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4. On the side $B C$ of triangle $A B C$, a point $M$ is taken such that $B M: M C=2: 7$. The bisector $B L$ of the given triangle and the segment $A M$ intersect at point $P$ at an angle of $90^{\circ}$.
a) Find the ratio of the area of triangle $A B P$ to the area of quadrilateral $L P M C$.
b) On the segment $M C$... | Answer. a) $11: 70$, b) $\arccos \frac{\sqrt{11}}{2 \sqrt{3}}$.
Solution. a) In triangle $A B M$, segment $B P$ is both a bisector and an altitude, so triangle $A B M$ is isosceles, and $B P$ is also its median. Let $B M=2 x$, then $A B=2 x, M C=7 x$. By the property of the bisector of a triangle, $A L: L C=A B: B C=2... | )\frac{11}{70},b)\arccos\frac{\sqrt{11}}{2\sqrt{3}} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
5. Find the number of pairs of integers $(x ; y)$ that satisfy the condition $6 x^{2}-7 x y+y^{2}=10^{100}$. | Answer: 19998.
Solution: By factoring the left and right sides of the equation, we get $(6 x-y)(x-y)=2^{100} \cdot 5^{100}$. Since each factor on the left side is an integer, it follows that
$$
\left\{\begin{array}{l}
6 x - y = 2 ^ { k } \cdot 5 ^ { l } , \\
x - y = 2 ^ { 1 0 0 - k } \cdot 5 ^ { 1 0 0 - l }
\end{arra... | 19998 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
6. Find all values of the parameter $b$, for each of which there exists a number $a$ such that the system
$$
\left\{\begin{array}{l}
x^{2}+y^{2}+2 b(b-x+y)=4 \\
y=\frac{9}{(x+a)^{2}+1}
\end{array}\right.
$$
has at least one solution $(x ; y)$. | Answer. $b \in[-11 ; 2)$.
“Phystech-2015”, 10th grade, solutions for ticket 6
Solution. The first equation of the system can be transformed into the form $(x-b)^{2}+(y+b)^{2}=2^{2}$, hence it represents a circle of radius 2 with center $(b ;-b)$.
Consider the function defined by the second equation when $a=0$. At th... | b\in[-11;2) | Algebra | math-word-problem | Yes | Yes | olympiads | false |
1. Solve the inequality $\sqrt{x^{2}-9} \cdot \sqrt{-2 x-1} \leq x^{2}-9$. | Answer. $x \in(-\infty ;-4] \cup\{-3\}$.
Solution. The domain of definition of the given inequality is the set $x \in(-\infty ;-5]$. Let's consider two cases.
a) When $x=-3$, the inequality is satisfied (we get $0=0$).
b) When $x<-3$, we divide both sides of the inequality by the positive number $\sqrt{x^{2}-9}$ and... | x\in(-\infty;-4]\cup{-3} | Inequalities | math-word-problem | Yes | Yes | olympiads | false |
2. Given the function $g(x)=\frac{4 \cos ^{4} x+5 \sin ^{2} x}{4 \sin ^{4} x+3 \cos ^{2} x}$. Find:
a) the roots of the equation $g(x)=\frac{4}{3}$;
b) the maximum and minimum values of the function $g(x)$. | Answer. a) $x=\frac{k \pi}{3}, k \in \mathrm{Z}$; b) $g_{\min }=\frac{5}{4}, g_{\max }=\frac{55}{39}$.
Solution. Transform the given function:
$$
g(x)=\frac{4 \cos ^{4} x+5-5 \cos ^{2} x}{4\left(1-\cos ^{2} x\right)^{2}+3 \cos ^{2} x}=\frac{4 \cos ^{4} x-5 \cos ^{2} x+5}{4 \cos ^{4} x-5 \cos ^{2} x+4}=1+\frac{1}{4 \c... | )\frac{k\pi}{3},k\in\mathrm{Z};b)g_{\}=\frac{5}{4},g_{\max}=\frac{55}{39} | Algebra | math-word-problem | Yes | Yes | olympiads | false |
3. Solve the system of equations $\left\{\begin{array}{l}\frac{1}{x}+\frac{1}{y+z}=1, \\ \frac{1}{y}+\frac{1}{x+z}=\frac{4}{3}, \\ \frac{1}{z}+\frac{1}{x+y}=-\frac{4}{5}\end{array}\right.$. | Answer: $(2 ; 3 ;-1)$.
Solution. Multiplying both sides of the first equation by $x(y+z)$, both sides of the second by $\frac{3}{4} y(x+z)$, and both sides of the third by $-\frac{5}{4} z(x+y)$, we obtain the system
$$
\left\{\begin{array}{l}
(x+y+z)=x y+x z \\
\frac{3}{4}(x+y+z)=x y+y z \\
-\frac{5}{4}(x+y+z)=x z+y ... | (2;3;-1) | Algebra | math-word-problem | Yes | Yes | olympiads | false |
4. On the side $B C$ of triangle $A B C$, a point $M$ is taken such that $B M: M C=3: 8$. The bisector $B L$ of the given triangle and the segment $A M$ intersect at point $P$ at an angle of $90^{\circ}$.
a) Find the ratio of the area of triangle $A B P$ to the area of quadrilateral $L P M C$.
b) On the segment $M C$... | Answer. a) $21: 100$, b) $\arccos \frac{2 \sqrt{7}}{\sqrt{33}}$.
Solution. a) In triangle $A B M$, segment $B P$ is both a bisector and an altitude, so triangle $A B M$ is isosceles, and $B P$ is also its median. Let $B M=3 x$, then $A B=3 x, M C=8 x$. By the property of the bisector of a triangle, $A L: L C=A B: B C=... | )\frac{21}{100},b)\arccos\frac{2\sqrt{7}}{\sqrt{33}} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
5. Find the number of pairs of integers $(x ; y)$ that satisfy the condition $x^{2}+6 x y+5 y^{2}=10^{100}$. | Answer: 19594.
Solution: By factoring the left and right sides of the equation, we get $(x+5 y)(x+y)=2^{100} \cdot 5^{100}$. Since each factor on the left side is an integer, it follows that
$$
\left\{\begin{array}{l}
x+5 y=2^{k} \cdot 5^{l}, \\
x+y=2^{100-k} \cdot 5^{100-l}
\end{array} \text { or } \left\{\begin{arr... | 19594 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
6. Find all values of the parameter $a$, for each of which there exists a number $b$ such that the system
$$
\left\{\begin{array}{l}
x^{2}+y^{2}+2 a(a-x-y)=64 \\
y=\frac{7}{(x+b)^{2}+1}
\end{array}\right.
$$
has at least one solution $(x ; y)$. | Answer. $a \in(-8 ; 15]$.
Solution. The first equation of the system can be transformed into the form $(x-a)^{2}+(y-a)^{2}=8^{2}$, hence it represents a circle with radius 8 and center at $(a ; a)$.
Consider the function defined by the second equation when $b=0$. At the point $x=0$, it attains its maximum value of 7.... | \in(-8;15] | Algebra | math-word-problem | Yes | Yes | olympiads | false |
1. Solve the inequality $\sqrt{x^{2}-4} \cdot \sqrt{2 x-1} \leq x^{2}-4$. | Answer. $x \in\{2\} \cup[3 ;+\infty)$.
Solution. The domain of definition of the given inequality is the set $x \in[2 ;+\infty)$. Consider two cases.
a) For $x=2$, the inequality is satisfied (we get $0=0$).
b) For $x>2$, divide both sides of the inequality by the positive number $\sqrt{x^{2}-4}$ and we get $\sqrt{2... | x\in{2}\cup[3;+\infty) | Inequalities | math-word-problem | Yes | Yes | olympiads | false |
2. Given the function $g(x)=\frac{4 \sin ^{4} x+7 \cos ^{2} x}{4 \cos ^{4} x+\sin ^{2} x}$. Find:
a) the roots of the equation $g(x)=4$;
b) the maximum and minimum values of the function $g(x)$. | Answer. a) $x= \pm \frac{\pi}{3}+k \pi, k \in \mathrm{Z}, x=\frac{\pi}{2}+k \pi, k \in \mathrm{Z}$; b) $g_{\min }=\frac{7}{4}, g_{\max }=\frac{63}{15}$.
Solution. Transform the given function:
$$
g(x)=\frac{4\left(1-\cos ^{2} x\right)^{2}+7 \cos ^{2} x}{4 \cos ^{4} x+1-\cos ^{2} x}=\frac{4 \cos ^{4} x-\cos ^{2} x+4}{... | )\\frac{\pi}{3}+k\pi,k\in\mathrm{Z},\frac{\pi}{2}+k\pi,k\in\mathrm{Z};b)g_{\}=\frac{7}{4},g_{\max}=\frac{63}{15} | Algebra | math-word-problem | Yes | Yes | olympiads | false |
3. Solve the system of equations $\left\{\begin{array}{l}\frac{1}{x}+\frac{1}{y+z}=\frac{6}{5} \text {, } \\ \frac{1}{y}+\frac{1}{x+z}=\frac{3}{4} \\ \frac{1}{z}+\frac{1}{x+y}=\frac{2}{3}\end{array}\right.$, | Answer. $(1 ; 2 ; 3)$.
Solution. Multiplying both sides of the first equation by $\frac{5}{6} x(y+z)$, both sides of the second by $\frac{4}{3} y(x+z)$, and both sides of the third by $\frac{3}{2} z(x+y)$, we obtain the system
$$
\left\{\begin{array}{l}
\frac{5}{6}(x+y+z)=x y+x z \\
\frac{4}{3}(x+y+z)=x y+y z \\
\fra... | (1;2;3) | Algebra | math-word-problem | Yes | Yes | olympiads | false |
4. On the side $B C$ of triangle $A B C$, a point $M$ is taken such that $B M: M C=3: 7$. The bisector $B L$ of the given triangle and the segment $A M$ intersect at point $P$ at an angle of $90^{\circ}$.
a) Find the ratio of the area of triangle $A B P$ to the area of quadrilateral $L P M C$.
b) On the segment $M C$... | Answer. a) $39: 161$, b) $\arccos \sqrt{\frac{13}{15}}$.
