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4. Quadrilateral $ABCD$ is inscribed in a circle of radius 7. The rays $AB$ and $DC$ intersect at point $P$, and the rays $BC$ and $AD$ intersect at point $Q$. It is known that triangles $ADP$ and $QAB$ are similar (vertices are not necessarily in the corresponding order).
a) Find $AC$.
b) Suppose additionally that t... | Answer. a) $A C=14 ;$ b) $\angle D A C=45^{\circ}, S_{A B C D}=94$.
Solution. a) Similarity of triangles is equivalent to the equality of all their angles. Since the angle at vertex $A$ is common to both triangles, there are two options: either $\angle A B Q=\angle A D P, \angle A Q B=\angle A P D$, or $\angle A B Q=\... | AC=14,\angleDAC=45,S_{ABCD}=94 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
5. Given the number $500 \ldots 005$ (80 zeros). It is required to replace some two zeros with non-zero digits so that after the replacement, the resulting number is divisible by 165. In how many ways can this be done? | Answer: 17280.
Solution. $165=3 \cdot 5 \cdot 11$. Divisibility by 5 is always satisfied since the number ends in a five. For divisibility by 11, it is significant on which positions the replaceable digits stand.
First case. We replace two zeros in positions of the same parity (both even or both odd). For divisibilit... | 17280 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
6. Find all values of the parameter $a$ for which there exists a value of the parameter $b$ such that the system
$$
\left\{\begin{array}{l}
\arcsin \left(\frac{a+y}{2}\right)=\arcsin \left(\frac{x+3}{3}\right) \\
x^{2}+y^{2}+6 x+6 y=b
\end{array}\right.
$$
has exactly two solutions. | Answer. $a \in\left(-\frac{7}{2} ; \frac{19}{2}\right)$.
Solution. The first equation on the domain of definition is equivalent to the equation $\frac{a+y}{2}=\frac{x+3}{3}, y=\frac{2 x}{3}+2-a$. The domain of definition is determined by the inequality $-1 \leqslant \frac{x+3}{3} \leqslant 1, -6 \leqslant x \leqslant ... | \in(-\frac{7}{2};\frac{19}{2}) | Algebra | math-word-problem | Yes | Yes | olympiads | false |
7. Consider quadrilateral pyramids $KABCD$ with the following properties: the base of the pyramid is a convex quadrilateral $ABCD$, where $AB = BC = 3$, $CD = DA = 4$, and each of the planes of the lateral faces $KAB$, $KBC$, $KCD$, $KDA$ forms an angle of $45^{\circ}$ with the plane of the base.
a) Find the volume of... | Answer: a) $\frac{4}{3}$; b) 48.
Let $KH$ be the height of the pyramid $(KH=h)$, and $P$ be the projection of $K$ onto the line $AB$. Then $KHP$ is a right triangle with an angle $\angle KPH=45^{\circ}$, from which $HP=h \cdot \operatorname{ctg} 45^{\circ}=h$. Similarly, it can be proven that the point $H$ is at a dis... | \frac{4}{3};48 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
2. It is known that the numbers $x, y, z$ form an arithmetic progression in the given order with a common difference $\alpha=\arccos \frac{2}{3}$, and the numbers $\frac{1}{\sin x}, \frac{6}{\sin y}, \frac{1}{\sin z}$ also form an arithmetic progression in the given order. Find $\sin ^{2} y$. | Answer: $\frac{5}{8}$.
Solution. Using the characteristic property of an arithmetic progression, we get $y=\frac{x+z}{2}$, $\frac{12}{\sin y}=\frac{1}{\sin x}+\frac{1}{\sin z}$. Transform the second equation:
$$
\frac{12}{\sin y}=\frac{\sin x+\sin z}{\sin x \sin z} \Leftrightarrow \frac{12}{\sin y}=\frac{2 \sin \frac... | \frac{5}{8} | Algebra | math-word-problem | Yes | Yes | olympiads | false |
3. Solve the inequality $8^{\log _{2}^{2} x}-2 \cdot x^{\log _{2} x} \geqslant \log _{6} 108+\left(4-\log _{6}^{2} 3\right) \log _{108} 6$. | Answer. $x \in\left(0 ; \frac{1}{2}\right] \cup[2 ;+\infty)$.
Solution. Transform the right side of the inequality:
$\log _{6} 108+\left(4-\log _{6}^{2} 3\right) \log _{108} 6=2+\log _{6} 3+\frac{\left(2-\log _{6} 3\right)\left(2+\log _{6} 3\right)}{\log _{6} 108}=2+\log _{6} 3+\frac{\left(2-\log _{6} 3\right)\left(2... | x\in(0;\frac{1}{2}]\cup[2;+\infty) | Inequalities | math-word-problem | Yes | Yes | olympiads | false |
4. Quadrilateral $ABCD$ is inscribed in a circle of radius 6. The rays $AB$ and $DC$ intersect at point $P$, and the rays $BC$ and $AD$ intersect at point $Q$. It is known that triangles $ADP$ and $QAB$ are similar (vertices are not necessarily in the corresponding order).
a) Find $AC$.
b) Suppose additionally that t... | Answer. a) $A C=12 ;$ b) $\angle D A C=45^{\circ}, S_{A B C D}=68$.
Solution. a) Similarity of triangles is equivalent to the equality of all their angles. Since the angle at vertex $A$ is common to both triangles, there are two options: either $\angle A B Q=\angle A D P, \angle A Q B=\angle A P D$, or $\angle A B Q=\... | AC=12,\angleDAC=45,S_{ABCD}=68 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
5. Given the number $200 \ldots 002$ (100 zeros). It is required to replace some two zeros with non-zero digits so that after the replacement, the resulting number is divisible by 66. In how many ways can this be done? | Answer: 27100.
Solution. $66=3 \cdot 2 \cdot 11$. Divisibility by 2 is always satisfied since the number ends in two. To investigate divisibility by 11, it is significant which positions the replaceable digits occupy.
First case. We replace two zeros in positions of the same parity (both even or both odd). For divisi... | 27100 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
6. Find all values of the parameter $a$ such that the system
$$
\left\{\begin{array}{l}
\arccos \left(\frac{4-y}{4}\right)=\arccos \left(\frac{a+x}{2}\right) \\
x^{2}+y^{2}+2 x-8 y=b
\end{array}\right.
$$
has no more than one solution for any value of the parameter $b$. | Answer. $a \in(-\infty ;-9] \cup[11 ;+\infty)$.
Solution. The first equation on the domain of definition is equivalent to the equation $\frac{4-y}{4}=\frac{a+x}{2}, y=4-2 a-2 x$. The domain of definition is determined by the inequality $-1 \leqslant \frac{4-y}{4} \leqslant 1,0 \leqslant y \leqslant 8$. Thus, the first... | \in(-\infty;-9]\cup[11;+\infty) | Algebra | math-word-problem | Yes | Yes | olympiads | false |
7. Consider the pyramids $T A B C D$ with the following properties: the base of the pyramid is a convex quadrilateral $A B C D$, where $A B=B C=1, C D=D A=2$, and each of the planes of the lateral faces $T A B, T B C, T C D, T D A$ forms an angle of $60^{\circ}$ with the plane of the base.
a) Find the volume of such a... | Answer: a) $\frac{4}{3 \sqrt{3}}$; b) $\frac{4}{\sqrt{3}}$.
Let $TH$ be the height of the pyramid, $(TH=h)$, and $P$ be the projection of $T$ onto the line $AB$. Then $THP$ is a right triangle with angle $\angle TPH=60^{\circ}$, from which $HP=h \cdot \operatorname{ctg} 60^{\circ}=\frac{h}{\sqrt{3}}$. Similarly, it ca... | \frac{4}{3\sqrt{3}};\frac{4}{\sqrt{3}} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
1. Which whole numbers from 1 to $8 \cdot 10^{20}$ (inclusive) are there more of, and by how many: those containing only even digits or those containing only odd digits? | Answer: the number of numbers containing only odd digits is greater by $\frac{5^{21}-5}{4}$.
Solution. Consider $k$-digit numbers ( $1 \leqslant k \leqslant 20$ ). The number of numbers consisting only of odd digits is $5^{k}$ (for each of the $k$ positions, any of the digits $1,3,5,7$, 9 can be chosen); the number of... | \frac{5^{21}-5}{4} | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
2. Given two linear functions $f(x)$ and $g(x)$ such that the graphs $y=f(x)$ and $y=g(x)$ are parallel lines, not parallel to the coordinate axes. Find the minimum value of the function $(g(x))^{2}+$ $2 f(x)$, if the minimum value of the function $(f(x))^{2}+2 g(x)$ is 5. | Answer: -7.
Solution. Let $f(x)=a x+b, g(x)=a x+c$, where $a \neq 0$. Consider $h(x)=(f(x))^{2}+2 g(x)$. Expanding the brackets, we get $h(x)=(a x+b)^{2}+2(a x+c)=a^{2} x^{2}+2 a(b+1) x+b^{2}+2 c$. The graph of $y=$ $h(x)$ is a parabola opening upwards, and the minimum value is attained at the vertex. The x-coordinate... | -7 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
3. The equation $x^{2}+a x+5=0$ has two distinct roots $x_{1}$ and $x_{2}$; in this case,
$$
x_{1}^{2}+\frac{250}{19 x_{2}^{3}}=x_{2}^{2}+\frac{250}{19 x_{1}^{3}}
$$
Find all possible values of $a$. | Answer: $a=10$.
Solution. For the equation to have roots, its discriminant must be positive, hence $a^{2}-20>0$. Under this condition, by Vieta's theorem, $x_{1}+x_{2}=-a, x_{1} x_{2}=5$. Then $x_{1}^{2}+$ $x_{1} x_{2}+x_{2}^{2}=\left(x_{1}+x_{2}\right)^{2}-x_{1} x_{2}=a^{2}-5$.
