, the edges $A D, B D$ and $C D$ are mutually perpendicular, and their lengths are $a, b, c$ respectively. Prove that for any point $M$, lying on one of the sides of the triangle $A B C$, the sum $S$ of the distances from the vertices $A, B$ and $C$ to the line $D M$ satisf... | 15.13. Let, for definiteness, point $M$ lie on side $A B$ of triangle $A B C$ and $\angle M D B=\varphi$ (Fig. 111). Then
$$
S=c+a \cos \varphi+b \sin \varphi
$$
and, denoting
$$
\begin{gathered}
d=a \cos \varphi+b \sin \varphi \\
\psi=\arccos \left(a / \sqrt{a^{2}+b^{2}}\right)
\end{gathered}
$$
we have the inequa... | S\leqslant\sqrt{2(^{2}+b^{2}+^{2})} | Inequalities | proof | Yes | Yes | olympiads | false | 29,073 |
15.14. (SFRY, 73; Austria - PDR, 80). Prove that for any point lying inside a tetrahedron, the sum of the angles under which its edges are seen from this point is greater than \(540^{\circ}\). | 15.14. Let point $O$ lie inside the tetrahedron $A B C D$. Denote by $P$ the point of intersection of the line $D O$ with the plane $A B C_{1}$, and by $Q$ the point of intersection of the line $B P$ with the side $A C$ (Fig. 112). Then, by Theorem 84, we have
$$
\begin{array}{r}
\angle A O B + \angle A O C = \angle A... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,074 |
15.15. (NPR, 76). Prove that the distances from some point in space to each of the four vertices of a regular tetrahedron with edge length 2 are simultaneously integers if and only if this point coincides with one of the vertices of the tetrahedron. | 15.15. We will prove that if the distances from a point \( M \) to the vertices of a tetrahedron \( ABCD \) with edge length 2 are integers, then at least one of the distances is 0 (the converse statement is beyond doubt). Note that if the point \( M \) is located on the line containing an edge of the tetrahedron, say ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,075 |
15.16. (CSSR, 73). Prove that for the heights \( h_{i} (i=1, 2,3,4) \) of any tetrahedron and the radii \( r_{i} \) of the exinscribed spheres, the following equality holds:
\[
2\left(\frac{1}{h_{1}}+\frac{1}{h_{2}}+\frac{1}{h_{3}}+\frac{1}{h_{4}}\right)=\frac{1}{r_{1}}+\frac{1}{r_{2}}+\frac{1}{r_{3}}+\frac{1}{r_{4}}
... | 15.16. Let $V$ and $S$ denote the volume and surface area of a tetrahedron, and let $S_{i}$ be the area of the face corresponding to the height $h_{i}$ of the tetrahedron, which touches the exsphere of radius $r_{i}$. Then we have the equalities
$$
3 V=h_{1} S_{1}=r_{1}\left(S_{2}+S_{3}+S_{4}-S_{1}\right)=r_{1}\left(S... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,076 |
15.17*. (PNR, 78). Prove that for the heights $h_{1}, h_{2}, h_{3}, h_{4}$ of any tetrahedron and the distances $d_{1}, d_{2}, d_{3}$ between pairs of its opposite edges, the following equality holds:
$$
\frac{1}{h_{1}^{2}}+\frac{1}{h_{2}^{2}}+\frac{1}{h_{3}^{2}}+\frac{1}{h_{4}^{2}}=\frac{1}{d_{1}^{2}}+\frac{1}{d_{2}^... | 15.17. Let in the tetrahedron $A_{1} A_{2} A_{3} A_{4}$ be denoted: $h_{i}$ - the height dropped from the vertex $A_{i}$, and $S_{i}$ - the area of the corresponding face, $d_{1}$, $d_{2}, d_{8}$ - the distances between the edges $A_{2} A_{3}, A_{1} A_{3}, A_{1} A_{2}$ and the opposite edges of the tetrahedron, respect... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,077 |
15.18*. (Jury, NRB, 81). A sphere touches the edges $A B$, $B C$, $C D$, $D A$ of the tetrahedron $A B C D$ at four points, which are the vertices of a square. Prove that if this sphere also touches the edge $A C$, then it touches the edge $B D$ as well. | 15.18. Let $K, L, M, N$ be the points of tangency of the edges $A B$, $B C, C D, D A$ with the sphere (Fig. 113). Through the center $O$ of the sphere, draw a line $l$ perpendicular to the plane of the square $K L M N$. The four planes tangent to the sphere at points $K, L, M, N$ form equal dihedral angles with the pla... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,078 |
15.19*. (NPR, 81). Planes $\alpha, \beta, \gamma, \delta$ touch the sphere circumscribed around the tetrahedron $A B C D$ at points $A, B, C$, $D$ respectively. Prove that if the line of intersection of planes $\alpha$ and $\beta$ lies in the same plane as line $C D$, then the line of intersection of planes $\gamma$ an... | 15.19. Let a sphere with center $O$ and radius $R$ be circumscribed around the tetrahedron $ABCD$. Denote
$$
\boldsymbol{a}=\overrightarrow{O A}, \boldsymbol{b}=\overrightarrow{O B}, \boldsymbol{c}=\overrightarrow{O C}, \boldsymbol{d}=\overrightarrow{O D}
$$
and, generally, $\boldsymbol{x}=\overrightarrow{O X}$ for a... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,079 |
16.1. (CSSR, 79). Cubes are considered whose centers coincide with the center of symmetry of a given rectangular parallelepiped with edges $a<b<c$, and whose faces are parallel to those of the latter. Find the edge of the cube that has the smallest difference between the volumes of the union and intersection with this ... | 16.1. If we denote the edge of the cube by $x$, then the difference in volumes specified in the problem is
$$
f(x)=\left\{\begin{array}{lc}
a b c-x^{3} & \text { for } \quad 00$, and its derivative is
$$
f^{\prime}(x)= \begin{cases}-3 x^{2} & \text { for } 03 b^{2}-2 a b>0$ ), and on the interval ( $a$; b) it either ... | \{43} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,080 |
16.2. (GDR, 83). Prove that inside a cube with edge a, two regular tetrahedra with edge $a$ can be placed without having any common points. | 16.2. Let's draw a plane through the center $O$ of the given cube $A_{1} A_{2} A_{3} A_{4} A_{1}^{\prime} A_{2}^{\prime} A_{3}^{\prime} A_{4}^{\prime}$, perpendicular to the diagonal $A_{1} A_{3}^{\prime}$ (Fig. 114). This plane passes through the midpoints $B_{1}, B_{2}, B_{3}$ of the edges $A_{1}^{\prime} A_{4}^{\pri... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,081 |
16.3. (SFRY, 73). In space, 5 points are arranged such that no 4 of them lie in the same plane. Prove that some line passing through 2 of them intersects the plane containing the other 3 points inside the triangle with vertices at these 3 points. | 16.3. Consider a tetrahedron with vertices at points \(A_{1}, A_{2}\), \(A_{8}, A_{4}\). Then the space is divided by the planes of its faces into 2 sets. The first of these sets combines 4 similar regions, each consisting of a trihedral angle at some vertex of the tetrahedron and its symmetric counterpart relative to ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,082 |