Solution. a) In triangle $A B M$, segment $B P$ is both a bisector and an altitude, so triangle $A B M$ is isosceles, and $B P$ is also its median. Let $B M=3 x$, then $A B=3 x, M C=7 x$. By the property of the bisector of a triangle, $A L: L C=A B: B C=3 x: 10 ... | 39:161,\arccos\sqrt{\frac{13}{15}} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
5. Find the number of pairs of integers $(x ; y)$ that satisfy the condition $x^{2}+7 x y+6 y^{2}=15^{50}$. | Answer: 4998.
Solution: Factoring the left and right sides of the equation, we get $(x+6 y)(x+y)=5^{50} \cdot 3^{50}$. Since each factor on the left side is an integer, it follows that
$$
\left\{\begin{array}{l}
x+6 y=5^{k} \cdot 3^{l}, \\
x+y=5^{50-k} \cdot 3^{50-l}
\end{array} \text { or } \left\{\begin{array}{l}
x... | 4998 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
6. Find all values of the parameter $b$, for each of which there exists a number $a$ such that the system
$$
\left\{\begin{array}{l}
x^{2}+y^{2}+2 b(b+x+y)=81 \\
y=\frac{5}{(x-a)^{2}+1}
\end{array}\right.
$$
has at least one solution $(x ; y)$. | Answer. $b \in[-14 ; 9)$.
Solution. The first equation of the system can be transformed into the form $(x+b)^{2}+(y+b)^{2}=9^{2}$, hence it represents a circle of radius 9 with center $(-b ;-b)$.
Consider the function defined by the second equation when $a=0$. At the point $x=0$, it takes the maximum value of 5. As $... | b\in[-14;9) | Algebra | math-word-problem | Yes | Yes | olympiads | false |
1. It is known that $\sin x=2 \cos y-\frac{5}{2} \sin y, \cos x=2 \sin y-\frac{5}{2} \cos y$. Find $\sin 2 y$. | Answer. $\sin 2 y=\frac{37}{40}$.
Solution. Squaring both equalities and adding them term by term, we get $1=4-20 \sin y \cos y+\frac{25}{4}$, from which $10 \sin 2 y=\frac{37}{4}, \sin 2 y=\frac{37}{40}$. | \frac{37}{40} | Algebra | math-word-problem | Yes | Yes | olympiads | false |
2. Find the number of natural numbers $k$, not exceeding 242400, such that $k^{2}+2 k$ is divisible by 303. Answer: 3200. | Solution. Factoring the dividend and divisor, we get the condition $k(k+2):(3 \cdot 101)$. This means that one of the numbers $k$ or $(k+2)$ is divisible by 101. Let's consider two cases.
a) $k: 101$, i.e., $k=101 p, p \in \mathrm{Z}$. Then we get $101 p(101 p+2):(3 \cdot 101) \Leftrightarrow p(101 p+2): 3$. The first... | 3200 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
3. Solve the system $\left\{\begin{array}{l}3 x \geq 2 y+16, \\ x^{4}+2 x^{2} y^{2}+y^{4}+25-26 x^{2}-26 y^{2}=72 x y .\end{array}\right.$ | Answer. $(6 ; 1)$.
Solution. Transform the equation of the system (add $36 x^{2}+36 y^{2}$ to both sides):
$$
\begin{gathered}
\left(x^{2}+y^{2}\right)^{2}+25+10 x^{2}+10 y^{2}=36 x^{2}+36 y^{2}+72 x y \Leftrightarrow\left(x^{2}+y^{2}+5\right)^{2}=(6 x+6 y)^{2} \Leftrightarrow \\
\Leftrightarrow\left[\begin{array} { ... | (6;1) | Algebra | math-word-problem | Yes | Yes | olympiads | false |
4. Find all values of the parameter $b$, for each of which there exists a number $a$ such that the system
$$
\left\{\begin{array}{l}
y=-b-x^{2}, \\
x^{2}+y^{2}+8 a^{2}=4+4 a(x+y)
\end{array}\right.
$$
has at least one solution $(x ; y)$. | Answer: $b \leq 2 \sqrt{2}+\frac{1}{4}$.
Solution: The second equation of the system can be transformed into $(x-2 a)^{2}+(y-2 a)^{2}=2^{2}$, hence it represents a circle with radius 2 and center at $(2 a ; 2 a)$. For all possible $a \in \mathrm{R}$, the graphs of these functions sweep out the strip $x-2 \sqrt{2} \leq... | b\leq2\sqrt{2}+\frac{1}{4} | Algebra | math-word-problem | Yes | Yes | olympiads | false |
5. Given a regular 20-gon $M$. Find the number of quadruples of vertices of this 20-gon that are the vertices of trapezoids. | Answer: 720.
Solution. Let's inscribe the given polygon $K_{1} K_{2} \ldots K_{20}$ in a circle. Each trapezoid is defined by a pair of parallel chords with endpoints at points $K_{1}, \ldots, K_{20}$.
Consider a chord connecting two adjacent vertices of the polygon, for example, $K_{6} K_{7}$. There are 9 more chord... | 720 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
6. Circles with centers \(O_{1}\) and \(O_{2}\) of equal radius are inscribed in the corners \(A\) and \(B\) of triangle \(ABC\), respectively, and point \(O\) is the center of the circle inscribed in triangle \(ABC\). These circles touch side \(AB\) at points \(K_{1}\), \(K_{2}\), and \(K\) respectively, with \(AK_{1}... | Answer. a) $A K=\frac{32}{5}$, b) $\angle C A B=2 \arcsin \frac{3}{5}=\arccos \frac{7}{25}$.
Solution. a) The lines $A O_{1}$ and $B O_{2}$ are the angle bisectors of angles $A$ and $B$ of the triangle, so they intersect at point $O$ - the center of the inscribed circle. Let the radii of the circles with centers $O_{1... | )AK=\frac{32}{5},b)\angleCAB=2\arcsin\frac{3}{5}=\arccos\frac{7}{25} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
1. It is known that $\sin y=\frac{3}{2} \sin x+\frac{2}{3} \cos x, \cos y=\frac{2}{3} \sin x+\frac{3}{2} \cos x$. Find $\sin 2 x$. | Answer. $\sin 2 x=-\frac{61}{72}$.
Solution. Squaring both equalities and adding them term by term, we get $1=\frac{9}{4}+4 \sin x \cos x+\frac{4}{9}$, from which $-2 \sin 2 x=\frac{61}{36}, \sin 2 x=-\frac{61}{72}$. | \sin2-\frac{61}{72} | Algebra | math-word-problem | Yes | Yes | olympiads | false |
2. Find the number of natural numbers $k$, not exceeding 353500, such that $k^{2}+k$ is divisible by 505. Answer: 2800. | Solution. Factoring the dividend and divisor, we get the condition $k(k+1):(5 \cdot 101)$. This means that one of the numbers $k$ or $(k+1)$ is divisible by 101. Let's consider two cases.
a) $k: 101$, i.e., $k=101 p, p \in \mathrm{Z}$. Then we get $101 p(101 p+1):(5 \cdot 101) \Leftrightarrow p(101 p+1): 5$. The first... | 2800 | Number Theory | math-word-problem | Yes | Incomplete | olympiads | false |
4. Find all values of the parameter $a$, for each of which there exists a number $b$ such that the system
$$
\left\{\begin{array}{l}
y=x^{2}-a \\
x^{2}+y^{2}+8 b^{2}=4 b(y-x)+1
\end{array}\right.
$$
has at least one solution $(x ; y)$. | Answer. $a \geq-\sqrt{2}-\frac{1}{4}$.
Solution. The second equation of the system can be transformed into the form $(x+2 b)^{2}+(y-2 b)^{2}=1^{2}$, hence it represents a circle of radius 1 with center $(-2 b ; 2 b)$. For all possible $b \in \mathbf{R}$, the graphs of these functions sweep out the strip $-x-\sqrt{2} \... | \geq-\sqrt{2}-\frac{1}{4} | Algebra | math-word-problem | Yes | Yes | olympiads | false |
5. Given a regular 16-gon $M$. Find the number of quadruples of vertices of this 16-gon that are the vertices of trapezoids. | Answer: 336.
Solution. Let's inscribe the given polygon $K_{1} K_{2} \ldots K_{16}$ in a circle. Each trapezoid is defined by a pair of parallel chords with endpoints at points $K_{1}, \ldots, K_{16}$.
Consider a chord connecting two adjacent vertices of the polygon, for example, $K_{6} K_{7}$. There are 7 more chord... | 336 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
6. In the corners $B$ and $C$ of triangle $ABC$, circles with centers $O_{1}$ and $O_{2}$ of equal radius are inscribed, and point $O$ is the center of the circle inscribed in triangle $ABC$. These circles touch side $BC$ at points $K_{1}, K_{2}$, and $K$ respectively, with $B K_{1}=4, C K_{2}=8$, and $B C=18$.
a) Fin... | Answer. a) $C K=12$, b) $\angle A B C=60^{\circ}$.
Solution. a) The lines $\mathrm{CO}_{2}$ and $\mathrm{BO}_{1}$ are the bisectors of angles $C$ and $B$ of the triangle, so they intersect at point $O$ - the center of the inscribed circle. Let the radii of the circles with centers $O_{1}$ and $O_{2}$ be $r$, and the r... | )CK=12,b)\angleABC=60 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
1. It is known that $\sin y=2 \cos x+\frac{5}{2} \sin x, \cos y=2 \sin x+\frac{5}{2} \cos x$. Find $\sin 2 x$. | Answer. $\sin 2 x=-\frac{37}{40}$.