Transform the given equality:
$$
x_{1... | 10 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
4. On each of the lines $x=0$ and $x=2$, there are 62 points with ordinates $1, 2, 3, \ldots, 62$. In how many ways can three points be chosen from the marked 124 points so that they form the vertices of a right triangle? | Answer: 7908.
Solution. There are two possibilities.
1) The hypotenuse of the triangle lies on one of the lines, and the vertex of the right angle is on the second line. Let $ABC$ be the given triangle with a right angle at vertex $C$, and $CH$ be its height dropped to the hypotenuse. From the proportionality of the ... | 7908 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
5. In a convex quadrilateral $ABCD$, diagonal $BD$ is drawn, and a circle is inscribed in each of the resulting triangles $ABD$ and $BCD$. A line passing through vertex $B$ and the center of one of the circles intersects side $DA$ at point $M$. Here, $AM=\frac{8}{5}$ and $MD=\frac{12}{5}$. Similarly, a line passing thr... | Answer: a) $A B: C D=4: 5, A B=4, C D=5$.
Solution. a) Since the bisector of a triangle divides its side proportionally to the other two sides, $A B: B D=A M: M D=2: 3$ and $B D: D C=B N: N C=6: 5$. Therefore, $A B: C D=\frac{2}{3} \cdot \frac{6}{5}=4: 5$.
b) Let the points of tangency of the incircle of triangle $A ... | AB=4,CD=5 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
6. Let's call the distance between numbers the absolute value of their difference. It is known that the sum of the distances from nine consecutive natural numbers to some number $a$ is 294, and the sum of the distances from these same nine numbers to some number $b$ is 1932. Find all possible values of $a$, given that ... | Answer: $a=\frac{13}{3}, a=\frac{755}{3}$.
Solution. Let the given consecutive natural numbers be denoted by $k, k+1, \ldots, k+8$. Notice that if some number lies on the segment $[k ; k+8]$, then the sum of the distances from it to the given nine numbers does not exceed $\frac{9}{2} \cdot 8=36$ (the sum of the distan... | =\frac{13}{3},=\frac{755}{3} | Algebra | math-word-problem | Yes | Yes | olympiads | false |
7. In triangle $A B C$, side $A C$ is equal to 6, and angle $A C B$ is $120^{\circ}$. Circle $\Omega$ with radius $\sqrt{3}$ touches sides $B C$ and $A C$ of triangle $A B C$ at points $K$ and $L$ respectively, and intersects side $A B$ at points $M$ and $N$ ($M$ lies between $A$ and $N$) such that segment $M K$ is par... | Answer: $C L=1, M K=3, A B=2 \sqrt{13}, S_{\triangle A L N}=\frac{125 \sqrt{3}}{52}$.
Solution. Let the center of the circle be denoted as $O$, the foot of the altitude from vertex $C$ of the triangle as $H$, and the angle $B A C$ as $\alpha$.
The center of the circle inscribed in an angle lies on the bisector of tha... | CL=1,MK=3,AB=2\sqrt{13},S_{\triangleALN}=\frac{125\sqrt{3}}{52} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
1. Which whole numbers from 1 to $4 \cdot 10^{25}$ (inclusive) are there more of, and by how many: those containing only even digits or those containing only odd digits? | Answer: the number of numbers containing only odd digits is greater by $\frac{5^{26}-5}{4}$.
Solution. Consider $k$-digit numbers ( $1 \leqslant k \leqslant 25$ ). The number of numbers consisting only of odd digits is $5^{k}$ (for each of the $k$ positions, any of the digits $1,3,5,7$, 9 can be chosen); the number of... | \frac{5^{26}-5}{4} | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
2. Given two linear functions $f(x)$ and $g(x)$ such that the graphs $y=f(x)$ and $y=g(x)$ are parallel lines, not parallel to the coordinate axes. Find the minimum value of the function $(g(x))^{2}+$ $8 f(x)$, if the minimum value of the function $(f(x))^{2}+8 g(x)$ is -29. | Answer: -3.
Solution. Let $f(x)=a x+b, g(x)=a x+c$, where $a \neq 0$. Consider $h(x)=(f(x))^{2}+8 g(x)$. Expanding the brackets, we get $h(x)=(a x+b)^{2}+8(a x+c)=a^{2} x^{2}+2 a(b+4) x+b^{2}+8 c$. The graph of $y=$ $h(x)$ is a parabola opening upwards, and the minimum value is attained at the vertex. The x-coordinate... | -3 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
3. The equation $x^{2}+a x+4=0$ has two distinct roots $x_{1}$ and $x_{2}$; in this case,
$$
x_{1}^{2}-\frac{20}{3 x_{2}^{3}}=x_{2}^{2}-\frac{20}{3 x_{1}^{3}}
$$
Find all possible values of $a$. | Answer: $a=-10$.
Solution. For the equation to have roots, its discriminant must be positive, hence $a^{2}-16>0$. Under this condition, by Vieta's theorem, $x_{1}+x_{2}=-a, x_{1} x_{2}=4$. Then $x_{1}^{2}+$ $x_{1} x_{2}+x_{2}^{2}=\left(x_{1}+x_{2}\right)^{2}-x_{1} x_{2}=a^{2}-4$.
Transform the given equation:
$$
x_{... | -10 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
4. On each of the lines $y=0$ and $y=2$, there are 64 points with abscissas $1,2,3, \ldots, 64$. In how many ways can three points be chosen from the marked 128 points so that they form the vertices of a right triangle? | Answer: 8420.
Solution. There are two possibilities.
1) The hypotenuse of the triangle lies on one of the lines, and the vertex of the right angle is on the second line. Let $ABC$ be the given triangle with a right angle at vertex $C$, and $CH$ be its height dropped to the hypotenuse. From the proportionality of the ... | 8420 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
5. In a convex quadrilateral $ABCD$, diagonal $BD$ is drawn, and a circle is inscribed in each of the resulting triangles $ABD$ and $BCD$. A line passing through vertex $B$ and the center of one of the circles intersects side $DA$ at point $M$. Here, $AM=\frac{25}{7}$ and $MD=\frac{10}{7}$. Similarly, a line passing th... | Answer: a) $A B: C D=5: 6 ;$ b) $A B=5, C D=6$.
Solution. a) Since the bisector of a triangle divides its side proportionally to the other two sides, $A B: B D=A M: M D=5: 2$ and $B D: D C=B N: N C=1: 3$. Therefore, $A B: C D=\frac{5}{2} \cdot \frac{1}{3}=5: 6$.
b) Let the points of tangency of the circle inscribed i... | AB=5,CD=6 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
6. Let's call the distance between numbers the absolute value of their difference. It is known that the sum of the distances from eight consecutive natural numbers to some number $a$ is 612, and the sum of the distances from these same eight numbers to some number $b$ is 240. Find all possible values of $a$, given that... | Answer: $a=27, a=-3$.
Solution. Let the given consecutive natural numbers be denoted as $k, k+1, \ldots, k+7$. Notice that if some number lies in the interval $[k ; k+8]$, then the sum of the distances from it to the given eight numbers does not exceed $4 \cdot 7=28$ (the sum of the distances to the two extreme number... | =27,=-3 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
7. In triangle $A B C$, side $B C$ is equal to 4, and angle $A C B$ is equal to $\frac{2 \pi}{3}$. Circle $\Gamma$ with radius $2 \sqrt{3}$ touches sides $B C$ and $A C$ of triangle $A B C$ at points $K$ and $L$ respectively, and intersects side $A B$ at points $M$ and $N$ ( $M$ lies between $A$ and $N$ ) such that seg... | Answer: $C K=2, M K=6, A B=4 \sqrt{13}, S_{\triangle C M N}=\frac{72 \sqrt{3}}{13}$.
Solution. Let the center of the circle be denoted as $O$, the foot of the altitude from vertex $C$ as $H$, and the angle $A B C$ as $\beta$.
The center of the circle inscribed in an angle lies on the angle bisector, so $\angle O C K ... | CK=2,MK=6,AB=4\sqrt{13},S_{\triangleCMN}=\frac{72\sqrt{3}}{13} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
1. Given 2117 cards, on which natural numbers from 1 to 2117 are written (each card has exactly one number, and the numbers do not repeat). It is required to choose two cards such that the sum of the numbers written on them is divisible by 100. In how many ways can this be done? | Answer: 22386.
Solution. We will take the cards in turn. There are several cases depending on the number written on the first card.
1) The number on the card ends in 00 (there are 21 such cards). For the sum to be divisible by 100, the second card must also end in 00. We get $C_{2} 1^{2}=\frac{21 \cdot 20}{2}=210$ op... | 22386 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
2. Given two linear functions $f(x)$ and $g(x)$ such that the graphs $y=f(x)$ and $y=g(x)$ are parallel lines, not parallel to the coordinate axes. It is known that the graph of the function $y=(f(x))^{2}$ touches the graph of the function $y=11 g(x)$. Find all values of $A$ such that the graph of the function $y=(g(x)... | Answer: $A=-11, A=0$.
Solution. Let $f(x)=a x+b, g(x)=a x+c$, where $a \neq 0$. The tangency of the graphs $y=(f(x))^{2}$ and $y=11 g(x)$ is equivalent to the equation $(f(x))^{2}=11 g(x)$ having exactly one solution. We get $(a x+b)^{2}=11(a x+c), a^{2} x^{2}+a(2 b-11) x+b^{2}-11 c=0$. The discriminant of this equati... | A=-11,A=0 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
3. The equation $x^{2}+a x+2=0$ has two distinct roots $x_{1}$ and $x_{2}$; in this case,
$$
x_{1}^{3}+\frac{14}{x_{2}^{2}}=x_{2}^{3}+\frac{14}{x_{1}^{2}}
$$
Find all possible values of $a$. | Answer: $a=4$.