16.4. (VNR, 80). Space is divided into 5 non-intersecting non-empty sets. Prove that some plane has common points with at least 4 sets. | 16.4. Suppose, contrary to the statement of the problem, that any plane intersects no more than 3 sets. Let's choose points $A, B, C, D, E$ from different sets. Then no 4 of them lie in the same plane, and, consequently, no 3 lie on the same line. Further, some plane passes through any 3 of them, relative to which the ... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 29,083 |
16.5. (Jury, USSR, 83). Prove that for any partition of space into 3 sets, at least one of the sets has the property that for each value $a>0$, it is possible to choose 2 points in it such that the distance between them is $a$. | 16.5. Let, contrary to the statement of the problem, the space is divided into 3 sets $M_{1}, M_{2}, M_{3}$ and there exist such positive numbers $a_{1} \leqslant a_{2} \leqslant a_{8}$, that for each value $i=1,2,3$ any two points of the set $M_{i}$ do not realize the distance $a_{i}$. Consider the tetrahedron $A B C ... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 29,084 |
16.6. (SRP, 80). Prove that for any vectors $a_{1}, a_{2}, a_{3}$ the following equality holds:
$$
\sum\left(\varepsilon_{1} a_{1}+\varepsilon_{2} a_{2}+\varepsilon_{3} a_{3}\right)^{2}=8 \sum_{k=1}^{3} a_{k}^{2}
$$
where the sum on the left is taken over all 8 different sets of numbers $\varepsilon_{1}, \varepsilon_... | 16.6. Let $A_{n}$ denote the set of $2^{n}$ all possible sequences
$$
\mathrm{e}=\left(\mathrm{e}_{1} ; \ldots ; \mathrm{e}_{n}\right),
$$
consisting of numbers $\varepsilon_{i} \in\{-1 ; 1\}(i=1, \ldots, n)$. We also introduce the notation
$$
a_{\varepsilon}=\sum_{k=1}^{n} \varepsilon_{k} a_{k}
$$
where $\varepsil... | proof | Algebra | proof | Yes | Yes | olympiads | false | 29,085 |
16.7. (Czechoslovakia, 62). Through the ends $A$ and $B$ of a segment of length $a$, lines are drawn perpendicular to each other and to the segment $A B$. Points $C$ and $D$ are taken on these lines such that the point of intersection of the segment $C D$ with the plane passing through the midpoint $O$ of the segment $... | 16.7. Let segment $C D$ intersect the plane specified in the problem at point $E$, and let points $C^{\prime}$ and $D^{\prime}$ be the projections of points $C$ and $D$ onto this plane (Fig. 117). From the equality of the right triangles $C C^{\prime} E$ and $D D^{\prime} E$
. A sphere $s$ of radius $r$ passes through the center of a sphere $S$ of radius $R$. Prove that if a chord $AB$ of sphere $S$ is tangent to sphere $s$ at point $C$, then
$$
A C^{2}+B C^{2} \leqslant 2 R^{2}+r^{2}
$$ | 16.8. Let's conduct a plane passing through points \( A, B \) and the center \( O \) of the sphere \( S \). The section of the sphere \( S \) will be a circle with center \( O_{1} \) and radius \( r_{1} \leq r \), touching the line \( AB \). Let \( OH \) and \( OK \) be the perpendiculars to the lines \( AB \) and \( O... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,087 |
16.9. (USA, 82). Each of two non-coinciding spheres of radii $r_{1}$ and $r_{2}$ touches the sphere $S$ of radius $R$ internally and passes through points $A, B, C$, which are arranged such that the perpendicular to the plane $ABC$, passing through point $A$, also passes through the center of sphere $S$. Prove that $r_... | 16.9. Draw a line $d$ through point $A$, perpendicular to the plane $ABC$. Then the centers $O_{1}$ and $O_{2}$ of the spheres touching sphere $S$ lie on the line $l$, which consists of points equidistant from points $A, B, C$, and thus is parallel to line $d$. Let $A^{\prime}$ be the point symmetric to point $A$ relat... | r_{1}+r_{2}=R | Geometry | proof | Yes | Yes | olympiads | false | 29,088 |
16.10. (CPR, 58). Let $S$ and $V$ denote the surface area and volume of a regular $n$-sided pyramid.
a) For given values of $n$ and $S$, find the maximum value of $V$.
b) Find the side lengths of the bases and the heights of all pyramids for which $n=4, S=144, V=64$. | 16.10. Let's introduce additional notations: $Q$ - the area of the base, $h$ - the height of the pyramid, $x$ - the cosine of the dihedral angle between the base and a lateral face, $a$ - the side of the base, $r$ - the radius of the circle inscribed in the base. Then the following equalities hold:
$$
\begin{aligned}
... | \frac{\sqrt{2}}{12}\cdot\frac{S^{3/2}}{\sqrt{n\operatorname{tg}(180/n)}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,089 |
16.11. (NPR, 82). Prove that for any value of $n \in \mathbf{N}$, greater than 1, among all regular $2 n$-sided prisms $A_{1} \ldots A_{2 n} A_{1}^{\prime} \ldots A_{2 n}^{\prime}$ with a fixed radius $R$ of the circle circumscribed around the base, the largest angle between the diagonal $A_{1} A_{n+1}^{\prime}$ and th... | 16.11. Let's denote the height of the prism by $h$. Then, in the tetrahedron
$$
A_{1} A_{3} A_{n+1}^{\prime} A_{n+2}^{\prime}
$$
(Fig. 120), the angle between the opposite edges $A_{1} A_{3}$ and $A_{n+1}^{\prime} A_{n+2}^{\prime}$ is equal to the angle
$$
\angle A_{3} A_{1} A_{2}=(1 / 2) \angle A_{3} O A_{2}=180^{\... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,090 |
16.12. (VNR, 79). Prove that if in a pyramid with equal lateral edges any two adjacent lateral faces form equal dihedral angles, and the base is a polygon with an odd number of sides, then this polygon is regular. | 16.12. From the equality of the lateral edges of the pyramid $S A_{1} \ldots A_{n}$, it follows that the projection $O$ of the vertex $S$ onto the base of the pyramid is equidistant from the vertices $A_{\mathbf{i}}, \ldots, A_{n}$, i.e., it is the center of the circle circumscribed around the polygon $A_{1} \ldots A_{... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,091 |
16.14. (PRA, 77). The areas of the bases of a truncated pyramid are $S_{1}$ and $S_{2}$, and the area of the lateral surface is $S$. Prove that if a certain plane, parallel to the bases, divides the pyramid into two truncated pyramids, each of which can have a sphere inscribed in it, then
$$
S=\left(\sqrt{S_{1}}+\sqrt... | 16.14. Let the larger base $M_{1}$ of the original truncated pyramid have area $S_{1}$, the smaller base $M_{2}$ have area $S_{2}$, and the common base $M_{0}$ of the two truncated pyramids that make up the original have area $S_{0}$. Extend the lateral edges of the pyramid to their intersection at point $T$ and denote... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,092 |