Solution. Squaring both equalities and adding them term by term, we get $1=4+20 \sin x \cos x+\frac{25}{4}$, from which $-10 \sin 2 x=\frac{37}{4}, \sin 2 x=-\frac{37}{40}$. | \sin2-\frac{37}{40} | Algebra | math-word-problem | Yes | Yes | olympiads | false |
2. Find the number of natural numbers $k$, not exceeding 333300, such that $k^{2}-2 k$ is divisible by 303. Answer: 4400. | Solution. Factoring the dividend and divisor, we get the condition $k(k-2):(3 \cdot 101)$. This means that one of the numbers $k$ or $(k-2)$ is divisible by 101. Let's consider two cases.
a) $k: 101$, i.e., $k=101 p, p \in \mathrm{Z}$. Then we get $101 p(101 p-2):(3 \cdot 101) \Leftrightarrow p(101 p-2): 3$. The first... | 4400 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
3. Solve the system $\left\{\begin{array}{l}2 x+y+8 \leq 0, \\ x^{4}+2 x^{2} y^{2}+y^{4}+9-10 x^{2}-10 y^{2}=8 x y .\end{array}\right.$ | Answer. $(-3 ;-2)$.
Solution. Transform the equation of the system (add $4 x^{2}+4 y^{2}$ to both sides):
$$
\begin{gathered}
\left(x^{2}+y^{2}\right)^{2}+9-6 x^{2}-6 y^{2}=4 x^{2}+4 y^{2}+8 x y \Leftrightarrow\left(x^{2}+y^{2}-3\right)^{2}=(2 x+2 y)^{2} \Leftrightarrow \\
\Leftrightarrow\left[\begin{array} { l }
{ ... | (-3,-2) | Algebra | math-word-problem | Yes | Yes | olympiads | false |
4. Find all values of the parameter $b$, for each of which there exists a number $a$ such that the system
$$
\left\{\begin{array}{l}
y=b-x^{2}, \\
x^{2}+y^{2}+2 a^{2}=4-2 a(x+y)
\end{array}\right.
$$
has at least one solution $(x ; y)$. | Answer. $b \geq-2 \sqrt{2}-\frac{1}{4}$.
Solution. The second equation of the system can be transformed into the form $(x+a)^{2}+(y+a)^{2}=2^{2}$, hence it represents a circle of radius 2 with center at $( -a ;-a )$. For all possible $a \in \mathrm{R}$, the graphs of these functions sweep out the strip $x-2 \sqrt{2} \... | b\geq-2\sqrt{2}-\frac{1}{4} | Algebra | math-word-problem | Yes | Yes | olympiads | false |
5. Given a regular 22-gon $M$. Find the number of quadruples of vertices of this 22-gon that are the vertices of trapezoids. | Answer: 990.
Solution. Let's inscribe the given polygon $K_{1} K_{2} \ldots K_{22}$ in a circle. Each trapezoid side is defined by a pair of parallel chords with endpoints at points $K_{1}, \ldots, K_{22}$.
Consider a chord connecting two adjacent vertices of the polygon, for example, $K_{6} K_{7}$. There are 10 more... | 990 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
6. In the corners $C$ and $B$ of triangle $ABC$, circles with centers $O_{1}$ and $O_{2}$ of equal radius are inscribed, and point $O$ is the center of the circle inscribed in triangle $ABC$. These circles touch side $BC$ at points $K_{1}, K_{2}$, and $K$ respectively, with $C K_{1}=3, B K_{2}=7$, and $B C=16$.
a) Fin... | Answer. a) $C K=\frac{24}{5}$, b) $\angle A C B=2 \arcsin \frac{3}{5}=\arccos \frac{7}{25}$.
Solution. a) The lines $\mathrm{CO}_{1}$ and $\mathrm{BO}_{2}$ are the angle bisectors of angles $C$ and $B$ of the triangle, so they intersect at point $O$ - the center of the inscribed circle. Let the radii of the circles wi... | )CK=\frac{24}{5},b)\angleACB=2\arcsin\frac{3}{5}=\arccos\frac{7}{25} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
1. It is known that $\sin x=\frac{3}{2} \sin y-\frac{2}{3} \cos y, \cos x=\frac{3}{2} \cos y-\frac{2}{3} \sin y$. Find $\sin 2 y$. | Answer. $\sin 2 y=\frac{61}{72}$.
Solution. Squaring both equalities and adding them term by term, we get $1=\frac{9}{4}-4 \sin y \cos y+\frac{4}{9}$, from which $2 \sin 2 y=\frac{61}{36}, \sin 2 y=\frac{61}{72}$. | \frac{61}{72} | Algebra | math-word-problem | Yes | Yes | olympiads | false |
2. Find the number of natural numbers $k$, not exceeding 454500, such that $k^{2}-k$ is divisible by 505. Answer: 3600. | Solution. Factoring the dividend and divisor, we get the condition $k(k-1):(5 \cdot 101)$. This means that one of the numbers $k$ or $(k-1)$ is divisible by 101. Let's consider two cases.
a) $k: 101$, i.e., $k=101 p, p \in \mathrm{Z}$. Then we get $101 p(101 p-1):(5 \cdot 101) \Leftrightarrow p(101 p-1): 5$. The first... | 3600 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
3. Solve the system $\left\{\begin{array}{l}x+3 y+14 \leq 0, \\ x^{4}+2 x^{2} y^{2}+y^{4}+64-20 x^{2}-20 y^{2}=8 x y .\end{array}\right.$ | Answer: $(-2 ;-4)$.
Solution. Transform the equation of the system (add $4 x^{2}+4 y^{2}$ to both sides):
$$
\begin{gathered}
\left(x^{2}+y^{2}\right)^{2}+64-16 x^{2}-16 y^{2}=4 x^{2}+4 y^{2}+8 x y \Leftrightarrow\left(x^{2}+y^{2}-8\right)^{2}=(2 x+2 y)^{2} \Leftrightarrow \\
\Leftrightarrow\left[\begin{array} { l } ... | (-2,-4) | Algebra | math-word-problem | Yes | Yes | olympiads | false |
4. Find all values of the parameter $a$, for each of which there exists a number $b$ such that the system
$$
\left\{\begin{array}{l}
y=x^{2}+a \\
x^{2}+y^{2}+2 b^{2}=2 b(x-y)+1
\end{array}\right.
$$
has at least one solution $(x ; y)$. | Answer. $a \leq \sqrt{2}+\frac{1}{4}$.
Solution. The second equation of the system can be transformed into the form $(x-b)^{2}+(y+b)^{2}=1^{2}$, hence it represents a circle of radius 1 with center $(b ;-b)$. For all possible $b \in \mathbf{R}$, the graphs of these functions sweep out the strip $-x-\sqrt{2} \leq y \le... | \leq\sqrt{2}+\frac{1}{4} | Algebra | math-word-problem | Yes | Yes | olympiads | false |
5. Given a regular 18-gon $M$. Find the number of quadruples of vertices of this 18-gon that are the vertices of trapezoids. | Answer: 504.
Solution. Let's inscribe the given polygon $K_{1} K_{2} \ldots K_{18}$ in a circle. Each trapezoid is defined by a pair of parallel chords with endpoints at points $K_{1}, \ldots, K_{18}$.
Consider a chord connecting two adjacent vertices of the polygon, for example, $K_{6} K_{7}$. There are 8 more chord... | 504 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
6. In the corners $C$ and $A$ of triangle $ABC$, circles with centers $O_{1}$ and $O_{2}$ of equal radius are inscribed, and point $O$ is the center of the circle inscribed in triangle $ABC$. These circles touch side $AC$ at points $K_{1}, K_{2}$, and $K$ respectively, with $C K_{1}=6, A K_{2}=8$, and $A C=21$.
a) Fin... | Answer. a) $C K=9$, b) $\angle A C B=60^{\circ}$.
Solution. a) The lines $\mathrm{CO}_{1}$ and $A O_{2}$ are the bisectors of angles $C$ and $A$ of the triangle, so they intersect at point $O$ - the center of the inscribed circle. Let the radii of the circles with centers $O_{1}$ and $O_{2}$ be $r$, and the radius of ... | )CK=9,b)\angleACB=60 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
11. Find the smallest natural $x$, for which from the fact that param1 is divisible by 31, it follows that param2 is also divisible by 31 ($m$ and $n$ - natural numbers).
| param1 | param2 | |
| :---: | :---: | :---: |
| $15 m+4 n$ | $17 m+x n$ | |
| $13 m+5 n$ | $23 m+x n$ | |
| $15 m+2 n$ | $16 m+x n$ | |
| $17 ... | 11. Find the smallest natural $x$, for which from the fact that param1 is divisible by 31, it follows that param2 is also divisible by 31 ($m$ and $n$ - natural numbers).
| param1 | param2 | Answer |
| :---: | :---: | :---: |
| $15 m+4 n$ | $17 m+x n$ | 19 |
| $13 m+5 n$ | $23 m+x n$ | 16 |
| $15 m+2 n$ | $16 m+x n$ |... | Number Theory | math-word-problem | Yes | Yes | olympiads | false | |
12. Find the number of integer solutions ( $x ; y ; z$ ) of the equation param 1 , satisfying the condition param 2.
| param1 | param2 | |
| :---: | :---: | :---: |
| $60^{x} \cdot\left(\frac{500}{3}\right)^{y} \cdot 360^{z}=2160$ | $\|x+y+z\| \leq 60$ | |
| $60^{x} \cdot\left(\frac{500}{3}\right)^{y} \cdot 360^{z}=... | 12. Find the number of integer solutions ( $x ; y ; z$ ) of the equation param 1 , satisfying the condition param 2.
| param1 | param2 | Answer |
| :---: | :---: | :---: |
| $60^{x} \cdot\left(\frac{500}{3}\right)^{y} \cdot 360^{z}=2160$ | $\|x+y+z\| \leq 60$ | 60 |
| $60^{x} \cdot\left(\frac{500}{3}\right)^{y} \cdot ... | 86 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
13. Two parabolas param1 and param2 touch at a point lying on the Ox axis. A vertical line through point $D$ - the second intersection point of the first parabola with the Ox axis - intersects the second parabola at point $A$, and the common tangent to the parabolas at point $B$. Find the ratio $B D: A B$.