Solution. For the equation to have roots, its discriminant must be positive, hence $a^{2}-8>0$. Under this condition, by Vieta's theorem, $x_{1}+x_{2}=-a, x_{1} x_{2}=2$. Then $x_{1}^{2}+x_{1} x_{2}+x_{2}^{2}=\left(x_{1}+x_{2}\right)^{2}-x_{1} x_{2}=a^{2}-2$.
Transform the given equation:
$$
x_{1}^{3}... | 4 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
4. Find the number of distinct quadratic trinomials (i.e., with the leading coefficient equal to 1) with integer coefficients such that they have two distinct roots, which are powers of the number 7 with integer non-negative exponents, and at the same time, their coefficients in absolute value do not exceed $343^{36}$. | Answer: 2969.
Solution. Such quadratic trinomials can be represented in the form $\left(x-7^{a}\right)\left(x-7^{b}\right)$, where $a \geqslant 0$, $b \geqslant 0$ are integers. To avoid repetitions, we assume that $a>b$. Expanding the brackets, we get $x^{2}-\left(7^{a}+7^{b}\right) x+7^{a+b}$. By the condition
$$
\... | 2969 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
5. The diagonals $A C$ and $B D$ of the quadrilateral $A B C D$, inscribed in a circle, intersect at point $P$. It is known that the distances from point $P$ to the sides $A B, B C, C D, D A$ are $4, \sqrt{3}, \frac{8}{\sqrt{19}}$ and $8 \sqrt{\frac{3}{19}}$ respectively (the bases of the perpendiculars dropped from po... | Answer: $A P: P C=4, B D=\frac{35}{\sqrt{19}}$.
Solution. Since inscribed angles subtended by the same arc are equal, $\angle P B C=$ $\angle P A D, \angle P C B=\angle P D A$. Therefore, triangles $P B C$ and $P D A$ are similar. Similarly, it can be proven that $\triangle A B P \sim \triangle D C P$. Corresponding e... | AP:PC=4,BD=\frac{35}{\sqrt{19}} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
6. Let's call the distance between numbers the absolute value of their difference. It is known that the sum of the distances from twenty consecutive natural numbers to some number $a$ is 4460, and the sum of the distances from these same twenty numbers to the number $a^{2}$ is 2755. Find all possible values of $a$. | Answer: $a=-\frac{37}{2}, a=\frac{39}{2}$.
Solution. Let the given consecutive natural numbers be denoted as $k, k+1, \ldots, k+19$. Notice that if some number lies on the segment $[k ; k+19]$, then the sum of the distances from it to the given twenty numbers does not exceed $10 \cdot 19=190$ (the sum of the distances... | =-\frac{37}{2},=\frac{39}{2} | Algebra | math-word-problem | Yes | Yes | olympiads | false |
7. Given a parallelogram $A B C D$. A circle $\Omega$ with a diameter of 13 is circumscribed around triangle $A B M$, where $M$ is the point of intersection of the diagonals of the given parallelogram. $\Omega$ intersects the ray $C B$ and the segment $A D$ at points $E$ and $K$ respectively. The length of arc $A E$ is... | Answer: $B C=13, B K=\frac{120}{13}, P_{A K M}=\frac{340}{13}$.
Solution. Let the degree measures of arcs $B M$ and $A E$ be $2 \alpha$ and $4 \alpha$ respectively. Then, by the inscribed angle theorem, $\angle A B E=\frac{1}{2} \cdot 4 \alpha=2 \alpha, \angle B A M=\frac{1}{2} \cdot 2 \alpha=\alpha$. Therefore, $\ang... | BC=13,BK=\frac{120}{13},P_{AKM}=\frac{340}{13} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
1. Given 2414 cards, on which natural numbers from 1 to 2414 are written (each card has exactly one number, and the numbers do not repeat). It is required to choose two cards such that the sum of the numbers written on them is divisible by 100. In how many ways can this be done? | Answer: 29112.
Solution. We will take cards in turn. There are several cases depending on the number written on the first card.
1) The number on the card ends in 00 (there are 24 such cards). For the sum to be divisible by 100, the second card must also end in 00. We get a total of $C_{2} 4^{2}=\frac{24 \cdot 23}{2}=... | 29112 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
2. Given two linear functions $f(x)$ and $g(x)$ such that the graphs $y=f(x)$ and $y=g(x)$ are parallel lines, not parallel to the coordinate axes. It is known that the graph of the function $y=(f(x))^{2}$ touches the graph of the function $y=-8 g(x)$. Find all values of $A$ such that the graph of the function $y=(g(x)... | Answer: $A=8, A=0$.
Solution. Let $f(x)=a x+b, g(x)=a x+c$, where $a \neq 0$. The tangency of the graphs $y=(f(x))^{2}$ and $y=-8 g(x)$ is equivalent to the equation $(f(x))^{2}=-8 g(x)$ having exactly one solution. We get $(a x+b)^{2}=-8(a x+c), a^{2} x^{2}+2 a(b+4) x+b^{2}+8 c=0$. A quarter of the discriminant of th... | A=8,A=0 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
3. The equation $x^{2}+a x+3=0$ has two distinct roots $x_{1}$ and $x_{2}$; in this case,
$$
x_{1}^{3}-\frac{99}{2 x_{2}^{2}}=x_{2}^{3}-\frac{99}{2 x_{1}^{2}}
$$
Find all possible values of $a$. | Answer: $a=-6$.
Solution. For the equation to have roots, its discriminant must be positive, hence $a^{2}-12>0$. Under this condition, by Vieta's theorem, $x_{1}+x_{2}=-a, x_{1} x_{2}=3$. Then $x_{1}^{2}+$ $x_{1} x_{2}+x_{2}^{2}=\left(x_{1}+x_{2}\right)^{2}-x_{1} x_{2}=a^{2}-3$.
Transform the given equation:
$$
x_{1... | -6 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
4. Find the number of distinct quadratic trinomials (i.e., with the leading coefficient equal to 1) with integer coefficients such that they have two distinct roots, which are powers of the number 5 with integer non-negative exponents, and at the same time, their coefficients in absolute value do not exceed $122^{20}$. | Answer: 5699.
Solution. Such quadratic trinomials can be represented in the form $\left(x-5^{a}\right)\left(x-5^{b}\right)$, where $a \geqslant 0$, $b \geqslant 0$ are integers. To avoid repetitions, we assume that $a>b$. Expanding the brackets, we get $x^{2}-\left(5^{a}+5^{b}\right) x+5^{a+b}$. According to the condi... | 5699 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
5. The diagonals $A C$ and $B D$ of the quadrilateral $A B C D$, inscribed in a circle, intersect at point $P$. It is known that the distances from point $P$ to the sides $A B, B C, C D, D A$ are $5, \sqrt{3}, \frac{5}{\sqrt{7}}$ and $5 \sqrt{\frac{3}{7}}$ respectively (the feet of the perpendiculars dropped from point... | Answer: $A P: P C=5, B D=\frac{24}{\sqrt{7}}$.
Solution. Since inscribed angles subtended by the same arc are equal, $\angle P B C=$ $\angle P A D, \angle P C B=\angle P D A$. Therefore, triangles $P B C$ and $P D A$ are similar. Similarly, it can be proven that $\triangle A B P \sim \triangle D C P$. Corresponding el... | AP:PC=5,BD=\frac{24}{\sqrt{7}} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
6. Let's call the distance between numbers the absolute value of their difference. It is known that the sum of the distances from twenty consecutive natural numbers to some number $a$ is 360, and the sum of the distances from these same twenty numbers to the number $a^{2}$ is 345. Find all possible values of $a$. | Answer: $a=-\frac{1}{2}, a=\frac{3}{2}$.
Solution. Let the given consecutive natural numbers be denoted as $k, k+1, \ldots, k+19$. Note that if some number lies on the segment $[k ; k+19]$, then the sum of the distances from it to the given twenty numbers does not exceed $10 \cdot 19=190$ (the sum of the distances to ... | =-\frac{1}{2},=\frac{3}{2} | Algebra | math-word-problem | Yes | Yes | olympiads | false |
7. Given a parallelogram $A B C D$. A circle $\Omega$ with a diameter of 5 is circumscribed around triangle $A B M$, where $M$ is the point of intersection of the diagonals of the given parallelogram. $\Omega$ intersects the ray $C B$ and the segment $A D$ at points $E$ and $K$ respectively. The length of arc $A E$ is ... | Answer: $B C=5, B K=\frac{24}{5}, P_{A K M}=\frac{42}{5}$.
Solution. Let the degree measures of arcs $B M$ and $A E$ be $2 \alpha$ and $4 \alpha$ respectively. Then, by the inscribed angle theorem, $\angle A B E=\frac{1}{2} \cdot 4 \alpha=2 \alpha, \angle B A M=\frac{1}{2} \cdot 2 \alpha=\alpha$. Therefore, $\angle A ... | BC=5,BK=\frac{24}{5},P_{AKM}=\frac{42}{5} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
1. It is known that the roots of the quadratic polynomial $f(x)=x^{2}+a x+b$ are each 3 more than the roots of the quadratic polynomial $g(x)$. Let $M$ be the sum of the coefficients of $f(x)$, and $N$ be the sum of the coefficients of the quadratic polynomial $g(x)$. What is the smallest value that $|M-N|$ can take?
... | 1. It is known that the roots of the quadratic polynomial $f(x)=x^{2}+a x+b$ are each 3 more than the roots of the quadratic polynomial $g(x)$. Let $M$ be the sum of the coefficients of $f(x)$, and $N$ be the sum of the coefficients of the quadratic polynomial $g(x)$. What is the smallest value that $|M-N|$ can take?