16.15. (England, 68). Find the maximum number of points that can be placed on a sphere of radius 1 so that the distance between any two of them is: a) not less than $\sqrt{2} ;$ b) greater than $\sqrt{2}$. | 16.15. a) Let's prove that if points $A_{1}, A_{2}, \ldots, A_{n}$ are located on a sphere with center $O$ and radius 1 such that the distance between any points $A_{i}, A_{j} (i \neq j)$ is at least $\sqrt{2}$, then $n \leqslant 6$. Indeed, let $n>6$. By the cosine theorem, we have
$$
A_{i} A_{j}^{2}=2-2 \cos \angle ... | 4 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,093 |
16.17. (USA, 79). Three circles lying on a sphere and having a common center $O$ with it pass through the point
A. On these circles, points $B, C, D$ are chosen respectively such that $\angle A O B=90^{\circ}$, and the line $O B$ is the bisector of angle $C O D$. Prove that if the rays $A B^{\prime}$, $A C^{\prime}, A... | 16.17. The angle $C^{\prime} A B^{\prime}$ represents the linear angle of the dihedral angle with edge $O A$ and planes passing through points $C$ and $B$ respectively, since the rays $A C^{\prime}$ and $A B^{\prime}$ are perpendicular to the common
. On the circumference of one base of a right circular cylinder, diametrically opposite points $A$ and $B$ are taken, and on the circumference of the other base - point $C$, which does not lie on the plane $A B O$, where $O$ is the midpoint of the cylinder's axis. Prove that the sum of the dihedr... | 16.18. Let point $C^{\prime}$ be symmetric to point $C$ with respect to the center of symmetry of the cylinder - point 0 (Fig. 125). Then, if the dihedral angles of the trihedral angle $O A B C$ at the edges $O A, O B, O C$ are $\alpha, \beta, \gamma$, the dihedral angles of the trihedral angle $O A B C^{\prime}$ at th... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,095 |
16.19. (USA, 81). The sum of the magnitudes of the plane angles of a convex polyhedral angle is equal to the sum of the magnitudes of its dihedral angles. Prove that this angle is trihedral. | 16.19. Note that the sum of the planar angles of the given angle with vertex $S$ and edges $S A_{i}, \ldots, S A_{n}$ is less than $360^{\circ}$. Take any internal point $O$ of this $n$-sided angle and drop perpendiculars $O H_{i}$ from it to the planes $S A_{i} A_{i+1}\left(i=1, \ldots, n ; A_{n+1}=\right.$ $=A_{1}$ )... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,096 |
16.20. (Jury, Belgium, 79). Can space be partitioned into 1979 equal non-overlapping subsets? | 16.20. Let's introduce a coordinate system in space and for each value $i=1, \ldots, 1979$ assign to the $i$-th set all points whose abscissas satisfy the equation
$$
[x] \equiv i(\bmod 1979)
$$
The constructed example shows that the answer to the question of the problem is positive. | proof | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 29,097 |
16.21. (England, 70). Find the smallest number of planes that divide a cube into at least 300 parts.
16.22\%. (SRP, 78). On planes $\alpha$ and $\alpha^{\prime}$, intersecting along a line $l$, three points each are chosen: $A, B, C$ and $A^{\prime}, B^{\prime}, C^{\prime}$ respectively. The plane $\alpha^{\prime}$ ro... | 16.21. We will prove by induction on $n$ that $n$ lines divide the plane into no more than
$$
p(n)=\frac{n(n+1)}{2}+1
$$
parts, and exactly $p(n)$ parts are obtained if no two lines are parallel and no three lines pass through the same point. Indeed, $p(0)=1$, and for any value of $n \in \mathbb{N}$, we have the ineq... | 13 | Geometry | proof | Yes | Yes | olympiads | false | 29,098 |
16.23*. (CSSR, 73). Consider rotations around different axes in space that translate vertex $A$ of the cube $A B C D A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ to vertex $B$. Find the geometric locus of points on the surface of this cube that are images of vertex $C$ under such rotations. | 16.23. A line \( l \) can serve as an axis of rotation that transforms point \( A \) into point \( B \) if and only if points \( A \) and \( B \) have a common projection \( O \) on this line and are equidistant from it, i.e., when the line \( l \) lies in the plane \( \alpha_{1} \), passing through the midpoint of seg... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 29,099 |
16.24*. (P.R.C., 82). A cube in space with a rectangular coordinate system is positioned such that the coordinates of some 4 of its vertices, not lying in the same plane, are integers. Prove that all vertices of the cube have integer coordinates. | 16.24. Suppose first that three of the four given vertices of the cube
$$
A_{1} A_{2} A_{3} A_{4} A_{1}^{\prime} A_{2}^{\prime} A_{3}^{\prime} A_{4}^{\prime}
$$
(see Fig. 114) lie in the same face. Let for definiteness these be the points $A_{1}, A_{2}, A_{3}$, then the vertex $A_{4}$ also has integer coordinates, si... | proof | Geometry | proof | Yes | Yes | olympiads | false | 29,100 |
17.1. (SFRY, 76). Calculate the sum $a_{1}+a_{2}+\ldots+a_{99}$, where it is denoted
$$
a_{n}=\frac{1}{(n+1) \sqrt{n}+n \sqrt{n+1}}
$$ | 17.1. Noting that for each $n=1,2, \ldots, 99$ the equalities hold
$$
\begin{aligned}
& a_{n}=\frac{1}{(n+1) \sqrt{n}+n \sqrt{n+1}}=\frac{(n+1) \sqrt{n}-n \sqrt{n+1}}{(n+1)^{2} n-n^{2}(n+1)}= \\
& \quad=\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}
\end{aligned}
$$
for the desired sum we get
$$
\begin{aligned}
& a_{1}+a_{... | \frac{9}{10} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 29,102 |
17.2. (CSSR, 72). Prove that there exist numbers $A$ and $B$, satisfying for any value of $n \in \mathbf{N}$ the equality
$$
a_{\mathbf{i}}+a_{2}+\ldots+a_{n}=A \operatorname{tg} n+B n
$$
where it is denoted
$$
a_{k}=\operatorname{tg} k \operatorname{tg}(k-1)
$$ | 17.2. Using the formula
$$
\operatorname{tg} 1=\frac{\operatorname{tg} k-\operatorname{tg}(k-1)}{1+\operatorname{tg} k \operatorname{tg}(k-1)}
$$
(note that due to the irrationality of the number $\pi$, the expression $\operatorname{tg} \boldsymbol{k}$ is defined for all $k \in N$), we obtain for any $n \in N$ the eq... | A=\frac{1}{\operatorname{tg}1},B=-1 | Algebra | proof | Yes | Yes | olympiads | false | 29,103 |
17.3. (New York, 74). Let
$$
a_{n}=\frac{1 \cdot 3 \cdot 5 \ldots(2 n-1)}{2 \cdot 4 \cdot 6 \ldots 2 n}, n \in \mathrm{N}
$$
Find $\lim a_{n}$.