| param1 | ... | 13. Two parabolas param1 and param2 touch at a point lying on the Ox axis. A vertical line through point $D$ - the second intersection point of the first parabola with the Ox axis - intersects the second parabola at point $A$, and the common tangent to the parabolas at point $B$. Find the ratio $B D: A B$.
| param1 | ... | 1.5 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
14. It is known that for all pairs of positive numbers ( $x ; y$ ), for which the equality $x+y=$ param 1 and the inequality $x^{2}+y^{2}>$ param 2 hold, the inequality $x^{4}+y^{4}>m$ also holds. What is the greatest value that $m$ can take?
| param1 | param2 | |
| :---: | :---: | :---: |
| 5 | 14 | |
| 7 | 28 | |... | 14. It is known that for all pairs of positive numbers ( $x ; y$ ), for which the equality $x+y=$ param 1 and the inequality $x^{2}+y^{2}>$ param 2 hold, the inequality $x^{4}+y^{4}>m$ also holds. What is the greatest value that $m$ can take?
| param1 | param2 | Answer |
| :---: | :---: | :---: |
| 5 | 14 | 135.5 |
| ... | Inequalities | math-word-problem | Yes | Yes | olympiads | false | |
15. Inside the acute angle $ABC$, points $M$ and $N$ are taken such that $\angle ABM = \angle MBN = \angle NBC$, $AM \perp BM$ and $AN \perp BN$. The line $MN$ intersects the ray $BC$ at point $K$. Find $MK$, if param1.
| param1 | |
| :---: | :--- |
| $BM=3 \sqrt{2}, BK=\sqrt{2}$ | |
| $BM=6, BK=\sqrt{11}$ | |
| $B... | 15. Inside the acute angle $ABC$, points $M$ and $N$ are taken such that $\angle ABM = \angle MBN = \angle NBC$, $AM \perp BM$ and $AN \perp BN$. The line $MN$ intersects the ray $BC$ at point $K$. Find $MK$, if param1.
| param1 | Answer |
| :---: | :---: |
| $BM=3 \sqrt{2}, BK=\sqrt{2}$ | 4 |
| $BM=6, BK=\sqrt{11}$ |... | Geometry | math-word-problem | Yes | Yes | olympiads | false | |
16. A circle passing through vertices $A$ and $C$ of triangle $A B C$ intersects sides $A B$ and $C B$ at points $Q$ and $P$ respectively. The bisector of angle $A B C$ intersects segment $A P$ at point $E$ and segment $C Q$ at point $F$. Find the length of $A E$, if param 1.
| param1 | | |
| :---: | :--- | :--- |
|... | 16. A circle passing through vertices $A$ and $C$ of triangle $ABC$ intersects sides $AB$ and $CB$ at points $Q$ and $P$ respectively. The bisector of angle $ABC$ intersects segment $AP$ at point $E$ and segment $CQ$ at point $F$. Find the length of $AE$, if param 1.
| param1 | Answer |
| :---: | :---: |
| $Q F=2, P E... | Geometry | math-word-problem | Incomplete | Yes | olympiads | false | |
17. It is known that the number $a$ satisfies the equation param1, and the number $b$ satisfies the equation param2. Find the greatest possible value of the sum $a+b$.
| param1 | param2 | |
| :---: | :---: | :---: |
| $x^{3}-3 x^{2}+5 x-17=0$ | $x^{3}-3 x^{2}+5 x+11=0$ | |
| $x^{3}+3 x^{2}+6 x-9=0$ | $x^{3}+3 x^{2}+... | 17. It is known that the number $a$ satisfies the equation param 1, and the number $b$ satisfies the equation param2. Find the greatest possible value of the sum $a+b$.
| param 1 | param2 | Answer |
| :---: | :---: | :---: |
| $x^{3}-3 x^{2}+5 x-17=0$ | $x^{3}-3 x^{2}+5 x+11=0$ | 2 |
| $x^{3}+3 x^{2}+6 x-9=0$ | $x^{3}... | 4 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
18. Given the sequence $x_{n}=n(n+1)$. It is known that the difference between two terms of this sequence with indices $k$ and l (param1) is divisible by param2. Find the smallest possible value of the sum $l+k$.
| param1 | param2 | |
| :---: | :---: | :---: |
| $l<70<k$ | $5^{8}$ | |
| $l<150<k$ | $5^{9}$ | |
| $l... | 18. Given the sequence $x_{n}=n(n+1)$. It is known that the difference between two terms of this sequence with indices $k$ and l (param1) is divisible by param2. Find the smallest possible value of the sum $l+k$.
| param1 | param2 | Answer |
| :---: | :---: | :---: |
| $l<70<k$ | $5^{8}$ | 3124 |
| $l<150<k$ | $5^{9}$... | notfound | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
19. In a football tournament held in a single round-robin format (each team must play each other exactly once), $N$ teams are participating. At some point in the tournament, the coach of team $A$ noticed that any two teams, different from $A$, have played a different number of games. It is also known that by this point... | 19. In a football tournament held in a single round-robin format (each team must play every other team exactly once), $N$ teams are participating. At a certain point in the tournament, the coach of team $A$ noticed that any two teams, different from $A$, have played a different number of games. It is also known that by... | 63 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
20. On the table, there are param 1 externally identical coins. It is known that among them, there are exactly param 2 counterfeit ones. You are allowed to point to any two coins and ask whether it is true that both these coins are counterfeit. What is the minimum number of questions needed to guarantee getting at leas... | 20. On the table, there are param 1 externally identical coins. It is known that among them, there are exactly param 2 counterfeit ones. You are allowed to point to any two coins and ask whether it is true that both these coins are counterfeit. What is the minimum number of questions needed to guarantee getting at leas... | 54 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
11. Find param 1 of all roots of the equation param 2.
| param1 | param2 | Answer |
| :---: | :---: | :---: |
| sum | $2^{3 x^{2}-6}-4 \cdot 2^{(x+2)^{2}}+2^{2 x(x+1)}-2^{6(x+2)}=0$ | |
| sum | $3^{3 x^{2}-2}-3^{x(x+6)}+3^{x(2 x+3)}-9 \cdot 27^{3 x}=0$ | |
| sum | $5^{3 x^{2}-4}-5^{x(x+2)}+5^{(x-1)(2 x+3)}-5 \cdot 1... | 11. Find param 1 of all roots of the equation param 2.
| param1 | param2 | Answer |
| :---: | :---: | :---: |
| sum | $2^{3 x^{2}-6}-4 \cdot 2^{(x+2)^{2}}+2^{2 x(x+1)}-2^{6(x+2)}=0$ | 2 |
| sum | $3^{3 x^{2}-2}-3^{x(x+6)}+3^{x(2 x+3)}-9 \cdot 27^{3 x}=0$ | 3 |
| sum | $5^{3 x^{2}-4}-5^{x(x+2)}+5^{(x-1)(2 x+3)}-5 \cdot... | notfound | Algebra | math-word-problem | Yes | Incomplete | olympiads | false |
12. Vasya listed all such numbers $x$ for which both numbers $x+\frac{1}{x}$ and the number 1 are integers. Find the sum of the squares of the numbers listed by Vasya.
| param1 | Answer |
| :---: | :---: |
| $x(x-7)$ | |
| $x(x-8)$ | |
| $x(x-9)$ | |
| $x(x-10)$ | | | 12. Vasya listed all such numbers $x$ for which both numbers $x+\frac{1}{x}$ and the number 1 are integers. Find the sum of the squares of the numbers listed by Vasya.
| param1 | Answer |
| :---: | :---: |
| $x(x-7)$ | 49 |
| $x(x-8)$ | 64 |
| $x(x-9)$ | 81 |
| $x(x-10)$ | 100 | | Algebra | math-word-problem | Yes | Incomplete | olympiads | false | |
13. Find the smallest natural $n$ such that param1.
| param1 | Answer |
| :---: | :---: |
| $\sin \left(n^{\circ}+80^{\circ}\right)+\sin \left(n^{\circ}-40^{\circ}\right)+\sin \left(n^{\circ}+70^{\circ}\right)-\cos \left(25^{\circ}\right)=0$ | |
| $\cos \left(n^{\circ}+20^{\circ}\right)-\cos \left(n^{\circ}+80^{\circ... | 13. Find the smallest natural $n$ such that ragat1.
| param 1 | Answer |
| :---: | :---: |
| $\sin \left(n^{\circ}+80^{\circ}\right)+\sin \left(n^{\circ}-40^{\circ}\right)+\sin \left(n^{\circ}+70^{\circ}\right)-\cos \left(25^{\circ}\right)=0$ | 105 |
| $\cos \left(n^{\circ}+20^{\circ}\right)-\cos \left(n^{\circ}+80^{\... | Algebra | math-word-problem | More than one problem | Yes | olympiads | false | |
14. The equation param 1 has a solution $x_{0}=a+b$. What is the greatest value that the product $a b$ can take?
| param1 | Answer |
| :---: | :---: |
| $(2 x+a)(2 x+b)=28$ | |
| $(3 x+a)(3 x+b)=49$ | |
| $(2 x+a)(2 x+b)=22$ | |
| $(4 x+a)(4 x+b)=81$ | | | 14. The equation param 1 has a solution $x_{0}=a+b$. What is the maximum value that the product $a b$ can take?
| param1 | Answer |
| :---: | :---: |
| $(2 x+a)(2 x+b)=28$ | 1,12 |
| $(3 x+a)(3 x+b)=49$ | 1 |
| $(2 x+a)(2 x+b)=22$ | 0,88 |
| $(4 x+a)(4 x+b)=81$ | 1 | | Algebra | math-word-problem | Yes | Incomplete | olympiads | false | |
15. Among the first hundred members of an arithmetic progression with a positive difference, there are numbers param 1. Find the smallest possible value of param 2 term of the progression.