... | Algebra | math-word-problem | Yes | Yes | olympiads | false | |
2. On the coordinate plane, the graph of the function param1 is drawn. Right-angled triangles are considered such that all three vertices lie on this graph, with the vertex of the right angle located at the origin. What param 2 value can the product of the distances from the vertices of the acute angles to the $O x$ ax... | 2. On the coordinate plane, the graph of the function param1 is drawn. Right triangles are considered such that all three vertices lie on this graph, with the right-angle vertex located at the origin. What param 2 value can the product of the distances from the vertices of the acute angles to the $O x$ axis take?
| pa... | notfound | Geometry | math-word-problem | Incomplete | Yes | olympiads | false |
3. Points $K, L$ and $M$ are the midpoints of sides $B C, C A$ and $A B$ of triangle $A B C$. Denote $R(X Y Z)$ - the radius of the circumcircle of triangle $X Y Z$. It turns out that $R(A K C): R(A K B)=$ param $1, R(C L B): R(B L A)=$ param2. Find the ratio $R(A M C): R(B M C)$.
| param1 | param2 | |
| :---: | :---... | 3. Points $K, L$ and $M$ are the midpoints of sides $B C, C A$ and $A B$ of triangle $A B C$. Denote $R(X Y Z)$ - the radius of the circumcircle of triangle $X Y Z$. It turns out that $R(A K C): R(A K B)=$ param1, $R(C L B): R(B L A)=$ param2. Find the ratio $R(A M C): R(B M C)$.
| param1 | param2 | Answer |
| :---: |... | Geometry | math-word-problem | Yes | Yes | olympiads | false | |
4. Let $f(x)$ be a quadratic trinomial. The graph of the parabola $y=f(x)$ touches the lines param 1 and param 2. Find the maximum possible value of the discriminant of this trinomial.
| param1 | param2 | |
| :---: | :---: | :---: |
| $y=x+3$ | $y=-3 x-9$ | |
| $y=x-2$ | $y=8-4 x$ | |
| $y=-3 x-15$ | $y=x+5$ | |
|... | 4. Let $f(x)$ be a quadratic trinomial. The graph of the parabola $y=f(x)$ touches the lines param1 and param2. Find the greatest possible value of the discriminant of this trinomial.
| param1 | param2 | Answer |
| :---: | :---: | :---: |
| $y=x+3$ | $y=-3 x-9$ | -3 |
| $y=x-2$ | $y=8-4 x$ | -4 |
| $y=-3 x-15$ | $y=x+... | -5 | Algebra | math-word-problem | Yes | Incomplete | olympiads | false |
5. The numbers $x$ and $y$ are solutions to the system of equations param 1, where $a$ is a parameter. What param 2 value does the expression param 3 take?
| param1 | param2 | param3 | |
| :---: | :---: | :---: | :---: |
| $\left\{\begin{array}{l}a x+y=a+1 \\ x+4 a y=3\end{array}\right.$ | maximum | $x^{2}-6 y^{2}$ |... | 5. The numbers $x$ and $y$ are solutions to the system of equations param 1, where $a$ is a parameter. What param 2 value does the expression param 3 take?
| param1 | param2 | param3 | Answer |
| :---: | :---: | :---: | :---: |
| $\left\{\begin{array}{l}a x+y=a+1 \\ x+4 a y=3\end{array}\right.$ | maximum | $x^{2}-6 y^... | 27 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
6. In how many ways can all asterisks be replaced with param1 even and param2 odd digits (not necessarily distinct) in the number param3 so that the resulting number is divisible by 12?
| param1 | param2 | param3 | |
| :---: | :---: | :---: | :---: |
| 3 | 3 | $2017^{*} 7^{* *} 13^{* *} 112^{*}$ | |
| 5 | 2 | $2017^... | 6. In how many ways can all asterisks be replaced with param1 even and param2 odd digits (not necessarily distinct) in the number param3 so that the resulting number is divisible by 12?
| param1 | param2 | param3 | Answer |
| :---: | :---: | :---: | :---: |
| 3 | 3 | $2017^{*} 7^{* *} 13^{* *} 112^{*}$ | 31250 |
| 5 |... | notfound | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
7. An acute triangle $A B C$ is inscribed in a circle $\Omega$. The extensions of the altitudes of triangle $A B C$, drawn from vertices $A$ and $C$, intersect $\Omega$ at points $M$ and $N$. Find the radius of the circle $\Omega$, if the ratio is 1.
| param1 | |
| :---: | :---: |
| $A C=7, M N=4 \sqrt{6}$ | |
| $A ... | 7. An acute triangle $ABC$ is inscribed in a circle $\Omega$. The extensions of the altitudes of triangle $ABC$, drawn from vertices $A$ and $C$, intersect $\Omega$ at points $M$ and $N$. Find the radius of the circle $\Omega$ if param1.
| param1 | Answer |
| :---: | :---: |
| $AC=7, MN=4 \sqrt{6}$ | 4.9 |
| $AC=11, M... | Geometry | math-word-problem | Yes | Yes | olympiads | false | |
8. param 1 people participated in a survey. They were given a list of $N$ movies. Each person was asked to name their favorite movies from this list. It turned out that everyone named at least two movies. Moreover, any pair of respondents had no more than one movie in common among those they named. Find the smallest po... | 8. param 1 people participated in a survey. They were given a list of $N$ movies. Each person was asked to name their favorite movies from this list. It turned out that everyone named at least two movies. Moreover, any pair of respondents had no more than one movie in common among the ones they named. Find the smallest... | 11 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
9. Given a regular param1. Find the number of its vertex quadruples that are vertices of a convex quadrilateral with exactly two angles equal to $90^{\circ}$. (Two quadruples of vertices that differ in the order of vertices are considered the same.)
| param1 | |
| :---: | :---: |
| 16-gon | |
| 18-gon | |
| 20-gon ... | 9. Given a regular param1. Find the number of its vertex quadruples that are vertices of a convex quadrilateral with exactly two angles equal to $90^{\circ}$. (Two quadruples of vertices that differ in the order of vertices are considered the same.)
| param1 | Answer |
| :---: | :---: |
| 16-gon | 336 |
| 18-gon | 504... | 504 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
10. Find the sum of the roots of the equation param1 that lie in the interval param2. Write the answer in degrees.
| param 1 | param2 | |
| :---: | :---: | :---: |
| $\sin x+\sin ^{2} x+\cos ^{3} x=0$ | $\left[360^{\circ} ; 720^{\circ}\right]$ | |
| $\cos x-\cos ^{2} x-\sin ^{3} x=0$ | $\left[180^{\circ} ; 540^{\cir... | 10. Find the sum of the roots of the equation param1 lying in the interval param2. Write the answer in degrees.
| param 1 | param2 | Answer |
| :---: | :---: | :---: |
| $\sin x+\sin ^{2} x+\cos ^{3} x=0$ | $\left[360^{\circ} ; 720^{\circ}\right]$ | 1800 |
| $\cos x-\cos ^{2} x-\sin ^{3} x=0$ | $\left[180^{\circ} ; 54... | -360 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
1. [5 points] $S$ is the sum of the first 10 terms of an increasing arithmetic progression $a_{1}, a_{2}, a_{3}, \ldots$, consisting of integers. It is known that $a_{6} a_{12}>S+1, a_{7} a_{11}<S+17$. Determine all possible values of $a_{1}$. | Answer: $-6; -5; -4; -2; -1; 0$.
Solution. Let the common difference of the progression be $d$. The inequalities given in the condition can be transformed as follows:
$$
\left\{\begin{array} { l }
{ ( a _ { 1 } + 5 d ) ( a _ { 1 } + 1 1 d ) > S + 1 , } \\
{ ( a _ { 1 } + 6 d ) ( a _ { 1 } + 1 0 d ) S+1 \\
a_{1}^{2}+... | -6;-5;-4;-2;-1;0 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
2. [5 points] Consider all possible tetrahedra $ABCD$ in which $AB=2$, $AC=CB=5$, and $AD=DB=6$. Each such tetrahedron is inscribed in a cylinder such that all vertices lie on its lateral surface, and the edge $CD$ is parallel to the axis of the cylinder. Choose the tetrahedron for which the radius of the cylinder is t... | Answer: $\sqrt{34} \pm \sqrt{23}$.
Solution. Let $E$ be the midpoint of $AB$. $CE$ and $DE$ are the medians of the isosceles triangles $ABC$ and $ABD$, and thus are also the angle bisectors and altitudes. Therefore, $AB \perp CE, AB \perp DE$. This means that the segment $AB$ is perpendicular to the plane $CDE$, hence... | \sqrt{34}\\sqrt{23} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
3. [7 points] Let $M$ be a figure on the Cartesian plane consisting of all points $(x ; y)$ such that there exists a pair of real numbers $a, b$, for which the system of inequalities is satisfied
$$
\left\{\begin{array}{l}
(x-a)^{2}+(y-b)^{2} \leqslant 2 \\
a^{2}+b^{2} \leqslant \min (2 a+2 b ; 2)
\end{array}\right.
$... | Answer: $6 \pi-\sqrt{3}$.
Solution. The second inequality is equivalent to the system of inequalities
$$
\left\{\begin{array}{l}
a^{2}+b^{2} \leqslant 2 a+2 b \\
a^{2}+b^{2} \leqslant 2
\end{array}\right.
$$
Thus, the original system is equivalent to the following:
$$
\left\{\begin{array} { l }
{ ( x - a ) ^ { 2 }... | 6\pi-\sqrt{3} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
4. [5 points] Find the number of triples of natural numbers $(a ; b ; c)$ that satisfy the system of equations
$$
\left\{\begin{array}{l}
\operatorname{GCD}(a ; b ; c)=6 \\
\operatorname{LCM}(a ; b ; c)=2^{15} \cdot 3^{16}
\end{array}\right.
$$ | Answer: 7560.