$$
n \rightarrow \infty
$$ | 17.3. Since
$$
\begin{aligned}
a_{n}^{2}=\frac{1^{2} \cdot 3^{2} \cdot \ldots \cdot(2 n-1)^{2}}{2^{2} \cdot 4^{2} \cdot \ldots \cdot(2 n)^{2}} & = \\
& =\frac{1 \cdot 3}{2^{2}} \cdot \frac{3 \cdot 5}{4^{2}} \cdots \frac{(2 n-1)(2 n+1)}{(2 n)^{2}} \cdot \frac{1}{2 n+1}<\frac{1}{2 n+1}
\end{aligned}
$$
for any $n \in \... | 0 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 29,104 |
17.4. (New York, 74). In the sequence of positive numbers $a_{0}, a_{i}, \ldots$ each term $a_{n}(n \in N)$ is either $a_{n-1} / 2$ or $\sqrt{a_{n-1}}$. Can this sequence have a limit that belongs to the interval $(0 ; 1)$? | 17.4. Let $A \in(0 ; 1)$ and $\lim _{n \rightarrow \infty} a_{n}=A$. Then there exists a natural number $N$ such that for all indices $n \geqslant N$ the estimates $2 A / 3 < a_{n} < 4 A / 3$ hold. If for all $n \geqslant N$ the equality $a_{n}=\sqrt{a_{n-1}}$ is satisfied, then passing to the limit as $n \rightarrow \... | proof | Algebra | math-word-problem | Yes | Yes | olympiads | false | 29,105 |
17.5. (USA, 80; SFRY, 81). For a given natural value $n \geqslant 3$, find the maximum possible number of increasing arithmetic progressions consisting of three terms that can be selected from any set containing exactly $n$ distinct numbers. | 17.5. Consider the number $a_{i}$ from the set $a_{1}<a_{2}<\ldots<a_{7}$. The number of three-term arithmetic progressions in which this number is the middle term does not exceed both $i-1$ and $n-i$, since the first term can only be one of the numbers $a_{1}, \ldots, a_{i-1}$, and the last term can be one of the numb... | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 29,106 | |
17.6. (SFRY, 81). The numbers 1, 9, 8, 1 are the first four terms of a sequence in which each subsequent term is equal to the last digit of the sum of the four preceding terms. Can the numbers 1, 2, 3, 4 appear consecutively in this sequence? | 17.6. Let \(a_{1}, a_{2}, a_{3}, \ldots\) be the sequence given in the problem. Consider the function defined on the set of integers:
$$
f(x)=\left\{\begin{array}{l}
0, \text { if } x \text { is even, } \\
1, \text { if } x \text { is odd, }
\end{array}\right.
$$
and define the sequence \(\left\{b_{n}\right\}\) by th... | proof | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 29,107 |
17.9. (Jury, France, 82). Let all members of the sequences $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$ be natural numbers. Prove that there exists a pair of indices $p<q$, for which the inequalities $a_{p} \leqslant a_{q}$ and $b_{p} \leqslant b_{q}$ hold. | 17.9. Let's construct an increasing sequence of indices $\left\{l_{n}\right\}$ as follows. Let $a_{i}$ be the smallest of the numbers $a_{1}, a_{2}, \ldots$ (it exists because all terms of the sequence $\left\{a_{n}\right\}$ are natural numbers); $a_{i_{2}}$ be the smallest of the numbers $a_{i_{1}+1}, a_{i_{1}+2}, \ld... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 29,109 |
17.10. (Beijing, 64). The sequence of positive numbers $a_{1}, a_{2}, \ldots$ satisfies the inequalities $a_{n}^{2} \leqslant a_{n}-a_{n+1}$ for $n \in \mathbf{N}$. Prove that for any value of $n \in \mathbf{N}$, the estimate $a_{n}<1 / n$ holds. | 17.10. We will prove the statement by induction on $n$. For $n=1$, we have $a_{1}^{2} \leqslant a_{1}-a_{2}<a_{1}$, from which it follows that $a_{i}<1$. Moreover,
$$
a_{2} \leqslant a_{1}-a_{1}^{2}=1 / 4-\left(a_{1}-1 / 2\right)^{2} \leq 1 / 4<1 / 2
$$
i.e., $a_{n}<1 / n$ for $n=2$. Suppose the statement is already ... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 29,110 |
17.12. (Austria - PNR, 80). Given a numerical sequence $\left\{a_{n}\right\}, \quad$ satisfying the inequalities $\left|a_{k+m}-a_{k}-a_{m}\right| \leqslant 1$ for $k, m \in \mathbf{N}$. Prove that for any $p, q \in \mathbf{N}$ the inequality
$$
\left|\frac{a_{p}}{p}-\frac{a_{q}}{q}\right|<\frac{1}{p}+\frac{1}{q}
$$
... | 17.12. From the condition, we have the inequalities
$$
a_{k+m}-1 \leqslant a_{k}+a_{m} \leqslant a_{k+m}+1, \quad k, m \in \mathbb{N}
$$
We will prove by induction on $q \in \mathbb{N}$ that for any $p, q \in \mathbb{N}$, the following inequalities hold:
$$
a_{p q}-(q-1) \leqslant q a_{p} \leqslant a_{p q}+(q-1)
$$
... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 29,112 |
17.13. (PDR, 78). For a given number $a_{i} \in \mathbf{R}$, define the sequence $a_{i}, a_{2}, \ldots$ as follows:
$$
a_{n+\hat{1}}=\left\{\begin{array}{cc}
(1 / 2)\left(a_{n}-1 / a_{n}\right), & \text { if } a_{n} \neq 0 \\
0 & , \text { if } a_{n}=0
\end{array}\right.