| param1 | param2 | Answer |
| :---: | :---: | :---: |
| $\frac{13}{6}, \frac{75}{2}$ and $\frac{389}{6}$ | second | |
| $\frac... | 15. Among the first hundred members of an arithmetic progression with a positive difference, there are numbers param1. Find the smallest possible value of param2 member of the progression.
| param1 | param2 | Answer |
| :---: | :---: | :---: |
| $\frac{13}{6}, \frac{75}{2}$ and $\frac{389}{6}$ | second | $-0.5$ |
| $... | -14.5 | Algebra | math-word-problem | Yes | Problem not solved | olympiads | false |
16. Given a rectangular grid of size param1. In how many ways can it be cut into grid rectangles of size $1 \times 2$ and $1 \times 7$?
| param1 | Answer |
| :---: | :---: |
| $1 \times 60$ | |
| $1 \times 61$ | |
| $1 \times 62$ | |
| $1 \times 58$ | |
| $1 \times 59$ | | | 16. Given a rectangular grid of size param1. In how many ways can it be cut into grid rectangles of size $1 \times 2$ and $1 \times 7$?
| param1 | Answer |
| :---: | :---: |
| $1 \times 60$ | 10196 |
| $1 \times 61$ | 12083 |
| $1 \times 62$ | 14484 |
| $1 \times 58$ | 7165 |
| $1 \times 59$ | 8547 |
| | | | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | |
17. From the ends of the diameter $A B$ of the circle $\Omega$, chords $A C$ and $B D$ are drawn. These chords intersect at point $M$. It is known that the value of $A C \cdot A M + B D \cdot B M$ is param 1, and the cosine of angle $A M B$ is param2. What is the maximum value that the product $A M \cdot B M$ can take?... | 17. From the ends of the diameter $A B$ of the circle $\Omega$, chords $A C$ and $B D$ are drawn. These chords intersect at point $M$. It is known that the value of $A C \cdot A M + B D \cdot B M$ is param 1, and the cosine of the angle $A M B$ is param2. What is the maximum value that the product $A M \cdot B M$ can t... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false |
18. For each natural $n$, which is not a perfect square, the number of values of the variable $x$ is calculated, for which both numbers $x+\sqrt{n}$ and $x^{2}+param1 \cdot \sqrt{n}$ are natural numbers less than param2. Find the total number of such values of $x$.
| param1 | param2 | answer |
| :---: | :---: | :---: ... | 18. For each natural $n$, which is not a perfect square, the number of values of the variable $x$ is calculated, for which both numbers $x+\sqrt{n}$ and $x^{2}+param1 \cdot \sqrt{n}$ are natural numbers less than param2. Find the total number of such values of $x$.
| param1 | param2 | answer |
| :---: | :---: | :---: ... | 108 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
19. In quadrilateral $A B C D$, where $B A=B C$ and $D A=D C$, the extensions of sides $B A$ and $C D$ intersect at point $N$, and the extensions of sides $B C$ and $A D$ intersect at point $M$. It is known that the difference in the lengths of two sides of quadrilateral $A B C D$ is equal to the radius of the circle i... | 19. In quadrilateral $A B C D$, where $B A=B C$ and $D A=D C$, the extensions of sides $B A$ and $C D$ intersect at point $N$, and the extensions of sides $B C$ and $A D$ intersect at point $M$. It is known that the difference in the lengths of two sides of quadrilateral $A B C D$ is equal to the radius of the circle i... | Geometry | math-word-problem | Yes | Yes | olympiads | false | |
20. Find param 1 given param 2.
| param1 | param2 | Answer |
| :---: | :---: | :---: |
| maximum $2 x+y$ | $\|4 x-3 y\|+5 \sqrt{x^{2}+y^{2}-20 y+100}=30$ | |
| maximum $x+2 y$ | $\|4 y-3 x\|+5 \sqrt{x^{2}+y^{2}+20 y+100}=40$ | |
| maximum $2 y-x$ | $\|4 y+3 x\|+5 \sqrt{x^{2}+y^{2}+10 x+25}=15$ | |
| maximum $x-5 y$... | 20. Find param 1 given param2.
| param1 | param2 | Answer |
| :---: | :---: | :---: |
| maximum $2 x+y$ | $\|4 x-3 y\|+5 \sqrt{x^{2}+y^{2}-20 y+100}=30$ | 16 |
| maximum $x+2 y$ | $\|4 y-3 x\|+5 \sqrt{x^{2}+y^{2}+20 y+100}=40$ | -12 |
| maximum $2 y-x$ | $\|4 y+3 x\|+5 \sqrt{x^{2}+y^{2}+10 x+25}=15$ | 8 |
| maximum $x... | -12 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
1. Find the sum of the roots of the equation param 1.
| param1 | Answer |
| :---: | :---: |
| $\log _{3}\left(9^{x+0.5}+54\right)-\log _{3}\left(2018-3^{x+1.5}\right)=x-0.5$ | |
| $\log _{3}\left(9^{x+0.5}+243\right)-\log _{3}\left(2\left(2018-3^{x-0.5}\right)\right)=x+1.5$ | |
| $\log _{3}\left(9^{x+1}+243\right)-\... | 1. Find the sum of the roots of the equation param 1.
| param1 | Answer |
| :---: | :---: |
| $\log _{3}\left(9^{x+0.5}+54\right)-\log _{3}\left(2018-3^{x+1.5}\right)=x-0.5$ | 2 |
| $\log _{3}\left(9^{x+0.5}+243\right)-\log _{3}\left(2\left(2018-3^{x-0.5}\right)\right)=x+1.5$ | 3 |
| $\log _{3}\left(9^{x+1}+243\right)... | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false |
2. Find the greatest possible value of the root of the equation $(x-a)(x-b)=(x-c)(x-d)$, given that the difference is 1 and $a \neq c$ (the numbers $a, b, c, d$ are not given).
| param1 | Answer |
| :---: | :---: |
| $a+d=b+c=1100$ | |
| $a+d=b+c=700$ | |
| $a+d=b+c=1300$ | |
| $a+d=b+c=850$ | |
| $a+d=b+c=1250$ |... | 2. Find the greatest possible value of the root of the equation $(x-a)(x-b)=(x-c)(x-d)$, given that the difference is 1 and $a \neq c$ (the numbers $a, b, c, d$ are not given).
| param1 | Answer |
| :---: | :---: |
| $a+d=b+c=1100$ | 550 |
| $a+d=b+c=700$ | 350 |
| $a+d=b+c=1300$ | 650 |
| $a+d=b+c=850$ | 425 |
| $a+d... | Algebra | math-word-problem | Yes | Yes | olympiads | false | |
3. Given an arithmetic progression $a_{1}, a_{2}, \ldots, a_{n}$, where param 1 . Find $n$, if it is known that param 2.
| param1 | param2 | Answer |
| :---: | :---: | :---: |
| $a_{1}=3, a_{n}=20$ | $\frac{1}{a_{1} a_{2}}+\frac{1}{a_{2} a_{3}}+\frac{1}{a_{3} a_{4}}+\ldots+\frac{1}{a_{n-1} a_{n}}=7$ | |
| $a_{1}=5, a... | 3. Given an arithmetic progression $a_{1}, a_{2}, \ldots, a_{n}$, where param1. Find $n$, if it is known that param 2.
| param1 | param2 | Answer |
| :---: | :---: | :---: |
| $a_{1}=3, a_{n}=20$ | $\frac{1}{a_{1} a_{2}}+\frac{1}{a_{2} a_{3}}+\frac{1}{a_{3} a_{4}}+\ldots+\frac{1}{a_{n-1} a_{n}}=7$ | 421 |
| $a_{1}=5, ... | Algebra | math-word-problem | Yes | Problem not solved | olympiads | false | |
4. The circle $\Gamma$, inscribed in triangle $ABC$, touches side $AB$ at point $K$, and side $AC$ at point $T$. On the smaller arc $TK$ of circle $\Gamma$, a point $P$ is chosen. The line passing through point $K$ parallel to line $AP$ intersects circle $\Gamma$ again at point $N$. Find $PK$, given that $AP=$ param $1... | 4. The circle $\Gamma$, inscribed in triangle $ABC$, touches side $AB$ at point $K$, and side $AC$ at point $T$. On the smaller arc $TK$ of circle $\Gamma$, a point $P$ is chosen. The line passing through point $K$ parallel to line $AP$ intersects circle $\Gamma$ again at point $N$. Find $PK$, given that $AP=$ param $1... | Geometry | math-word-problem | Incomplete | Yes | olympiads | false | |
5. On the base $AD$ of the trapezoid $ABCD$, a point $K$ was chosen. It turned out that $AB = BK$ and $KC = CD$. It is known that the ratio of the areas of triangles $ABK$ and $KCD$ is param1, and the radii of the circles inscribed in triangles $ABK$ and $KCD$ are param2 and param3, respectively. Find the radius of the... | 5. On the base $AD$ of trapezoid $ABCD$, a point $K$ was chosen. It turned out that $AB = BK$ and $KC = CD$. It is known that the ratio of the areas of triangles $ABK$ and $KCD$ is param1, and the radii of the circles inscribed in triangles $ABK$ and $KCD$ are param2 and param3, respectively. Find the radius of the cir... | Geometry | math-word-problem | Yes | Yes | olympiads | false | |
6. Find the sum of the roots of the equation param 1 that lie in the interval param 2.
| param1 | param2 | Answer |
| :---: | :---: | :---: |
| $\sin ^{4}\left(\frac{\pi x}{3}\right)+\sqrt{3} \sin ^{3}\left(\frac{\pi x}{3}\right) \cos \left(\frac{\pi x}{3}\right)-$ $-2 \sin ^{2}\left(\frac{\pi x}{3}\right) \cos ^{2}\... | 6. Find the sum of the roots of the equation param 1 that lie in the interval param 2.