Solution. Let $a=2^{\alpha_{1}} \cdot 3^{\alpha_{2}}, b=2^{\beta_{1}} \cdot 3^{\beta_{2}}, c=2^{\gamma_{1}} \cdot 3^{\gamma_{2}}$ (the numbers $a$, $b$, $c$ cannot contain any other prime factors - otherwise the second condition of the system is violated). From this,
$$
\operatorname{LCM}(a ; b ; c)=2^{... | 7560 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
5. [5 points] Given the numbers $\log _{\sqrt{5 x-1}}(4 x+1), \log _{4 x+1}\left(\frac{x}{2}+2\right)^{2}, \log _{\frac{x}{2}+2}(5 x-1)$. For which $x$ are two of these numbers equal, and the third one less than them by 1? | Answer: $x=2$.
Solution. From the condition, it follows that the functions $4 x+1, \frac{x}{2}+2,5 x-1$ are positive and do not take the value 1 for all $x$ from the domain of admissible values. Let $a=\log _{\sqrt{5 x-1}}(4 x+1), b=\log _{4 x+1}\left(\frac{x}{2}+2\right)^{2}, c=$ $\log _{\frac{x}{2}+2}(5 x-1)$. Then
... | 2 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
6. [7 points] An acute triangle $ABC$ is inscribed in a circle $\omega$ with center $O$. The circle passing through points $A, O$, and $C$ intersects the segment $BC$ at point $P$. The tangents to $\omega$ drawn through points $A$ and $C$ intersect at point $T$. Segment $TP$ intersects side $AC$ at point $K$. It is kno... | Answer: a) $S_{A B C}=25 ;$ b) $A C=\frac{50}{\sqrt{42}}$.
Solution. Since lines $T C$ and $T A$ are tangents to $\omega$, they are perpendicular to the radii drawn to the points of tangency, and $\angle O C T=\angle O A T=90^{\circ}$. Therefore, points $A$ and $C$ lie on a circle with diameter $O T$ (let's call this ... | S_{ABC}=25;AC=\frac{50}{\sqrt{42}} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
1. [5 points] $S$ - the sum of the first 7 terms of an increasing arithmetic progression $a_{1}, a_{2}, a_{3}, \ldots$, consisting of integers. It is known that $a_{7} a_{12}>S+20, a_{9} a_{10}<S+44$. Indicate all possible values of $a_{1}$. | Answer: $-9; -8; -7; -6; -4; -3; -2; -1$.
Solution. Let the common difference of the progression be $d$. The inequalities given in the condition can be transformed as follows:
$$
\left\{\begin{array} { l }
{ ( a _ { 1 } + 6 d ) ( a _ { 1 } + 11 d ) > S + 20 , } \\
{ ( a _ { 1 } + 8 d ) ( a _ { 1 } + 9 d ) S+20 \\
a_... | -9;-8;-7;-6;-4;-3;-2;-1 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
2. [5 points] Consider all possible tetrahedra $ABCD$ in which $AB=2$, $AC=CB=5$, and $AD=DB=7$. Each such tetrahedron is inscribed in a cylinder such that all vertices lie on its lateral surface, and the edge $CD$ is parallel to the axis of the cylinder. Choose the tetrahedron for which the radius of the cylinder is t... | Answer: $\sqrt{47} \pm \sqrt{23}$.
Solution. Let $E$ be the midpoint of $AB$. $CE$ and $DE$ are the medians of the isosceles triangles $ABC$ and $ABD$, and thus are also the angle bisectors and altitudes. Therefore, $AB \perp CE, AB \perp DE$. This means that the segment $AB$ is perpendicular to the plane $CDE$, hence... | \sqrt{47}\\sqrt{23} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
3. [7 points] Let $M$ be a figure on the Cartesian plane consisting of all points $(x, y)$ such that there exists a pair of real numbers $a, b$ for which the system of inequalities is satisfied
$$
\left\{\begin{array}{l}
(x-a)^{2}+(y-b)^{2} \leqslant 5 \\
a^{2}+b^{2} \leqslant \min (4 a-2 b ; 5)
\end{array}\right.
$$
... | Answer: $15 \pi-\frac{5 \sqrt{3}}{2}$.
Solution. The second inequality is equivalent to the system of inequalities
$$
\left\{\begin{array}{l}
a^{2}+b^{2} \leqslant 4 a-2 b \\
a^{2}+b^{2} \leqslant 5
\end{array}\right.
$$
Thus, the original system is equivalent to the following:
$$
\left\{\begin{array} { l }
{ ( x ... | 15\pi-\frac{5\sqrt{3}}{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
4. [5 points] Find the number of triples of natural numbers $(a ; b ; c)$ that satisfy the system of equations
$$
\left\{\begin{array}{l}
\operatorname{GCD}(a ; b ; c)=15 \\
\operatorname{LCM}(a ; b ; c)=3^{15} \cdot 5^{18}
\end{array}\right.
$$ | Answer: 8568.
Solution. Let $a=3^{\alpha_{1}} \cdot 5^{\alpha_{2}}, b=3^{\beta_{1}} \cdot 5^{\beta_{2}}, c=3^{\gamma_{1}} \cdot 5^{\gamma_{2}}$ (the numbers $a$, $b$, $c$ cannot contain any other prime factors - otherwise the second condition of the system is violated). From this,
$$
\operatorname{LCM}(a ; b ; c)=3^{... | 8568 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
5. [5 points] Given the numbers $\log _{\sqrt{\frac{x}{3}+3}}(6 x-14), \log _{6 x-14}(x-1)^{2}, \log _{x-1}\left(\frac{x}{3}+3\right)$. For which $x$ are two of these numbers equal, and the third one less than them by 1? | Answer: $x=3$.
Solution. From the condition, it follows that the functions $6 x-14, x-1, \frac{x}{3}+3$ are positive and do not take the value 1 for all $x$ in the domain of admissible values. Let $a=\log _{\sqrt{\frac{x}{3}+3}}(6 x-14), b=\log _{6 x-14}(x-1)^{2}, c=$ $\log _{x-1}\left(\frac{x}{3}+3\right)$. Then
$$
... | 3 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
6. [7 points] An acute triangle $ABC$ is inscribed in a circle $\omega$ with center $O$. The circle passing through points $A, O$, and $C$ intersects the segment $BC$ at point $P$. The tangents to $\omega$ drawn through points $A$ and $C$ intersect at point $T$. Segment $TP$ intersects side $AC$ at point $K$. It is kno... | Answer: a) $S_{A B C}=\frac{121}{5}$; b) $A C=\frac{5 \sqrt{33}}{6}$.
Solution. Since the lines $T C$ and $T A$ are tangents to $\omega$, they are perpendicular to the radii drawn to the points of tangency, and $\angle O C T=\angle O A T=90^{\circ}$. Therefore, points $A$ and $C$ lie on a circle with diameter $O T$ (l... | S_{ABC}=\frac{121}{5};AC=\frac{5\sqrt{33}}{6} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
1. [5 points] $S$ is the sum of the first 14 terms of an increasing arithmetic progression $a_{1}, a_{2}, a_{3}, \ldots$, consisting of integers. It is known that $a_{9} a_{17}>S+12, a_{11} a_{15}<S+47$. Indicate all possible values of $a_{1}$. | Answer: $-9; -8; -7; -6; -4; -3; -2; -1$.
Solution. Let the common difference of the progression be $d$. The inequalities given in the conditions can be transformed as follows:
$$
\left\{\begin{array} { l }
{ ( a _ { 1 } + 8 d ) ( a _ { 1 } + 1 6 d ) > S + 1 2 , } \\
{ ( a _ { 1 } + 1 0 d ) ( a _ { 1 } + 1 4 d ) S+1... | -9;-8;-7;-6;-4;-3;-2;-1 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
2. [5 points] Consider all possible tetrahedra $ABCD$ in which $AB=2$, $AC=CB=6$, and $AD=DB=7$. Each such tetrahedron is inscribed in a cylinder such that all vertices lie on its lateral surface, and the edge $CD$ is parallel to the axis of the cylinder. Choose the tetrahedron for which the radius of the cylinder is t... | Answer: $\sqrt{47} \pm \sqrt{34}$.
Solution. Let $E$ be the midpoint of $AB$. $CE$ and $DE$ are the medians of the isosceles triangles $ABC$ and $ABD$, and thus are also the angle bisectors and altitudes. Therefore, $AB \perp CE, AB \perp DE$. This means that the segment $AB$ is perpendicular to the plane $CDE$, hence... | \sqrt{47}\\sqrt{34} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
3. [7 points] Let $M$ be a figure on the Cartesian plane consisting of all points $(x, y)$ such that there exists a pair of real numbers $a, b$ for which the system of inequalities is satisfied
$$
\left\{\begin{array}{l}
(x-a)^{2}+(y-b)^{2} \leqslant 25 \\
a^{2}+b^{2} \leqslant \min (-8 a-6 b ; 25)
\end{array}\right.
... | Answer: $75 \pi-\frac{25 \sqrt{3}}{2}$.
Solution. The second inequality is equivalent to the system of inequalities
$$
\left\{\begin{array}{l}
a^{2}+b^{2} \leqslant-8 a-6 b \\
a^{2}+b^{2} \leqslant 25
\end{array}\right.
$$
Thus, the original system is equivalent to the following:
$$
\left\{\begin{array} { l }
{ ( ... | 75\pi-\frac{25\sqrt{3}}{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
4. [5 points] Find the number of triples of natural numbers $(a ; b ; c)$ that satisfy the system of equations
$$
\left\{\begin{array}{l}
\operatorname{GCD}(a ; b ; c)=21 \\
\operatorname{LCM}(a ; b ; c)=3^{17} \cdot 7^{15}
\end{array}\right.
$$ | Answer: 8064.