$$
for $n \in \mathbf{N}$. Prove that this seq... | 17.13. Suppose the set of non-positive terms is finite. Then there exists a natural number $N$ such that $a_{n}>0$ for all $n \geqslant N$. In this case, for all $n \geqslant N$ we also have $a_{n}>1$ (if $a_{n} \leqslant 1$, then $a_{n+1}=(1 / 2)\left(a_{n}-1 / a_{n}\right) \leqslant 0$). On the other hand,
\[
\begin... | proof | Algebra | proof | Yes | Yes | olympiads | false | 29,113 |
17.15. (Austria, 72; NRB, 78). Prove that the sequence of non-zero numbers \(a_{1}, a_{2}, \ldots\), satisfying for some number \(a\) the conditions
\[
a_{1}, a_{2} \in \mathbf{Z},\left(a_{1}^{2}+a_{2}^{2}+a\right) /\left(a_{1} a_{2}\right) \in \mathbf{Z}, a_{n+3}=\left(a_{n+1}^{2}+a\right) / a_{n}
\]
for \(n \in \ma... | 17.15. Note that for the sequence specified in the problem, the following relations hold
$$
a_{n+1} a_{n+3}+a_{n+1}=a_{n+2}^{2}+a+a_{n+1}^{2}=a_{n+2}^{2}+a_{n} a_{n+2} \quad n \in N
$$
from which we obtain the equality
$$
\left(a_{n+3}+a_{n+1}\right) / a_{n+2}=\left(a_{n+2}+a_{n}\right) / a_{n+1}
$$
Therefore, if w... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 29,114 |
17.16. (CSSR, 68). Prove that each member of the sequence
$$
a_{n}=\left((2+\sqrt{3})^{n}-(2-\sqrt{3})^{n}\right) /(2 \sqrt{3}) \quad(n \in \mathbf{Z})
$$
is an integer. Find all values of $n \in \mathbf{Z}$ for which the number $a_{n}$ is divisible by 3. | 17.16. Note that $a_{0}=0, a_{1}=1$, and let's prove the validity of the equality $a_{n+2}=4 a_{n+1}-a_{n}$ for each $n \in \mathbf{Z}^{+}$ (from which we will sequentially obtain that each of the numbers $a_{2}, a_{3}, a_{4}, \ldots$ is also an integer). Indeed, let
$$
\alpha=(2+\sqrt{3})^{n} /(2 \sqrt{3}), \quad \be... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 29,115 |
17.18*. (Jury, England, 82). The sequence $a_{0}, a_{1}, \ldots$ satisfies for some parameter $a \in \mathbf{N}$ the relations
$$
a_{0}=0, a_{1}=1, a_{n+1}=2 a_{n}+(a-1) a_{n-i}
$$
for $n \in \mathrm{N}$. For a fixed prime number $p_{0}>2$, find the smallest value of $a$ for which the following two statements are tru... | 17.18. Note that the two sequences
$$
a_{n}=(1+\sqrt{\bar{a}})^{n} \text { and } a_{n}=(1-\sqrt{ } \bar{a})^{n}
$$
satisfy the condition
$$
a_{n+1}=2 a_{n}+(a-1) a_{n-1}, \quad n \in \mathrm{N}
$$
Indeed, we have
$$
\begin{aligned}
& a_{n+1}-2 a_{n}+(1-a) a_{n-1}=(1 \pm \sqrt{ })^{2} a_{n-1}-2(1 \pm \sqrt{a}) a_{n... | notfound | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 29,117 |
17.21*. (England, 83). Prove that for the sequence of numbers (Fibonacci) \(a_{1}, a_{2}, \ldots\), defined by the relations
\[
a_{1}=a_{2}=1, a_{n+2}=a_{n+1}+a_{n} \quad \text { for } \quad n \in \mathbf{N} \text {, }
\]
there exists a unique triplet of numbers \(a, b, c \in \mathbf{N}\), satisfying the conditions: ... | 17.21. Let's prove that the triple of numbers $a, b, c \in \mathrm{N}$, where $b>1$, satisfies the condition (2) as well, since
$$
\begin{gathered}
a_{n}=n b c^{n}(\bmod a) \\
a_{n-1} \equiv(n-1) b c^{n-1}(\bmod a), \quad a_{n+1} \equiv(n+1) b c^{n+1}(\bmod a), \\
a_{n}+a_{n-1}=a_{n+1} \cdot
\end{gathered}
$$
Now let... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 29,118 |
18.1. (GDR, 73). Find all pairs $(x ; y)$ of positive numbers at which the minimum value of the function
$$
f(x, y)=\frac{x^{4}}{y^{4}}+\frac{y^{4}}{x^{4}}-\frac{x^{2}}{y^{2}}-\frac{y^{2}}{x^{2}}+\frac{x}{y}-\frac{y}{x}
$$
is achieved, and find this minimum value. | 18.1. The minimum value of the function $f(x, y)$ for $x, y > 0$ is 2, since the following relations hold:
$$
\begin{aligned}
& f(x, y)-2= \\
& =\left(\frac{x^{2}}{y^{2}}-1\right)^{2}+\left(\frac{y^{2}}{x^{2}}-1\right)^{2}+\left(\frac{x}{y}-\frac{y}{x}\right)^{2}+\left(\frac{x}{y}-2+\frac{y}{x}\right) \geqslant \\
& \... | ytheminimumvalueis2 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 29,119 |
18.2. (Jury, Sweden, 79). Find the maximum value of the product $x^{2} y^{2} z^{2} u$ given that $x, y, z, u \geqslant 0$ and
$$
2 x+x y+z+y z u=1
$$ | 18.2. Using the theorem of means, we have
$$
\sqrt[4]{2 x^{2} y^{2} z^{2} u}=\sqrt[4]{2 x \cdot x y \cdot z \cdot y z u} \leqslant \frac{2 x+x y+z+y z u}{4}=\frac{1}{4}
$$
i.e., $x^{2} y^{2} z^{2} u \leqslant 1 / 512$. Equality is achieved if $2 x=x y=z=y z u=1 / 4$, i.e., when $x=1 / 8, y=2, z=1 / 4, u=1 / 2$. Thus,... | \frac{1}{512} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 29,120 |
18.3. (GDR, 78; CSSR, 80). For the given numbers $a_{1}<a_{2}<\ldots<a_{n}$, determine whether there exist points $x \in \mathbf{R}$ at which the function
$$
f(x)=\sum_{i=1}^{n}\left|x-a_{i}\right|
$$
attains its minimum value. If such points exist, find all such points, as well as the minimum value of the function $... | 18.3. Let initially $n=2 k$, where $k \in \mathbb{N}$. By the triangle inequality, we have
$$
\left\{\begin{array}{c}
\left|x-a_{1}\right|+\left|x-a_{n}\right| \geqslant a_{n}-a_{1} \\
\left|x-a_{2}\right|+\left|x-a_{n-1}\right| \geqslant a_{n-1}-a_{2} \\
\left|x-a_{k}\right|+\left|x-a_{k+1}\right| \geqslant a_{k+1}-a... | proof | Algebra | math-word-problem | Yes | Yes | olympiads | false | 29,121 |
18.4. (Jury, GDR, 79). For a given number $n \geqslant 2$ find the greatest and the least values of the product
$$
x_{1} x_{2} \ldots x_{n}
$$
under the condition that $x_{i} \geqslant 1 / n(i=1,2, \ldots, n)$ and $x_{1}^{2}+x_{2}^{2}+\ldots$ $\cdots+x_{n}^{2}=1$. | 18.4. a) Let's find the minimum value of the product $x_{1} x_{2} \ldots x_{n}$. Let an arbitrary set $(x_{1}; \ldots; x_{n})$ satisfy the condition of the problem. Consider a new set