| param1 | param2 | Answer |
| :---: | :---: | :---: |
| $\sin ^{4}\left(\frac{\pi x}{3}\right)+\sqrt{3} \sin ^{3}\left(\frac{\pi x}{3}\right) \cos \left(\frac{\pi x}{3}\right)-$ $-2 \sin ^{2}\left(\frac{\pi x}{3}\right) \cos ^{2}\... | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false |
7. We took natural numbers param 1 and wrote these param 2 along the top side of the table param3 (one number above each column), and wrote the same param2 along the left side of the table (one number to the left of each row). In the cells of the table, we wrote the products of the corresponding numbers (a "multiplicat... | 7. We took natural numbers param 1 and wrote these param 2 along the top side of the table param3 (one number above each column), and wrote the same param2 along the left side of the table (one number to the left of each row). In the cells of the table, we wrote the products of the corresponding numbers (a "multiplicat... | notfound | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
8. According to the results of the volleyball tournament, which was held in a single round-robin (i.e., each team played one game against each other), it turned out that the first ten teams won against each of the remaining teams, and the total points scored by the first ten teams were param1 more than the total points... | 8. According to the results of a volleyball tournament held in a single round-robin format (i.e., each team played one game against each other), it turned out that the top ten teams won against each of the other teams, and the total points scored by the top ten teams were param1 more than the total points scored by the... | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | |
9. Find param1, if param2 for all real $x$, and param3.
| param1 | param2 | param3 | Answer |
| :---: | :---: | :---: | :---: |
| $f(250)$ | $f(x+3)=f(x)+x^{2}+x-7$ | $f(1)=1$ | |
| $f(400)$ | $f(x+3)=f(x)+x^{2}-x+3$ | $f(1)=2$ | |
| $f(430)$ | $f(x+3)=f(x)+x^{2}+2 x+1$ | $f(1)=15$ | |
| $f(310)$ | $f(x+3)=f(x)-x^{... | 9. Find param1, if param2 for all real $x$, and param3.
| param1 | param2 | param3 | Answer |
| :---: | :---: | :---: | :---: |
| $f(250)$ | $f(x+3)=f(x)+x^{2}+x-7$ | $f(1)=1$ | 1714698 |
| $f(400)$ | $f(x+3)=f(x)+x^{2}-x+3$ | $f(1)=2$ | 7005245 |
| $f(430)$ | $f(x+3)=f(x)+x^{2}+2 x+1$ | $f(1)=15$ | 8803238 |
| $f(310... | Algebra | math-word-problem | Yes | Yes | olympiads | false | |
10. In the cells of a rectangular board param1, chips were placed. It turned out that for any chip, either in the column or in the row where this chip is located, there are no more chips. What is the maximum number of chips that can be on the board?
| param1 | Answer |
| :---: | :---: |
| $20 \times 50$ | |
| $25 \ti... | 10. In the cells of a rectangular board param1, chips were placed. It turned out that for any chip, either in the column or in the row where this chip is located, there are no more chips. What is the maximum number of chips that can be on the board?
| param1 | Answer |
| :---: | :---: |
| $20 \times 50$ | 68 |
| $25 \... | Combinatorics | math-word-problem | Yes | Problem not solved | olympiads | false | |
1. Given quadratic trinomials $f_{1}(x)=x^{2}-2 a x+3, f_{2}(x)=x^{2}+x+b, f_{3}(x)=3 x^{2}+(1-4 a) x+6+b$ and $f_{4}(x)=3 x^{2}+(2-2 a) x+3+2 b$. Let the differences of their roots be $A, B, C$ and $D$ respectively. It is known that $|A| \neq|B|$. Find the ratio $\frac{C^{2}-D^{2}}{A^{2}-B^{2}}$. The values of $A, B, ... | Answer: $\frac{1}{3}$.
Solution. Let $\alpha x^{2}+\beta x+\gamma$ be a quadratic trinomial with a positive discriminant $T$. Then its roots are determined by the formula $x_{1,2}=\frac{-b \pm \sqrt{T}}{2 a}$, so $\left|x_{2}-x_{1}\right|=\left|\frac{-b+\sqrt{T}-(-b-\sqrt{T})}{2 a}\right|=\frac{\sqrt{T}}{|a|}$. Applyi... | \frac{1}{3} | Algebra | math-word-problem | Yes | Yes | olympiads | false |
2. Find all values of the variable $x$, for each of which both expressions $f(x)=\sqrt{21-x^{2}-4 x}$ and $g(x)=|x+2|$ are defined, and $\min (f(x) ; g(x))>\frac{x+4}{2}$. | Answer: $x \in\left[-7 ;-\frac{8}{3}\right) \cup(0 ; 2)$.
Solution. Both functions are defined when $21-x^{2}-4 x \geqslant 0 \Leftrightarrow -7 \leqslant x \leqslant 3$. Note that the following two statements are equivalent: "the smaller of two numbers is greater than $A$" and "both numbers are greater than $A$". The... | x\in[-7;-\frac{8}{3})\cup(0;2) | Algebra | math-word-problem | Yes | Yes | olympiads | false |
3. Find the first term and the common ratio of an infinite decreasing geometric progression, if the ratio of the sum of the cubes of all its terms to the sum of all terms of this progression is $\frac{48}{7}$, and the ratio of the sum of the fourth powers of the terms to the sum of the squares of the terms of this prog... | Answer: $b_{1}= \pm 3, q=\frac{1}{4}$.
Solution. It is known that the sum of the first $n$ terms of a geometric progression with the first term $b_{1}$ and common ratio $q$ is $\frac{b_{1}\left(1-q^{n}\right)}{1-q}$. For a decreasing infinite geometric progression, $|q|<1$, so as $n$ approaches infinity, $q^{n}$ appro... | b_{1}=\3,q=\frac{1}{4} | Algebra | math-word-problem | Yes | Yes | olympiads | false |
4. Given an isosceles trapezoid \(ABCD (AD \parallel BC, AD > BC)\). A circle \(\Omega\) is inscribed in angle \(BAD\), touches segment \(BC\) at point \(C\), and intersects \(CD\) again at point \(E\) such that \(CE = 9, ED = 7\). Find the radius of the circle \(\Omega\) and the area of trapezoid \(ABCD\). | Answer: $R=6, S_{A B C D}=96+24 \sqrt{7}$.
Solution. Let the points of tangency of the circle with the sides $A B$ and $A D$ of the trapezoid be denoted as $K$ and $W$ respectively. By the tangent-secant theorem, $D W^{2}=D E \cdot D C=7 \cdot 16, D W=4 \sqrt{7}$. Since $C$ and $W$ are points of tangency of the circle... | R=6,S_{ABCD}=96+24\sqrt{7} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
5. On the table, there are 140 different cards with numbers $3,6,9, \ldots 417,420$ (each card has exactly one number, and each number appears exactly once). In how many ways can 2 cards be chosen so that the sum of the numbers on the selected cards is divisible by $7?$ | Answer: 1390.
Solution. The given numbers, arranged in ascending order, form an arithmetic progression with a common difference of 3. Therefore, the remainders when these numbers are divided by 7 alternate. Indeed, if one of these numbers is divisible by 7, i.e., has the form $7k$, where $k \in \mathbb{N}$, then the n... | 1390 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
6. On the coordinate plane, consider a figure $M$ consisting of all points with coordinates $(x ; y)$ that satisfy the system of inequalities
$$
\left\{\begin{array}{l}
|x-1|+|5-x| \leqslant 4 \\
\frac{x^{2}-6 x+2 y+7}{y+x-4} \leqslant 0
\end{array}\right.
$$
Sketch the figure $M$ and find its area. | Answer: 4.
Consider the first inequality. To open the absolute values, we consider three possible cases.
1) $x<1$. Then $1-x+5-x \leqslant 4 \Leftrightarrow x \geqslant 1$, i.e., there are no solutions.
2) $1 \leqslant x \leqslant 5$. Then $x-1+5-x \leqslant 4 \Leftrightarrow 4 \leqslant 4$, which is always true, so ... | 4 | Inequalities | math-word-problem | Yes | Yes | olympiads | false |
7. Circles $\omega$ and $\Omega$ touch externally at point $F$, and their common external tangent touches circles $\omega$ and $\Omega$ at points $A$ and $B$, respectively. Line $\ell$ passes through point $B$, intersects circle $\Omega$ again at point $C$, and intersects $\omega$ at points $D$ and $E$ (point $D$ is be... | Answer: $H P=8 \sqrt{31}, r=6 \sqrt{\frac{93}{5}}, R=2 \sqrt{465}$.
Solution. We apply the theorem of the tangent and the secant three times:
$$
\begin{gathered}
H F^{2}=H C \cdot H B=2 \cdot 62 \Rightarrow H F=2 \sqrt{31} \\
H F^{2}=H D \cdot H E \Rightarrow H E=\frac{H F^{2}}{H D}=62 \\
B A^{2}=B D \cdot B E=64 \cd... | HP=8\sqrt{31},r=6\sqrt{\frac{93}{5}},R=2\sqrt{465} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
1. Given quadratic trinomials $f_{1}(x)=x^{2}-x+2 a, f_{2}(x)=x^{2}+2 b x+3, f_{3}(x)=4 x^{2}+(2 b-3) x+6 a+3$ and $f_{4}(x)=4 x^{2}+(6 b-1) x+9+2 a$. Let the differences of their roots be $A, B, C$ and $D$ respectively. It is known that $|A| \neq|B|$. Find the ratio $\frac{C^{2}-D^{2}}{A^{2}-B^{2}}$. The values of $A,... | Answer: $\frac{1}{2}$.