Solution. Let $a=3^{\alpha_{1}} \cdot 7^{\alpha_{2}}, b=3^{\beta_{1}} \cdot 7^{\beta_{2}}, c=3^{\gamma_{1}} \cdot 7^{\gamma_{2}}$ (the numbers $a$, $b, c$ cannot contain any other prime factors - otherwise the second condition of the system is violated). From this,
$$
\operatorname{LCM}(a ; b ; c)=3^{\m... | 8064 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
5. [5 points] Given the numbers $\log _{\left(\frac{x}{2}-1\right)^{2}}\left(\frac{x}{2}-\frac{1}{4}\right), \log _{\sqrt{x-\frac{11}{4}}}\left(\frac{x}{2}-1\right), \log _{\frac{x}{2}-\frac{1}{4}}\left(x-\frac{11}{4}\right)^{2}$. For which $x$ are two of these numbers equal, and the third one greater than them by 1? | Answer: $x=5$.
Solution. From the condition, it follows that the functions $\frac{x}{2}-1, \frac{x}{2}-\frac{1}{4}, x-\frac{11}{4}$ are positive and do not take the value 1 for all $x$ in the domain of admissible values. Let $a=\log _{\sqrt{x-\frac{11}{4}}}\left(\frac{x}{2}-1\right), b=\log _{\frac{x}{2}-\frac{1}{4}}\... | 5 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
6. [7 points] An acute triangle $ABC$ is inscribed in a circle $\omega$ with center $O$. The circle passing through points $A, O$, and $C$ intersects the segment $BC$ at point $P$. The tangents to $\omega$ drawn through points $A$ and $C$ intersect at point $T$. Segment $TP$ intersects side $AC$ at point $K$. It is kno... | Answer: a) $S_{A B C}=\frac{128}{3}$; b) $A C=\frac{4 \sqrt{26}}{\sqrt{3}}$.
Solution. Since lines $T C$ and $T A$ are tangents to $\omega$, they are perpendicular to the radii drawn to the points of tangency, and $\angle O C T=\angle O A T=90^{\circ}$. It follows that points $A$ and $C$ lie on a circle with diameter ... | S_{ABC}=\frac{128}{3};AC=\frac{4\sqrt{26}}{\sqrt{3}} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
1. [5 points] $S$ - the sum of the first 5 terms of an increasing arithmetic progression $a_{1}, a_{2}, a_{3}, \ldots$, consisting of integers. It is known that $a_{6} a_{11}>S+15, a_{9} a_{8}<S+39$. Indicate all possible values of $a_{1}$. | Answer: $-9; -8; -7; -6; -4; -3; -2; -1$.
Solution. Let the common difference of the progression be $d$. The inequalities given in the condition can be transformed as follows:
$$
\left\{\begin{array} { l }
{ ( a _ { 1 } + 5 d ) ( a _ { 1 } + 10 d ) > S + 15 , } \\
{ ( a _ { 1 } + 8 d ) ( a _ { 1 } + 7 d ) S+15 \\
a_... | -9;-8;-7;-6;-4;-3;-2;-1 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
2. [5 points] Consider all possible tetrahedra $ABCD$ in which $AB=2$, $AC=CB=7$, and $AD=DB=8$. Each such tetrahedron is inscribed in a cylinder such that all vertices lie on its lateral surface, and the edge $CD$ is parallel to the axis of the cylinder. Choose the tetrahedron for which the radius of the cylinder is t... | Answer: $\sqrt{62} \pm \sqrt{47}$.
Solution. Let $E$ be the midpoint of $AB$. $CE$ and $DE$ are the medians of the isosceles triangles $ABC$ and $ABD$, and thus are also the angle bisectors and altitudes. Therefore, $AB \perp CE, AB \perp DE$. This means that the segment $AB$ is perpendicular to the plane $CDE$, hence... | \sqrt{62}\\sqrt{47} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
3. [7 points] Let $M$ be a figure on the Cartesian plane consisting of all points $(x, y)$ such that there exists a pair of real numbers $a, b$ for which the system of inequalities is satisfied
$$
\left\{\begin{array}{l}
(x-a)^{2}+(y-b)^{2} \leqslant 13 \\
a^{2}+b^{2} \leqslant \min (-4 a-6 b ; 13)
\end{array}\right.
... | Answer: $39 \pi-\frac{13 \sqrt{3}}{2}$.
Solution. The second inequality is equivalent to the system of inequalities
$$
\left\{\begin{array}{l}
a^{2}+b^{2} \leqslant-4 a-6 b \\
a^{2}+b^{2} \leqslant 13
\end{array}\right.
$$
Thus, the original system is equivalent to the following:
$$
\left\{\begin{array} { l }
{ ( ... | 39\pi-\frac{13\sqrt{3}}{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
4. [5 points] Find the number of triples of natural numbers $(a ; b ; c)$ that satisfy the system of equations
$$
\left\{\begin{array}{l}
\operatorname{GCD}(a ; b ; c)=10 \\
\operatorname{LCM}(a ; b ; c)=2^{17} \cdot 5^{16}
\end{array}\right.
$$ | Answer: 8640.
Solution. Let $a=2^{\alpha_{1}} \cdot 5^{\alpha_{2}}, b=2^{\beta_{1}} \cdot 5^{\beta_{2}}, c=2^{\gamma_{1}} \cdot 5^{\gamma_{2}}$ (the numbers $a$, $b$, $c$ cannot contain any other prime factors - otherwise the second condition of the system is violated). From this,
$$
\operatorname{LCM}(a ; b ; c)=2^{... | 8640 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
5. [5 points] Given the numbers $\log _{\sqrt{2 x-8}}(x-4), \log _{(x-4)^{2}}(5 x-26), \log _{\sqrt{5 x-26}}(2 x-8)$. For which $x$ are two of these numbers equal, and the third one greater than them by 1? | Answer: $x=6$.
Solution. From the condition, it follows that the functions $x-4, 5x-26$ are positive and do not take the value 1 for all $x$ in the domain of admissible values. Let $a=\log _{\sqrt{2 x-8}}(x-4), b=\log _{\sqrt{5 x-26}}(2 x-8), c=\log _{(x-4)^{2}}(5 x-26)$. Then
$$
\begin{aligned}
& a b c=\log _{\sqrt{... | 6 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
6. [7 points] An acute triangle $ABC$ is inscribed in a circle $\omega$ with center $O$. The circle passing through points $A, O$, and $C$ intersects the segment $BC$ at point $P$. The tangents to $\omega$ drawn through points $A$ and $C$ intersect at point $T$. Segment $TP$ intersects side $AC$ at point $K$. It is kno... | Answer: a) $S_{A B C}=\frac{81}{2}$; b) $A C=\frac{3 \sqrt{17}}{2}$.
Solution. Since the lines $T C$ and $T A$ are tangents to $\omega$, they are perpendicular to the radii drawn to the points of tangency, and $\angle O C T=\angle O A T=90^{\circ}$. It follows that points $A$ and $C$ lie on a circle with diameter $O T... | S_{ABC}=\frac{81}{2};AC=\frac{3\sqrt{17}}{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
1. [5 points] $S$ - the sum of the first 7 terms of an increasing arithmetic progression $a_{1}, a_{2}, a_{3}, \ldots$, consisting of integers. It is known that $a_{8} a_{17}>S+27, a_{11} a_{14}<S+60$. Indicate all possible values of $a_{1}$. | Answer: $-11; -10; -9; -7; -6; -5$.
Solution. Let the common difference of the progression be $d$. The inequalities given in the problem can be transformed as follows:
$$
\left\{\begin{array} { l }
{ ( a _ { 1 } + 7 d ) ( a _ { 1 } + 1 6 d ) > S + 2 7 , } \\
{ ( a _ { 1 } + 1 0 d ) ( a _ { 1 } + 1 3 d ) S+27 \\
a_{1... | -11;-10;-9;-7;-6;-5 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
2. [5 points] Consider all possible tetrahedra $ABCD$ in which $AB=4, AC=CB=5, AD=DB=6$. Each such tetrahedron is inscribed in a cylinder so that all vertices lie on its lateral surface, and the edge $CD$ is parallel to the axis of the cylinder. We select the tetrahedron for which the radius of the cylinder is the smal... | Answer: $2 \sqrt{7} \pm \sqrt{17}$.
Solution. Let $E$ be the midpoint of $AB$. $CE$ and $DE$ are the medians of the isosceles triangles $ABC$ and $ABD$, and thus are also the angle bisectors and altitudes. Therefore, $AB \perp CE, AB \perp DE$. This means that the segment $AB$ is perpendicular to the plane $CDE$, henc... | 2\sqrt{7}\\sqrt{17} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
3. [7 points] Let $M$ be a figure on the Cartesian plane consisting of all points $(x, y)$ such that there exists a pair of real numbers $a, b$ for which the system of inequalities is satisfied
$$
\left\{\begin{array}{l}
(x-a)^{2}+(y-b)^{2} \leqslant 20 \\
a^{2}+b^{2} \leqslant \min (8 a-4 b ; 20)
\end{array}\right.
$... | Answer: $60 \pi-10 \sqrt{3}$.
Solution. The second inequality is equivalent to the system of inequalities
$$
\left\{\begin{array}{l}
a^{2}+b^{2} \leqslant 8 a-4 b \\
a^{2}+b^{2} \leqslant 20
\end{array}\right.
$$
Thus, the original system is equivalent to the following:
$$
\left\{\begin{array} { l }
{ ( x - a ) ^ ... | 60\pi-10\sqrt{3} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
4. [5 points] Find the number of triples of natural numbers $(a ; b ; c)$ that satisfy the system of equations
$$
\left\{\begin{array}{l}
\operatorname{GCD}(a ; b ; c)=35 \\
\operatorname{LCM}(a ; b ; c)=5^{18} \cdot 7^{16}
\end{array}\right.
$$ | Answer: 9180.