$$
\left(x_{1}^{(1)}; x_{2}^{(1)}; \ldots; x_{n}^{(1)}\right)
$$
where
$$
\begin{aligned}
& x_{1}^{(1)}=x_{1}, x_{2}^{(1)}=x_{2}, \ld... | n^{-n/2} | Inequalities | math-word-problem | Yes | Yes | olympiads | false | 29,122 |
18.5. (Jury, Sweden, 79). For given numbers $n \geqslant 2$ and $a>0$, find the maximum value of the sum $\sum_{i=1}^{n-1} x_{i} x_{i+1}$ under the condition that $x_{i} \geqslant 0(i=1, \ldots, n)$ and $x_{1}+\ldots+x_{n}=a$. | 18.5. Let $\max \left(x_{1}, \ldots, x_{n}\right)=x_{k}$. Then
$$
\begin{aligned}
& \sum_{i=1}^{n-1} x_{i} x_{i+1}=\sum_{i=1}^{k-1} x_{i} x_{i+1}+\sum_{i=k}^{n-1} x_{i} x_{i+1} \leqslant \\
& \leqslant x_{k} \sum_{i=1}^{k-1} x_{i}+x_{k} \sum_{i=k+1}^{n} x_{i}=x_{k}\left(a-x_{k}\right) \leqslant\left(\left(x_{k}+a-x_{k... | \frac{^2}{4} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 29,123 |
18.6. (SRP, 79). Given positive numbers $a_{1}<a_{2}<\ldots$ $\ldots<a_{n}$. For which permutation $\left(b_{1} ; b_{2} ; \ldots ; b_{n}\right)$ of these numbers is the product
$$
\prod_{i=1}^{n}\left(a_{i}+1 / b_{i}\right)
$$
maximal? | 18.6. Let $A=\prod_{i=1}^{n} a_{i}$. For numbers $0<a_{i}<\ldots<a_{n}$, we have
\[
\prod_{i=1}^{n}\left(a_{i}+1 / b_{i}\right)=\prod_{i=1}^{n}\left(\left(a_{i} b_{i}+1\right) / b_{i}\right)=(1 / A) \prod_{i=1}^{n}\left(a_{i} b_{i}+1\right) \leq
\]
\[
\leq(1 / A) \prod_{i=1}^{n}\left(a_{i}^{2}+1\right)
\]
(to prove... | (b_{1};\ldots;b_{n})=(a_{1};\ldots;a_{n}) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 29,124 |
18.7. (CSSR, 63). For each value of $k \in \mathbf{N}$, represent the number $2 k$ as the sum of two coprime numbers $x$ and $y$ such that the product $x y$ is the largest possible. | 18.7. Without loss of generality, we can assume that $x \geqslant y$. Let $p=x-k$. Then $x=k+p, y=2 k-x=k-p, p \geqslant 0$, and the product
$$
x y=(k+p)(k-p)=k^{2}-p^{2}
$$
attains its maximum value at the smallest possible value of $p$. Let $p=0$, then $x=y=k$ and the numbers $x, y$ are coprime only when $k=1$. Let... | k\1,k\1 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 29,125 |
18.8. (CSSR, 83). For given numbers $n \in \mathbf{N}$ and $a \in[0 ; n]$, find the maximum value of the expression
$$
\left|\sum_{i=1}^{n} \sin 2 x_{i}\right| \text { given }
$$
the condition that
$$
\sum_{i=1}^{n} \sin ^{2} x_{i}=a
$$ | 18.8. We have
$$
a=\sum_{i=1}^{n} \sin ^{2} x_{i}=\sum_{i=1}^{n}\left(1-\cos 2 x_{i}\right) / 2
$$
i.e.
$$
\sum_{i=1}^{n} \cos 2 x_{i}=n-2 a
$$
Next, consider the vectors on the plane
$$
\left(\cos 2 x_{i} ; \quad \sin 2 x_{i}\right)
$$
of unit length. Their sum has a length not greater than $n$, and thus the ine... | 2\sqrt{(n-)} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 29,126 |
18.9. (Yugoslavia, 74). For each number $n \in \mathbf{N}$, find the greatest value that the product of natural numbers with a fixed sum $n$ can take.
18.10\%. (England, 81). Find the smallest value of the quantity $\left|12^{m}-5^{n}\right|$ for $m, n \in \mathbf{N}$.
## § 19. Various Properties of Functions
## (se... | 18.9. Note that any fixed number $n \in N$ can be decomposed into a sum of natural numbers in only a finite number of ways. 298
Therefore, among these representations, there will be one (perhaps not unique) decomposition $n=m_{1}+m_{2}+\ldots+m_{k}$, where $m_{1} \leqslant m_{2} \leqslant \ldots \leqslant m_{k}$, for ... | f(1)=1,f(3)=3^{},f(3-1)=2\cdot3^{-1},f(3+1)=4\cdot3^{-1} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 29,127 |
19.2. (SRP, 81). Does there exist a function $f: \mathbb{R} \rightarrow \mathbb{R}$, satisfying for all $x \in \mathbb{R}$ - the inequality
$$
f\left(x^{2}\right)-(f(x))^{2} \geqslant 1 / 4
$$
and not taking any value more than once? | 19.2. Let's prove that such a function does not exist. Indeed, otherwise we have: $f(0)-(f(0))^{2} \geqslant 1 / 4$, i.e., $(f(0)-1 / 2)^{2} \leqslant 0$, from which $f(0)=1 / 2$. Similarly, we obtain that $f(1)=1 / 2$, i.e., $f(0)=$ $=f(1)$, which is impossible. | proof | Algebra | math-word-problem | Yes | Yes | olympiads | false | 29,128 |
19.3. (PRC, 79; Jury, USA, 79). Prove that if the function \( f : \mathbf{R} \rightarrow \mathbf{R} \) satisfies for all \( x, y \in \mathbf{R} \) the inequalities
\[
f(x) \leqslant x, \quad f(x+y) \leqslant f(x)+f(y)
\]
then the identity
\[
f(x) \equiv x, \quad x \in \mathbb{R}
\]
holds. | 19.3. Let's prove the required identity. Substituting the values \( x = y = 0 \) into the inequality
$$
f(x+y) \leqslant f(x) + f(y)
$$
we get \( f(0) \leqslant 2 f(0) \), or \( f(0) \geqslant 0 \). From this and the inequality \( f(0) < 0 \), it follows that \( f(0) = 0 \). Further, for any \( x \in \mathbf{R} \) we... | proof | Algebra | proof | Yes | Yes | olympiads | false | 29,129 |
19.5. (Canada, 81). Let continuous functions $f(x)$ and $g(x)$ satisfy the identity
$$
f(g(x)) \equiv g(f(x)), \quad x \in \mathbf{R}
$$
Prove that if the equation $f(x)=g(x)$ has no real solutions, then the equation $f(f(x))=g(g(x))$ also has no real solutions. | 19.5. Since the equation $f(x)=g(x)$ has no real solutions, the function
$$
h(x)=f(x)-g(x)
$$
takes either only positive or only negative values for all $x \in \mathbf{R}$. Therefore, the function
$$
f(f(x))-g(g(x))=f(f(x))-g(f(x))+f(g(x))-g(g(x))=\left(\begin{array}{l}
=h(f(x))+h(g(x))
\end{array}\right.