Solution. Let $\alpha x^{2}+\beta x+\gamma-$ be a quadratic trinomial with a positive discriminant $T$. Then its roots are determined by the formula $x_{1,2}=\frac{-b \pm \sqrt{T}}{2 a}$, so $\left|x_{2}-x_{1}\right|=\left|\frac{-b+\sqrt{T}-(-b-\sqrt{T})}{2 a}\right|=\frac{\sqrt{T}}{|a|}$. Apply... | \frac{1}{2} | Algebra | math-word-problem | Yes | Yes | olympiads | false |
2. Find all values of the variable $x$, for each of which both expressions $f(x)=\sqrt{16-x^{2}+6 x}$ and $g(x)=|x-3|$ are defined, and $\min (f(x) ; g(x))>\frac{5-x}{2}$. | Answer: $x \in(-1 ; 1) \cup\left(\frac{11}{3} ; 8\right]$.
Solution. Both functions are defined when $16-x^{2}+6 x \geqslant 0 \Leftrightarrow -2 \leqslant x \leqslant 8$. Note that the following two statements are equivalent: "the smaller of two numbers is greater than $A$" and "both numbers are greater than $A$". Th... | x\in(-1;1)\cup(\frac{11}{3};8] | Algebra | math-word-problem | Yes | Yes | olympiads | false |
3. It is known that the ratio of the sum of all terms of an infinitely decreasing geometric progression to the sum of the cubes of all terms of the same progression is $\frac{1}{12}$, and the ratio of the sum of the fourth powers of all terms to the sum of the squares of all terms of this progression is $\frac{36}{5}$.... | Answer: $b_{1}= \pm 3, q=-\frac{1}{2}$.
Solution. It is known that the sum of the first $n$ terms of a geometric progression with the first term $b_{1}$ and common ratio $q$ is $\frac{b_{1}\left(1-q^{n}\right)}{1-q}$. For a decreasing infinite geometric progression, $|q|<1$, so as $n$ approaches infinity, $q^{n}$ appr... | b_{1}=\3,q=-\frac{1}{2} | Algebra | math-word-problem | Yes | Yes | olympiads | false |
4. Given an isosceles trapezoid \(ABCD (AD \parallel BC, AD > BC)\). A circle \(\Omega\) is inscribed in angle \(BAD\), touches segment \(BC\) at point \(C\), and intersects \(CD\) again at point \(E\) such that \(CE = 7\), \(ED = 9\). Find the radius of the circle \(\Omega\) and the area of trapezoid \(ABCD\). | Answer: $R=2 \sqrt{7}, S_{A B C D}=56 \sqrt{7}$.
Solution. Let the points of tangency of the circle with the sides $A B$ and $A D$ of the trapezoid be denoted as $K$ and $W$ respectively. By the tangent-secant theorem, $D W^{2}=D E \cdot D C=9 \cdot 16, D W=12$. Since $C$ and $W$ are points of tangency of the circle w... | R=2\sqrt{7},S_{ABCD}=56\sqrt{7} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
5. On the table, there are 210 different cards with numbers $2,4,6, \ldots 418,420$ (each card has exactly one number, and each number appears exactly once). In how many ways can 2 cards be chosen so that the sum of the numbers on the selected cards is divisible by $7?$ | Answer: 3135.
Solution. The given numbers, arranged in ascending order, form an arithmetic progression with a difference of 2. Therefore, the remainders of these numbers when divided by 7 alternate. Indeed, if one of these numbers is divisible by 7, i.e., has the form $7k$, where $k \in \mathbb{N}$, then the following... | 3135 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
6. On the coordinate plane, consider a figure $M$ consisting of all points with coordinates $(x ; y)$ that satisfy the system of inequalities
$$
\left\{\begin{array}{l}
|y|+|4-y| \leqslant 4 \\
\frac{y^{2}+x-4 y+1}{2 y+x-7} \leqslant 0
\end{array}\right.
$$
Sketch the figure $M$ and find its area. | Answer: 8.
Solution. Consider the first inequality. To open the absolute values, we consider three possible cases.
1) $y < 0$. Then $-y-4+y \leqslant 4 \Leftrightarrow -4 \leqslant 4$, which is always true, so $y \in (-\infty, 0)$.
2) $0 \leqslant y \leqslant 4$. Then $y-4+y \leqslant 4 \Leftrightarrow 2y \leqslant 8... | 8 | Inequalities | math-word-problem | Yes | Yes | olympiads | false |
7. Circles $\omega$ and $\Omega$ touch externally at point $F$, and their common external tangent touches circles $\omega$ and $\Omega$ at points $A$ and $B$, respectively. Line $\ell$ passes through point $B$, intersects circle $\Omega$ again at point $C$, and intersects $\omega$ at points $D$ and $E$ (point $D$ is be... | Answer: $H P=3 \sqrt{7}, r=2 \sqrt{21}, R=6 \sqrt{21}$.
Solution. We apply the tangent-secant theorem three times:
$$
\begin{gathered}
H F^{2}=H D \cdot H E=3 \cdot 21 \Rightarrow H F=3 \sqrt{7} \\
H F^{2}=H C \cdot H B \Rightarrow H B=\frac{H F^{2}}{H C}=21 \\
B A^{2}=B D \cdot B E=24 \cdot 42=\Rightarrow B A=12 \sq... | HP=3\sqrt{7},r=2\sqrt{21},R=6\sqrt{21} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
1. Given a linear function $f(x)$. It is known that the distance between the points of intersection of the graphs $y=$ $x^{2}-1$ and $y=f(x)$ is $\sqrt{30}$, and the distance between the points of intersection of the graphs $y=x^{2}$ and $y=f(x)+3$ is $\sqrt{46}$. Find the distance between the points of intersection of... | Answer: $\sqrt{38}$.
Solution. Let $f(x)=a x+b$. Then the abscissas of the points of intersection of the graphs in the first case are determined from the equation $x^{2}-1=a x+b$, and in the second case - from the equation $x^{2}=a x+b+3$.
Consider the first case in more detail. The equation has the form $x^{2}-a x-(... | \sqrt{38} | Algebra | math-word-problem | Yes | Yes | olympiads | false |
2. Solve the inequality $\frac{\sqrt[3]{\frac{x^{2}}{2}}-\sqrt[6]{4 x^{2}}}{\left(x^{2}-4|x|\right)^{2}-8 x^{2}+32|x|-48} \geqslant 0$. | Answer: $x \in(-\infty ;-6) \cup[-4 ;-2) \cup(-2 ; 2) \cup(2 ; 4] \cup(6 ;+\infty)$.
Solution. Consider the denominator of the fraction. It can be written as $\left(x^{2}-4|x|\right)^{2}-8\left(x^{2}-4|x|\right)-48$, or, if we let $x^{2}-4|x|=t$, in the form $t^{2}-8 t-48=(t-12)(t+4)$. If we return to the variable $x$... | x\in(-\infty;-6)\cup[-4;-2)\cup(-2;2)\cup(2;4]\cup(6;+\infty) | Inequalities | math-word-problem | Yes | Yes | olympiads | false |
3. Given an infinitely decreasing geometric progression. The sum of all its terms with odd indices is 2 more than the sum of all terms with even indices. And the difference between the sum of the squares of all terms in odd positions and the sum of the squares of all terms in even positions is $\frac{36}{5}$. Find the ... | Answer: $b_{1}=3, q=\frac{1}{2}$.
Solution. It is known that the sum of the first $n$ terms of a geometric progression with the first term $b_{1}$ and common ratio $q$ is $\frac{b_{1}\left(1-q^{n}\right)}{1-q}$. For a decreasing infinite geometric progression, $|q|<1$, so as $n$ approaches infinity, $q^{n}$ approaches... | b_{1}=3,q=\frac{1}{2} | Algebra | math-word-problem | Yes | Yes | olympiads | false |
4. A trapezoid $ABCD (AD \| BC)$ and a rectangle $A_{1}B_{1}C_{1}D_{1}$ are inscribed in a circle $\Omega$ with radius 13, such that $AC \perp B_{1}D_{1}, BD \perp A_{1}C_{1}$. Find the ratio of the area of $ABCD$ to the area of $A_{1}B_{1}C_{1}D_{1}$, given that $AD=10, BC=24$. | Answer: $\frac{289}{338}$ or $\frac{1}{2}$.
Solution. Draw a line through the center of the circle $O$ perpendicular to the bases of the trapezoid. Let it intersect $A D$ and $B C$ at points $N$ and $M$ respectively. Since a diameter perpendicular to a chord bisects the chord, $B M=M C=12, A N=N D=5$. By the Pythagore... | \frac{289}{338} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
5. There are 200 different cards with numbers $2,3,2^{2}, 3^{2}, \ldots, 2^{100}, 3^{100}$ (each card has exactly one number, and each number appears exactly once). In how many ways can 2 cards be chosen so that the product of the numbers on the chosen cards is a cube of an integer? | Answer: 4389.
Solution. To obtain the cube of a natural number, it is necessary and sufficient for each factor to enter the prime factorization of the number in a power that is a multiple of 3.
Suppose two cards with powers of two are chosen. We have 33 exponents divisible by 3 $(3,6,9, \ldots, 99)$, 34 exponents giv... | 4389 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
6. On the coordinate plane, consider a figure $M$ consisting of all points with coordinates $(x ; y)$ that satisfy the system of inequalities
$$
\left\{\begin{array}{l}
y-x \geqslant|x+y| \\
\frac{x^{2}+8 x+y^{2}+6 y}{2 y-x-8} \leqslant 0
\end{array}\right.
$$
Sketch the figure $M$ and find its area. | Answer: 8.