Solution. Let $a=5^{\alpha_{1}} \cdot 7^{\alpha_{2}}, b=5^{\beta_{1}} \cdot 7^{\beta_{2}}, c=5^{\gamma_{1}} \cdot 7^{\gamma_{2}}$ (the numbers $a$, $b, c$ cannot contain any other prime factors - otherwise the second condition of the system is violated). From this,
$$
\operatorname{LCM}(a ; b ; c)=5^{\m... | 9180 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
5. [5 points] Given the numbers $\log _{\sqrt{2 x-3}}(x+1), \log _{2 x^{2}-3 x+5}(2 x-3)^{2}, \log _{x+1}\left(2 x^{2}-3 x+5\right)$. For which $x$ are two of these numbers equal, and the third one less than them by 1? | Answer: $x=4$.
Solution. From the condition, it follows that the functions $x+1, 2x-3$ are positive and do not take the value 1 for all $x$ in the domain of admissible values. Let $a=\log _{\sqrt{2 x-3}}(x+1), b=\log _{2 x^{2}-3 x+5}(2 x-3)^{2}, c=\log _{x+1}\left(2 x^{2}-3 x+5\right)$. Then
$$
\begin{aligned}
a b c=... | 4 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
6. [7 points] An acute triangle $ABC$ is inscribed in a circle $\omega$ with center $O$. The circle passing through points $A, O$, and $C$ intersects the segment $BC$ at point $P$. The tangents to $\omega$ drawn through points $A$ and $C$ intersect at point $T$. Segment $TP$ intersects side $AC$ at point $K$. It is kno... | Answer: a) $S_{A B C}=49$; b) $A C=\frac{7 \sqrt{5}}{\sqrt{6}}$.
Solution. Since lines $T C$ and $T A$ are tangents to $\omega$, they are perpendicular to the radii drawn to the points of tangency, and $\angle O C T=\angle O A T=90^{\circ}$. Therefore, points $A$ and $C$ lie on a circle with diameter $O T$ (let's call... | S_{ABC}=49;AC=\frac{7\sqrt{5}}{\sqrt{6}} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
1. [5 points] $S$ is the sum of the first 15 terms of an increasing arithmetic progression $a_{1}, a_{2}, a_{3}, \ldots$, consisting of integers. It is known that $a_{7} a_{16}>S-24, a_{11} a_{12}<S+4$. Determine all possible values of $a_{1}$. | Answer: $-5; -4; -2; -1$.
Solution. Let the common difference of the progression be $d$. The inequalities given in the condition can be transformed as follows:
$$
\left\{\begin{array} { l }
{ ( a _ { 1 } + 6 d ) ( a _ { 1 } + 1 5 d ) > S - 2 4 , } \\
{ ( a _ { 1 } + 1 0 d ) ( a _ { 1 } + 1 1 d ) S-24 \\
a_{1}^{2}+21... | -5;-4;-2;-1 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
2. [5 points] Consider all possible tetrahedra $ABCD$ in which $AB=4, AC=CB=5, AD=$ $DB=7$. Each such tetrahedron is inscribed in a cylinder so that all vertices lie on its lateral surface, and the edge $CD$ is parallel to the axis of the cylinder. We select the tetrahedron for which the radius of the cylinder is the s... | Answer: $\sqrt{41} \pm \sqrt{17}$.
Solution. Let $E$ be the midpoint of $AB$. $CE$ and $DE$ are the medians of the isosceles triangles $ABC$ and $ABD$, and thus are also the angle bisectors and altitudes. Therefore, $AB \perp CE, AB \perp DE$. This means that the segment $AB$ is perpendicular to the plane $CDE$, hence... | \sqrt{41}\\sqrt{17} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
3. [7 points] Let $M$ be a figure on the Cartesian plane consisting of all points $(x, y)$ such that there exists a pair of real numbers $a, b$ for which the system of inequalities is satisfied
$$
\left\{\begin{array}{l}
(x-a)^{2}+(y-b)^{2} \leqslant 50 \\
a^{2}+b^{2} \leqslant \min (14 a+2 b ; 50)
\end{array}\right.
... | Answer: $150 \pi-25 \sqrt{3}$.
Solution. The second inequality is equivalent to the system of inequalities
$$
\left\{\begin{array}{l}
a^{2}+b^{2} \leqslant 14 a+2 b \\
a^{2}+b^{2} \leqslant 50
\end{array}\right.
$$
Thus, the original system is equivalent to the following:
$$
\left\{\begin{array} { l }
{ ( x - a ) ... | 150\pi-25\sqrt{3} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
4. [5 points] Find the number of triples of natural numbers $(a ; b ; c)$, satisfying the system of equations
$$
\left\{\begin{array}{l}
\text { GCD }(a ; b ; c)=14 \\
\text { LCM }(a ; b ; c)=2^{17} \cdot 7^{18}
\end{array}\right.
$$ | Answer: 9792.
Solution. Let $a=2^{\alpha_{1}} \cdot 7^{\alpha_{2}}, b=2^{\beta_{1}} \cdot 7^{\beta_{2}}, c=2^{\gamma_{1}} \cdot 7^{\gamma_{2}}$ (the numbers $a$, $b, c$ cannot contain any other prime factors - otherwise the second condition of the system is violated). From this,
$$
\operatorname{LCM}(a ; b ; c)=2^{\m... | 9792 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
5. [5 points] Given the numbers $\log _{\left(\frac{x}{2}+1\right)^{2}}\left(\frac{7 x}{2}-\frac{17}{4}\right), \log _{\sqrt{\frac{7 x}{2}-\frac{17}{4}}}\left(\frac{3 x}{2}-6\right)^{2}, \log _{\sqrt{\frac{3 x}{2}-6}}\left(\frac{x}{2}+1\right)$. For which $x$ are two of these numbers equal, and the third one less than ... | Answer: $x=7$.
Solution. From the condition, it follows that the functions $\left(\frac{x}{2}+1\right),\left(\frac{7 x}{2}-\frac{17}{4}\right),\left(\frac{3 x}{2}-6\right)$ are positive and do not take the value 1 for all $x$ in the domain of admissible values. Let $a=\log _{\left(\frac{x}{2}+1\right)^{2}}\left(\frac{... | 7 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
6. [7 points] An acute triangle $ABC$ is inscribed in a circle $\omega$ with center $O$. The circle passing through points $A, O$, and $C$ intersects the segment $BC$ at point $P$. The tangents to $\omega$ drawn through points $A$ and $C$ intersect at point $T$. Segment $TP$ intersects side $AC$ at point $K$. It is kno... | Answer: a) $S_{A B C}=\frac{144}{5} ;$ b) $A C=\frac{4 \sqrt{17}}{\sqrt{7}}$.
Solution. Since lines $T C$ and $T A$ are tangents to $\omega$, they are perpendicular to the radii drawn to the points of tangency, and $\angle O C T=\angle O A T=90^{\circ}$. It follows that points $A$ and $C$ lie on a circle with diameter... | S_{ABC}=\frac{144}{5};AC=\frac{4\sqrt{17}}{\sqrt{7}} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
1. [5 points] $S$ - the sum of the first 6 terms of an increasing arithmetic progression $a_{1}, a_{2}, a_{3}, \ldots$, consisting of integers. It is known that $a_{10} a_{16}>S+39, a_{11} a_{15}<S+55$. Indicate all possible values of $a_{1}$. | Answer: $-12; -11; -10; -8; -7; -6$.
Solution. Let the common difference of the progression be $d$. The inequalities given in the problem can be transformed as follows:
$$
\left\{\begin{array} { l }
{ ( a _ { 1 } + 9 d ) ( a _ { 1 } + 1 5 d ) > S + 3 9 , } \\
{ ( a _ { 1 } + 1 0 d ) ( a _ { 1 } + 1 4 d ) S+39 \\
a_{... | -12;-11;-10;-8;-7;-6 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
2. [5 points] Consider all possible tetrahedra $ABCD$ in which $AB=4, AC=CB=6, AD=$ $DB=7$. Each such tetrahedron is inscribed in a cylinder such that all vertices lie on its lateral surface, and the edge $CD$ is parallel to the axis of the cylinder. Choose the tetrahedron for which the radius of the cylinder is the sm... | Answer: $\sqrt{41} \pm 2 \sqrt{7}$.
Solution. Let $E$ be the midpoint of $AB$. $CE$ and $DE$ are the medians of the isosceles triangles $ABC$ and $ABD$, and thus are also the angle bisectors and altitudes. Therefore, $AB \perp CE, AB \perp DE$. This means that segment $AB$ is perpendicular to the plane $CDE$, hence $A... | \sqrt{41}\2\sqrt{7} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
3. [7 points] Let $M$ be a figure on the Cartesian plane consisting of all points $(x, y)$ such that there exists a pair of real numbers $a, b$ for which the system of inequalities is satisfied
$$
\left\{\begin{array}{l}
(x-a)^{2}+(y-b)^{2} \leqslant 8 \\
a^{2}+b^{2} \leqslant \min (-4 a+4 b ; 8)
\end{array}\right.
$$... | Answer: $24 \pi-4 \sqrt{3}$.
Solution. The second inequality is equivalent to the system of inequalities
$$
\left\{\begin{array}{l}
a^{2}+b^{2} \leqslant-4 a+4 b \\
a^{2}+b^{2} \leqslant 8
\end{array}\right.
$$
Thus, the original system is equivalent to the following:
$$
\left\{\begin{array} { l }
{ ( x - a ) ^ { ... | 24\pi-4\sqrt{3} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
4. [5 points] Find the number of triples of natural numbers $(a ; b ; c)$ that satisfy the system of equations
$$
\left\{\begin{array}{l}
\operatorname{GCD}(a ; b ; c)=22 \\
\operatorname{LCM}(a ; b ; c)=2^{16} \cdot 11^{19}
\end{array}\right.
$$ | Answer: 9720.