$$
does n... | proof | Algebra | proof | Yes | Yes | olympiads | false | 29,130 |
19.6. (SRP, 81). Let $f:[0 ;+\infty) \rightarrow[0 ;+\infty)$ be a continuous function. Prove that
a) if $\lim _{x \rightarrow+\infty} f(f(x))=+\infty$, then $\lim _{x \rightarrow+\infty} f(x)=+\infty$;
b) the result of part a) is not true for a function $f: (0 ;+\infty) \rightarrow(0 ;+\infty)$. | 19.6. a) Let $f:[0 ;+\infty) \rightarrow[0 ;+\infty)$ be a continuous function and
$$
\lim _{x \rightarrow+\infty} f(f(x))=+\infty
$$
Suppose the statement is false. Then there exists a value $N>0$ such that for any $n \in \mathbf{N}$, there is a number $x_{n}>n$ satisfying the condition $f\left(x_{n}\right) \in[0 ; ... | proof | Calculus | proof | Yes | Yes | olympiads | false | 29,131 |
19.7. (SRP, 79). Prove that there does not exist a continuous function $f: \mathbf{R} \rightarrow \mathbf{R}$, possessing the following property: the number $f(x)$ is rational for those and only those values of $x \in \mathbf{R}$ for which the number $f(x+1)$ is irrational. | 19.7. Let the function $f(x)$ described in the problem exist. Consider the continuous functions
$$
g(x)=f(x+1)-f(x) \text { and } h(x)=f(x+1)+f(x) .
$$
They cannot both be constant, since otherwise the function $f(x)=(h(x)-g(x)) / 2$ would be constant. Suppose, for example, that $h(x)$ is not a constant function, i.e... | proof | Calculus | proof | Yes | Yes | olympiads | false | 29,132 |
19.8. (New York, 79). Does there exist a non-constant function $f: \mathbf{R} \rightarrow \mathbf{R}$, satisfying for all $x, y \in \mathbf{R}$ the inequality
$$
(f(x)-f(y))^{2} \leqslant|x-y|^{3 ?}
$$ | 19.8. From the inequality $(f(x)-f(y))^{2} \leqslant|x-y|^{3}$ for $x \neq y$, it follows that
$$
\left|\frac{f(x)-f(y)}{x-y}\right| \leq|x-y|^{1 / 2}
$$
from which
$$
\lim _{x \rightarrow y}\left|\frac{f(x)-f(y)}{x-y}\right|=0
$$
(since $\lim _{x \rightarrow y}|x-y|^{1 / 2}=0$). Therefore, the function $f(x)$ is d... | proof | Inequalities | math-word-problem | Yes | Yes | olympiads | false | 29,133 |
19.9. (New York, 76). Let a continuous function
$$
f:[0 ; 1] \rightarrow[0 ; 1]
$$
be differentiable on the interval $(0 ; 1)$, and $f(0)=0$, $f(1)=1$. Prove that there exist numbers $a, b \in(0 ; 1)$ such that
$$
a \neq b \text { and } f^{\prime}(a) f^{\prime}(b)=1
$$ | 19.9. Let the function $f(x)$ satisfy the condition of the problem. Consider the function $g(x)=f(x)+x-1$, defined on the interval $[0 ; 1]$. Since it is continuous (because $f(x)$ is continuous), and $g(0)=-1, g(1)=1$, there exists a number $c \in(0 ; 1)$ such that $g(c)=0$, i.e., $f(c)=1-c$. By Lagrange's theorem, th... | proof | Calculus | proof | Yes | Yes | olympiads | false | 29,134 |
19.10*. (Australia, 82). Find all numbers $d \in(0 ; 1]$, possessing the following property: if $f(x)$ is any continuous function defined for $x \in[0 ; 1]$, and $f(0)=f(1)$, then there exists a number $x_{0} \in[0 ; 1-d]$, such that
$$
f\left(x_{0}\right)=f\left(x_{0}+d\right)
$$ | 19.10. We will prove that any number $d=1 / k$, where $k \in \mathbb{N}$, satisfies the condition of the problem. Let's take an arbitrary continuous function $f(x)$ and a number $k>1$ (the number $d=1$ satisfies the condition, since $f(0) = f(1)$). Consider the function
$$
g(x)=f(x+1 / k)-f(x)
$$
defined on the inter... | proof | Calculus | math-word-problem | Yes | Yes | olympiads | false | 29,135 |
19.11*. (CPP, 78). Let the function $f: \mathbf{R} \rightarrow \mathbf{R}$ be defined as follows: $f(x)=0$, if $x$ is irrational; $f(p / q)=1 / q^{3}$, if $p \in \mathbf{Z}, q \in \mathbf{N}$ and the fraction $p / q$ is irreducible. Prove that this function is differentiable at each point $x=\sqrt{k}$, where $k$ is a n... | 19.11. Let the number $k \in \mathbb{N}$ not be a square of an integer. We will show that $f'(\sqrt{k})=0$. Since $\sqrt{k} \notin \mathbb{Q}$, then $f(\sqrt{k})=0$, and it remains to prove that the limit
$$
\lim _{x \rightarrow \sqrt{k}} \frac{f(x)}{x-\sqrt{k}}
$$
exists and equals 0. Take any $\varepsilon>0$. There... | proof | Calculus | proof | Yes | Yes | olympiads | false | 29,136 |
19.12*. (Jury, PNR, 76). Let $I=(0 ; 1]$. For a given value $a \in(0 ; 1)$, define the function $f: I \rightarrow I$ as follows:
$$
f(x)= \begin{cases}x+-(1-a) & \text { for } 0<x \leqslant a \\ x-a & \text { for } a<x \leqslant 1\end{cases}
$$
Prove that for any interval $J \subset I$ there exists a number $n \in \m... | 19.12. Suppose there exists an interval $J \subset I$ of length $d$, for which
$$
f^{n}(J) \cap J=\varnothing \text { for any } n \in \mathrm{N} .
$$
Then for any $m, n \in \mathbf{N}$ we have
$$
f^{m+n}(J) \cap f^{m}(J)=f^{m}\left(f^{n}(J) \cap J\right)=f^{m}(\varnothing)=\varnothing,
$$
therefore the sets $f(J), ... | proof | Algebra | proof | Yes | Yes | olympiads | false | 29,137 |
20.1. (New York, 78). The function $f: \mathbf{R} \rightarrow \mathbf{R}$ satisfies the identity
$$
f(x y) \equiv \frac{f(x)+f(y)}{x+y}, \quad x, y \in \mathbf{R}, \quad x+y \neq 0
$$
Does there exist a value $x \in \mathbb{R}$ for which $f(x) \neq 0$? | 20.1. Let $y=1$ in the identity satisfied by the function $f(x)$. Then we have the identity
$$
f(x)=(f(x)+f(1)) /(x+1) \quad(x \neq-1)
$$
i.e., $x f(x)=f(1)$. For $x=0$, we get $f(1)=0$, and thus for $x \notin \{-1 ; 0\}$ we have $f(x)=0$. Next, substituting the values $y=0, x=2$ into the original identity, we get $f... | f(x)=0,x\in{R} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 29,139 |
20.2. (NBR, 68). Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$, satisfying the identity
$$
x f(y) + y f(x) \equiv (x + y) f(x) f(y), \quad x, y \in \mathbb{R}
$$ | 20.2. Let's set \( x = y = 1 \) in the original identity, then we get \( 2 f(1) = 2 (f(1))^2 \), i.e., \( f(1) = 0 \) or \( f(1) = 1 \). Let's consider each of these cases:
a) If \( f(1) = 0 \), then by setting \( y = 1 \) in the identity, we get the identity \( f(x) \equiv 0 \).