Solution. The first inequality is equivalent to the system ${ }^{1}$ $\left\{\begin{array}{l}x+y \leqslant y-x, \\ x+y \geqslant x-y\end{array} \Leftrightarrow\left\{\begin{array}{l}x \leqslant 0, \\ y \geqslant 0 .\end{array}\right.\right.$
Consider the second inequality. It can be written as $\frac{(x+4)... | 8 | Inequalities | math-word-problem | Yes | Yes | olympiads | false |
7. Chords $AB$ and $CD$ of circle $\Gamma$ with center $O$ have a length of 4. The extensions of segments $BA$ and $CD$ beyond points $A$ and $D$ respectively intersect at point $P$. Line $PO$ intersects segment $AC$ at point $L$, such that $AL: LC = 1: 4$.
a) Find $AP$.
b) Suppose additionally that the radius of cir... | Answer: $A P=\frac{4}{3}, P T=\frac{\sqrt{145}}{3}-3, S_{\triangle A P C}=\frac{128 \sqrt{5}}{87}$.
Solution. a) Drop perpendiculars $O H$ and $O N$ from point $O$ to the chords $C D$ and $A B$ respectively. Since these chords are equal, the distances from the center of the circle to them are also equal, so $O H=O N$.... | AP=\frac{4}{3},PT=\frac{\sqrt{145}}{3}-3,S_{\triangleAPC}=\frac{128\sqrt{5}}{87} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
1. Given a linear function $f(x)$. It is known that the distance between the points of intersection of the graphs $y=$ $x^{2}-1$ and $y=f(x)+1$ is $3 \sqrt{10}$, and the distance between the points of intersection of the graphs $y=x^{2}$ and $y=f(x)+3$ is $3 \sqrt{14}$. Find the distance between the points of intersect... | Answer: $3 \sqrt{2}$.
Solution. Let $f(x)=a x+b$. Then the abscissas of the points of intersection of the graphs in the first case are determined from the equation $x^{2}-1=a x+b+1$, and in the second case - from the equation $x^{2}=a x+b+3$.
Consider the first case in more detail. The equation has the form $x^{2}-a ... | 3\sqrt{2} | Algebra | math-word-problem | Yes | Yes | olympiads | false |
2. Solve the inequality $\frac{\sqrt[6]{4 x^{2}}-\sqrt[3]{x^{2}}}{\left(x^{2}-2|x|\right)^{2}+4|x|-2 x^{2}-3} \leqslant 0$. | Answer: $x \in(-\infty ;-3) \cup[-2 ;-1) \cup(-1 ; 1) \cup(1 ; 2] \cup(3 ;+\infty)$.
Solution. Consider the denominator of the fraction. It can be written as $\left(x^{2}-2|x|\right)^{2}-2\left(x^{2}-2|x|\right)-3$, or, if we let $x^{2}-2|x|=t$, in the form $t^{2}-2 t-3=(t-3)(t+1)$. If we return to the variable $x$, w... | x\in(-\infty;-3)\cup[-2;-1)\cup(-1;1)\cup(1;2]\cup(3;+\infty) | Inequalities | math-word-problem | Yes | Yes | olympiads | false |
3. Given an infinitely decreasing geometric progression. The sum of all its terms with odd indices is 10 more than the sum of all terms with even indices. And the difference between the sum of the squares of all terms at odd positions and the sum of the squares of all terms at even positions is 20. Find the first term ... | Answer: $b_{1}=5, q=-\frac{1}{2}$.
Solution. It is known that the sum of the first $n$ terms of a geometric progression with the first term $b_{1}$ and common ratio $q$ is $\frac{b_{1}\left(1-q^{n}\right)}{1-q}$. For a decreasing infinite geometric progression, $|q|<1$, so as $n$ approaches infinity, $q^{n}$ approache... | b_{1}=5,q=-\frac{1}{2} | Algebra | math-word-problem | Yes | Yes | olympiads | false |
4. A trapezoid $ABCD (AD \| BC)$ and a rectangle $A_{1}B_{1}C_{1}D_{1}$ are inscribed in a circle $\Omega$ with radius 17, such that $AC \perp B_{1}D_{1}, BD \perp A_{1}C_{1}$. Find the ratio of the area of $ABCD$ to the area of $A_{1}B_{1}C_{1}D_{1}$, given that $AD=30, BC=16$. | Answer: $\frac{529}{578}$ or $\frac{1}{2}$.
Solution. Draw a line through the center of the circle $O$ perpendicular to the bases of the trapezoid. Let it intersect $A D$ and $B C$ at points $N$ and $M$ respectively. Since a diameter perpendicular to a chord bisects the chord, $B M=M C=8, A N=N D=15$. By the Pythagore... | \frac{529}{578} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
5. There are 100 different cards with numbers $2,5,2^{2}, 5^{2}, \ldots, 2^{50}, 5^{50}$ (each card has exactly one number, and each number appears exactly once). In how many ways can 2 cards be chosen so that the product of the numbers on the chosen cards is a cube of an integer? | Answer: 1074.
Solution. To obtain the cube of a natural number, it is necessary and sufficient for each factor to enter the prime factorization of the number in a power that is a multiple of 3.
Suppose two cards with powers of two are chosen. We have 16 exponents that are divisible by 3 $(3,6,9, \ldots, 48)$, 17 expo... | 1074 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
6. On the coordinate plane, consider a figure $M$ consisting of all points with coordinates $(x ; y)$ that satisfy the system of inequalities
$$
\left\{\begin{array}{l}
x+y+|x-y| \leqslant 0 \\
\frac{x^{2}+6 x+y^{2}-8 y}{x+3 y+6} \geqslant 0
\end{array}\right.
$$
Sketch the figure $M$ and find its area. | Answer: 3.
Solution. The first inequality is equivalent to the system $\left\{\begin{array}{l}x-y \leqslant-x-, \\ x-y \geqslant x+y\end{array} \Leftrightarrow\left\{\begin{array}{l}x \leqslant 0, \\ y \leqslant 0 .\end{array}\right.\right.$
Consider the second inequality. It can be written as $\frac{(x+3)^{2}+(y-4)^... | 3 | Inequalities | math-word-problem | Yes | Yes | olympiads | false |
7. Chords $AB$ and $CD$ of circle $\Gamma$ with center $O$ have a length of 6. The extensions of segments $BA$ and $CD$ beyond points $A$ and $D$ respectively intersect at point $P$. Line $PO$ intersects segment $AC$ at point $L$, such that $AL: LC = 1: 2$.
a) Find $AP$.
b) Suppose additionally that the radius of cir... | Answer: $A P=6, P T=2 \sqrt{22}-4, S_{\triangle A P C}=\frac{81 \sqrt{7}}{11}$.
Solution. a) Drop perpendiculars $O H$ and $O N$ from point $O$ to the chords $C D$ and $A B$ respectively. Since these chords are equal, the distances from the center of the circle to them are also equal, so $O H=O N$. Right triangles $O ... | AP=6,PT=2\sqrt{22}-4,S_{\triangleAPC}=\frac{81\sqrt{7}}{11} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
1. Two parabolas param 1 and param 2 touch at a point lying on the $O x$ axis. Through point $D$, the second intersection point of the first parabola with the $O x$ axis, a vertical line is drawn, intersecting the second parabola at point $A$ and the common tangent to the parabolas at point $B$. Find the ratio $D A: D ... | 1. Two parabolas param 1 and param 2 touch at a point lying on the $O x$ axis. Through point $D$, the second intersection point of the first parabola with the $O x$ axis, a vertical line is drawn, intersecting the second parabola at point $A$ and the common tangent to the parabolas at point $B$. Find the ratio $D A: D ... | 4 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
2. Inside the acute angle $ABC$, points $M$ and $N$ are taken such that $\angle ABM = \angle MBN = \angle NBC$, $AM \perp BM$ and $AN \perp BN$. The line $MN$ intersects the ray $BC$ at point $K$. Find $BN$, if param1.
| param1 | |
| :---: | :---: |
| $BM=24, BK=3$ | |
| $BM=18, BK=7$ | |
| $BM=50, BK=14$ | |
| $B... | 2. Inside the acute angle $ABC$, points $M$ and $N$ are taken such that $\angle ABM = \angle MBN = \angle NBC$, $AM \perp BM$ and $AN \perp BN$. The line $MN$ intersects the ray $BC$ at point $K$. Find $BN$ if param1.
| param1 | Answer |
| :---: | :---: |
| $BM=24, BK=3$ | 4 |
| $BM=18, BK=7$ | 8.4 |
| $BM=50, BK=14$ ... | Geometry | math-word-problem | Incomplete | Yes | olympiads | false | |
3. Find the number of integer solutions $(a ; b ; c)$ of the equation param1, satisfying the condition param2.
| param1 | param2 | |
| :---: | :---: | :---: |
| $150^{a} \cdot\left(\frac{200}{3}\right)^{b} \cdot 2250^{c}=33750$ | $\|a+b+c\| \leq 120$ | |
| $150^{a} \cdot\left(\frac{200}{3}\right)^{b} \cdot 2250^{c}=... | 3. Find the number of integer solutions ( $a ; b ; c$ ) of the equation param1, satisfying the condition param2.
| param1 | param2 | Answer |
| :---: | :---: | :---: |
| $150^{a} \cdot\left(\frac{200}{3}\right)^{b} \cdot 2250^{c}=33750$ | $\|a+b+c\| \leq 120$ | 120 |
| $150^{a} \cdot\left(\frac{200}{3}\right)^{b} \cdo... | 120,90,250,112 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
4. Given the sequence $y_{n}=n(n+1)$. It is known that the difference between two terms of this sequence with indices $k$ and l (param1) is divisible by param2. Find the smallest possible value of the sum $l+k$.
| param1 | param2 |
| :---: | :---: |
| $l<100<k$ | $3^{10}$ |
| $l<115<k$ | $3^{11}$ |
| $l<125<k$ | $3^{1... | 4. Given the sequence $y_{n}=n(n+1)$. It is known that the difference between two terms of this sequence with indices $k$ and l (param1) is divisible by param2. Find the smallest possible value of the sum $l+k$.
| param1 | param2 | Answer |
| :---: | :---: | :---: |
| $l<100<k$ | $3^{10}$ | 728 |
| $l<115<k$ | $3^{11}... | notfound | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
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