Solution. Let $a=2^{\alpha_{1}} \cdot 11^{\alpha_{2}}, b=2^{\beta_{1}} \cdot 11^{\beta_{2}}, c=2^{\gamma_{1}} \cdot 11^{\gamma_{2}}$ (the numbers $a, b, c$ cannot contain any other prime factors - otherwise the second condition of the system is violated). From this,
$$
\operatorname{LCM}(a ; b ; c)=2^{\... | 9720 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
5. [5 points] Given the numbers $\log _{\sqrt{x+34}}(2 x+23), \log _{(x+4)^{2}}(x+34), \log _{\sqrt{2 x+23}}(-x-4)$. For which $x$ are two of these numbers equal, and the third one greater than them by 1? | Answer: $x=-9$.
Solution. From the condition, it follows that the functions $2 x+23, x+34,-x-4$ are positive and do not take the value 1 for all $x$ in the domain of admissible values. Let $a=\log _{\sqrt{x+34}}(2 x+23), b=\log _{(x+4)^{2}}(x+34), c=\log _{\sqrt{2 x+23}}(-x-4)$. Then
$$
\begin{aligned}
& a b c=\log _... | -9 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
6. [7 points] An acute triangle $ABC$ is inscribed in a circle $\omega$ with center $O$. The circle passing through points $A, O$, and $C$ intersects the segment $BC$ at point $P$. The tangents to $\omega$ drawn through points $A$ and $C$ intersect at point $T$. Segment $TP$ intersects side $AC$ at point $K$. It is kno... | Answer: a) $S_{A B C}=\frac{784}{13}$; b) $A C=\frac{14}{\sqrt{3}}$.
Solution. Since lines $T C$ and $T A$ are tangents to $\omega$, they are perpendicular to the radii drawn to the points of tangency, and $\angle O C T=\angle O A T=90^{\circ}$. It follows that points $A$ and $C$ lie on a circle with diameter $O T$ (l... | S_{ABC}=\frac{784}{13};AC=\frac{14}{\sqrt{3}} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
1. [5 points] $S$ - the sum of the first 9 terms of an increasing arithmetic progression $a_{1}, a_{2}, a_{3}, \ldots$, consisting of integers. It is known that $a_{5} a_{18}>S-4, a_{13} a_{10}<S+60$. Indicate all possible values of $a_{1}$. | Answer: $-10; -9; -8; -7; -5; -4; -3; -2$.
Solution. Let the common difference of the progression be $d$. The inequalities given in the condition can be transformed as follows:
$$
\left\{\begin{array} { l }
{ ( a _ { 1 } + 4 d ) ( a _ { 1 } + 1 7 d ) > S - 4 , } \\
{ ( a _ { 1 } + 1 2 d ) ( a _ { 1 } + 9 d ) S-4 \\
... | -10;-9;-8;-7;-5;-4;-3;-2 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
2. [5 points] Consider all possible tetrahedra $ABCD$ in which $AB=4$, $AC=CB=7$, and $AD=DB=8$. Each such tetrahedron is inscribed in a cylinder such that all vertices lie on its lateral surface, and the edge $CD$ is parallel to the axis of the cylinder. Choose the tetrahedron for which the radius of the cylinder is t... | Answer: $2 \sqrt{14} \pm \sqrt{41}$.
Solution. Let $E$ be the midpoint of $AB$. $CE$ and $DE$ are the medians of the isosceles triangles $ABC$ and $ABD$, and thus are also the angle bisectors and altitudes. Therefore, $AB \perp CE, AB \perp DE$. This means that the segment $AB$ is perpendicular to the plane $CDE$, hen... | 2\sqrt{14}\\sqrt{41} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
3. [7 points] Let $M$ be a figure on the Cartesian plane consisting of all points $(x, y)$ such that there exists a pair of real numbers $a, b$ for which the system of inequalities is satisfied
$$
\left\{\begin{array}{l}
(x-a)^{2}+(y-b)^{2} \leqslant 10 \\
a^{2}+b^{2} \leqslant \min (-6 a-2 b ; 10)
\end{array}\right.
... | Answer: $30 \pi-5 \sqrt{3}$.
Solution. The second inequality is equivalent to the system of inequalities
$$
\left\{\begin{array}{l}
a^{2}+b^{2} \leqslant-6 a-2 b \\
a^{2}+b^{2} \leqslant 10
\end{array}\right.
$$
Thus, the original system is equivalent to the following:
$$
\left\{\begin{array} { l }
{ ( x - a ) ^ {... | 30\pi-5\sqrt{3} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
4. [5 points] Find the number of triples of natural numbers $(a ; b ; c)$ that satisfy the system of equations
$$
\left\{\begin{array}{l}
\text { GCD }(a ; b ; c)=33, \\
\text { LCM }(a ; b ; c)=3^{19} \cdot 11^{15} .
\end{array}\right.
$$ | Answer: 9072.
Solution. Let $a=3^{\alpha_{1}} \cdot 11^{\alpha_{2}}, b=3^{\beta_{1}} \cdot 11^{\beta_{2}}, c=3^{\gamma_{1}} \cdot 11^{\gamma_{2}}$ (the numbers $a, b, c$ cannot contain any other prime factors - otherwise the second condition of the system is violated). From this,
$$
\operatorname{LCM}(a ; b ; c)=3^{\... | 9072 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
5. [5 points] Given the numbers $\log _{\sqrt{29-x}}\left(\frac{x}{7}+7\right), \log _{(x+1)^{2}}(29-x), \log _{\sqrt{\frac{x}{7}+7}}(-x-1)$. For which $x$ are two of these numbers equal, and the third one greater than them by 1? | Answer: $x=-7$.
Solution. From the condition, it follows that the functions $\frac{x}{7}+7, 29-x, -x-1$ are positive and do not take the value 1 for all $x$ in the domain of admissible values. Let $a=\log _{\sqrt{29-x}}\left(\frac{x}{7}+7\right), b=\log _{(x+1)^{2}}(29-x), c=$ $\log _{\sqrt{\frac{x}{7}+7}}(-x-1)$. The... | -7 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
6. [7 points] An acute triangle $ABC$ is inscribed in a circle $\omega$ with center $O$. The circle passing through points $A, O$, and $C$ intersects the segment $BC$ at point $P$. The tangents to $\omega$ drawn through points $A$ and $C$ intersect at point $T$. Segment $TP$ intersects side $AC$ at point $K$. It is kno... | Answer. a) $S_{A B C}=\frac{450}{7}$; b) $A C=\frac{5 \sqrt{41}}{\sqrt{14}}$.
Solution. Since lines $T C$ and $T A$ are tangents to $\omega$, they are perpendicular to the radii drawn to the points of tangency, and $\angle O C T=\angle O A T=90^{\circ}$. It follows that points $A$ and $C$ lie on a circle with diameter... | S_{ABC}=\frac{450}{7};AC=\frac{5\sqrt{41}}{\sqrt{14}} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
1. Which whole numbers from 1 to 60000 (inclusive) are there more of, and by how many: those containing only even digits or those containing only odd digits? | Answer: There are 780 more numbers containing only odd digits.
Solution. Consider $k$-digit numbers ( $1 \leqslant k \leqslant 4$ ). The number of numbers consisting only of odd digits is $5^{k}$ (for each of the $k$ positions, any of the digits $1,3,5,7$, 9 can be chosen); the number of numbers consisting only of eve... | 780 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
2. Given two linear functions $f(x)$ and $g(x)$ such that the graphs $y=f(x)$ and $y=g(x)$ are parallel lines, not parallel to the coordinate axes. Find the minimum value of the function $(g(x))^{2}+$ $f(x)$, if the minimum value of the function $(f(x))^{2}+g(x)$ is -6. | # Answer: $\frac{11}{2}$.
Solution. Let $f(x)=a x+b, g(x)=a x+c$, where $a \neq 0$. Consider $h(x)=(f(x))^{2}+g(x)$. Expanding the brackets, we get $h(x)=(a x+b)^{2}+(a x+c)=a^{2} x^{2}+a(2 b+1) x+b^{2}+c$. The graph of $y=h(x)$ is a parabola opening upwards, and the minimum value is attained at the vertex. The x-coor... | \frac{11}{2} | Algebra | math-word-problem | Yes | Yes | olympiads | false |
3. The equation $x^{2}+a x+6=0$ has two distinct roots $x_{1}$ and $x_{2}$; in this case,
$$
x_{1}-\frac{72}{25 x_{2}^{3}}=x_{2}-\frac{72}{25 x_{1}^{3}}
$$
Find all possible values of $a$. | Answer: $a= \pm 9$.
Solution. For the equation to have roots, its discriminant must be positive, from which $a^{2}-24>0$. Under this condition, by Vieta's theorem, $x_{1}+x_{2}=-a, x_{1} x_{2}=6$. Then $x_{1}^{2}+$ $x_{1} x_{2}+x_{2}^{2}=\left(x_{1}+x_{2}\right)^{2}-x_{1} x_{2}=a^{2}-6$.
Transform the given equality:... | \9 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
4. On each of the lines $y=3$ and $y=4$, there are 73 points with abscissas $1,2,3, \ldots, 73$. In how many ways can three points be chosen from the marked 146 so that they form the vertices of a right triangle? | Answer: 10654.
Solution. There are two possibilities.
1) The hypotenuse of the triangle lies on one of the lines, and the vertex of the right angle is on the second line. Let $ABC$ be the given triangle with a right angle at vertex $C$, and $CH$ be its height dropped to the hypotenuse. From the proportionality of the... | 10654 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
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