b) If \( f(1) = 1 \), then by setting... | f(x)\equiv0 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 29,140 |
20.3. (Jury, GDR, 82). Let $M$ be the set of functions $f: \mathbf{Z} \rightarrow \mathbf{R}$, satisfying the condition $f(0) \neq 0$ and the identity
$$
f(n) f(m) \equiv f(n+m) + f(n-m), \quad n, m \in \mathbf{Z}
$$
Find: a) all functions $f(n) \in M$, for which $f(1)=5 / 2$; b) all functions $f(n) \in M$, for which... | 20.3. Substituting the initial identity for the function $f(n) \in M$ with values $n=m=0$, we get $(f(0))^{2}=2 f(0)$. But $f(0) \neq 0$, so $f(0)=2$. Further, substituting the value $m=1$ into the identity, we get the identity $f(n) f(1)=f(n+1)+f(n-1), n \in \mathbf{Z}$. If the values of the function $f(n)$ at points ... | proof | Algebra | math-word-problem | Yes | Yes | olympiads | false | 29,141 |
20.4. (Austria-Poland, 79). Find all functions $f: \mathbf{Z}^{+} \rightarrow \mathbf{R}$, satisfying the identity
$$
f(n+m)+f(n-m) \equiv f(3 n), \quad n, m \in \mathbf{Z}^{+}, \quad n \geqslant m
$$ | 20.4. Setting $m=0$ in the original identity for the function $f(n)$, we get $2 f(n) \equiv f(3 n)\left(n \in \mathbb{Z}^{+}\right)$, and for $n=m=0$ we have $f(0)=0$. Further,
setting $n=m$ in the identity, we get
$$
f(2 n)+f(0) \equiv f(3 n), \text{ i.e. } f(2 n) \equiv f(3 n)
$$
From this, on the one hand, for an... | f(n)=0 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 29,142 |
20.5. (New York, 76). Functions $f, g: R \rightarrow R$ are non-constant and satisfy two identities
$$
\begin{aligned}
& f(x+y)=f(x) g(y)+g(x) f(y) \\
& g(x+y)=g(x) g(y)-f(x) f(y)
\end{aligned}
$$
$x, y \in \mathbf{R}$. Find all possible values of $f(0)$ and $g(0)$. | 20.5. Substituting $x=y=0$ into each of the two identities satisfied by the non-constant functions $f(x), g(x)$, we obtain two equations
$$
f(0)=2 f(0) g(0) \text { and } g(0)=(g(0))^{2}-(f(0))^{2}
$$
Since $g(0) \neq 1 / 2$ (otherwise, from the second equation it would follow that $\left.(f(0))^{2}=1 / 4-1 / 2<0\rig... | f(0)=0,(0)=1 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 29,143 |
20.6. (MMS, Luxembourg, 80). Find all functions $f: \mathbf{Q} \rightarrow \mathbf{Q}$, satisfying the condition $f(1)=2$ and the identity
$$
f(x y) \equiv f(x) f(y)-f(x+y)+1, \quad x, y \in \mathbf{Q}
$$ | 20.6. Putting \( y=1 \) in the original identity, we get the identity
\[
f(x) \equiv f(x) f(1)-f(x+1)+1 \quad (x \in \mathbb{Q})
\]
i.e.
\[
f(x+1) \equiv f(x)+1
\]
From this, for all \( x \in \mathbb{Q}, n \in \mathbb{Z} \) we have
\[
f(x+n)=f(x)+n
\]
therefore
\[
f(n)=f(1)+n-1=n+1
\]
Next, substituting into th... | f(x)=x+1 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 29,144 |
20.7. (SFROI, 83). The function $\mathrm{f:} \mathbf{Z} \rightarrow \mathbf{R}$ satisfies the conditions
$$
f(n)= \begin{cases}n-10, & \text { if } n>100 \\ f(f(n+11)), & \text { if } n \leqslant 100\end{cases}
$$
for $n \in \mathbf{Z}$. Prove that for any value $n \leqslant 100$, the equality $f(n)=91$ holds. | 20.7. Let first $n \leqslant 100$ and $n+11>100$, i.e., $90 \leqslant n \leqslant 100$.
$$
f(n)=f(f(n+11))=f(n+11-10)=f(n+1),
$$
therefore
$$
f(90)=f(91)=\ldots=f(100)=f(101)=91
$$
Now let $n<90$. Choose such a number $m \in \mathbb{N}$ so that the estimates
$$
90<n+11 m \leqslant 100
$$
are satisfied. Then we ha... | proof | Algebra | proof | Yes | Yes | olympiads | false | 29,145 |
20.8. (SRP, 79). Functions $f, g, h: \mathbf{N} \rightarrow \mathbf{N}$ satisfy the following three conditions:
a) the function $h(n)$ does not take any value more than once for $n \in \mathbf{N}$;
b) the set of values of the function $g(n)$ is $\mathbf{N}$;
c) $f(n)=g(n)-h(n)+1, n \in \mathbf{N}$.
Prove that the i... | 20.8. We will prove the identity $g(n)=h(n) \quad(n \in N)$, from which, due to condition b), it will follow that
$$
f(n) \equiv g(n)-h(n)+1 \equiv 1, \quad n \in \mathrm{N} .
$$
For any $n \in \mathbf{N}$, we have
$$
h(n)=g(n)+-1-f(n) \leqslant g(n)
$$
(since $f(n) \geqslant 1$). Suppose that for some value $n \in... | proof | Algebra | proof | Yes | Yes | olympiads | false | 29,146 |
20.9. (SRP, 78). Prove that there exists a function $f: \mathbf{N} \rightarrow \mathbf{N}$, satisfying the identity
$$
f(f(n)) \equiv n^{2}, \quad n \in \mathrm{N}
$$ | 20.9. Let the sequence $n_{i}=2, n_{2}=3, n_{3}=5, \ldots$ enumerate in increasing order all natural numbers that are not squares of integers. We define
$$
n_{k, m}=\left(n_{k}\right)^{2^{m}}, \text { where } k \in \mathbb{N}, m \in \mathbf{Z}^{+}
$$
Then $n_{k, m+1}=\left(n_{k, m}\right)^{2}$, and each value $n>1$ c... | proof | Algebra | proof | Yes | Yes | olympiads | false | 29,147 |
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