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falin_algebra.1 | 1 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Faculty of Computational Mathematics and Cybernetics, oral, 2006) Find the sum of the digits of the number $10^{2006} - 2006$. | 18\,047 | \boxed{18\,047} | unverifiable | unclassified | 11 | |||
falin_algebra.1002 | 1002 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Mech-Math, 2006, No. 2) Solve the inequality
$$\frac{\sqrt{1+3^{-x}}}{\sqrt{1+3^{-x}} - \sqrt{1-3^{-x}}} - \frac{3^{-x}-1}{\sqrt{1-9^{-x}} + 3^{-x} - 1} \geq \frac{1+\sqrt{1-9^{-x}}}{3^{-x}}.$$ | x > 0. | \boxed{x > 0.} | verified | 1 interval(s) checked | 124 | |||
falin_algebra.1003 | 1003 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (ВМК, 2004, июль, №2) Solve the inequality
$$\frac{x^2 - 4}{\sqrt{15 + 2x - x^2}} \geq |x| - 2.$$ | (-3; -2] \cup \left[\frac{3 - \sqrt{31}}{2}; \frac{\sqrt{23} - 1}{2}\right] \cup [2; 5). | \boxed{(-3; -2] \cup \left[\frac{3 - \sqrt{31}}{2}; \frac{\sqrt{23} - 1}{2}\right] \cup [2; 5).} | unverifiable | unparsed answer | 125 | |||
falin_algebra.1005 | 1005 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (ВМК, 1990, июль, №4) Solve the inequality
$$\sqrt{9v^2 - 48v - 21} + \sqrt{9v^2 - 51v - 15} \leq |3v - 6|.$$ | \left[\frac{27 - 4\sqrt{66}}{9}; \frac{8 - \sqrt{85}}{3}\right] \cup \left[\frac{17 + \sqrt{349}}{6}; \frac{27 + 4\sqrt{66}}{9}\right]. | \boxed{\left[\frac{27 - 4\sqrt{66}}{9}; \frac{8 - \sqrt{85}}{3}\right] \cup \left[\frac{17 + \sqrt{349}}{6}; \frac{27 + 4\sqrt{66}}{9}\right].} | unverifiable | no valid sample points | 125 | |||
falin_algebra.1007 | 1007 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (ВМК, 2001, июль, №6) The function $f(x)$ is defined, increasing and negative on the entire number line. Solve the inequality
$$\frac{2 \cdot f(x^2 - 2x - 112) + \left|f(x^2 - 2x - 112) - 3 \cdot f(-2x\sqrt{32 - 2x})\right|}{\left(3 \cdot f(-2x\sqrt{32 - 2x} - 112) - 2 \cdot f(-2x\sqrt{32 - 2x})\right)^7} > 0.$$ | (-13 - \sqrt{57}; \, 8). | \boxed{(-13 - \sqrt{57}; \, 8).} | unverifiable | no valid sample points | 125 | |||
falin_algebra.1009 | 1009 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (ВМК, 2009, №9) Solve the inequality
$$\sqrt{5 - x} - 7x \leq x \cdot |x - 3| - 7.$$ | [1; 5]. | \boxed{[1; 5].} | verified | 1 interval(s) checked | 125 | |||
falin_algebra.1010 | 1010 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (хим., 2006, №1) Solve the inequality
$$\sqrt{1 - |x|} \geq x - 2.$$ | no-gold | 125 | ||||||
falin_algebra.1011 | 1011 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | 1011. (Geology, oral, 2008, July) Solve the inequality $\sqrt{x} + \sqrt{x^2 - 2x} \leq 2\sqrt{3}$. | \{0\} \cup [2; 3] | \boxed{\{0\} \cup [2; 3]} | verified | 2 interval(s) checked | 126 | |||
falin_algebra.1013 | 1013 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | 1013. (ICA, 1994, No. 5) Solve the inequality $$\left|x - 4^{1+\sqrt{3-x}}\right| \leq \frac{5}{3}x - 4 \cdot 4^{\sqrt{3-x}}.$$ | x = 3 | \boxed{x = 3} | verified | 1 integer(s) checked | 126 | |||
falin_algebra.1014 | 1014 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | 1014. (PhB, 2008, No. 3) Solve the inequality $$x - \sqrt{6x - x^2 - 8} \leq 2 - \sqrt{x^2 - 2x - 3}.$$ | x = 3 | \boxed{x = 3} | verified | 1 integer(s) checked | 126 | |||
falin_algebra.1016 | 1016 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | 1016. (VMK, oral, 2001) Solve the inequality $$4\sqrt{1 - \frac{1}{x}} + \sqrt{x - \frac{16}{x}} < x.$$ | [4; +\infty) \setminus \left\{\frac{1+\sqrt{65}}{2}\right\} | \boxed{[4; +\infty) \setminus \left\{\frac{1+\sqrt{65}}{2}\right\}} | unverifiable | unparsed answer | 126 | |||
falin_algebra.1017 | 1017 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | 1017. (VMK, oral, 2003) Solve the inequality $$x + y^2 + \sqrt{x - y^2 - 1} \leq 1.$$ | x = 1; y = 0 | \boxed{x = 1; y = 0} | unverifiable | no valid sample points | 126 | |||
falin_algebra.1018 | 1018 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | 1018. (Geology, oral, 2005) Solve the inequality $$\sqrt{x^2 + x - 1} + \sqrt{x - y^2 - 1} \leq 1.$$ | x = 1; y = 0 | \boxed{x = 1; y = 0} | unverifiable | no valid sample points | 126 | |||
falin_algebra.102 | 102 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Geology, 1994, No. 2) Simplify the following expression to an integer:
$$2\frac{\sqrt{6+2\sqrt{5}}}{\sqrt{6-2\sqrt{5}}} - \frac{\sqrt{7+\sqrt{5}}}{\sqrt{7-\sqrt{5}}} \cdot 2\sqrt{11}.$$ | -4. | \boxed{-4.} | verified | numeric(8 pts) | 21 | |||
falin_algebra.1021 | 1021 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (BMK, 2006, April, No. 2) Solve the inequality
$$\sqrt{\frac{x^{2}-22x+121}{x^{2}-24x+140}} \geq 50x - 2x^{2} - 309.$$ | $x < 10,\ x = 11,\ x > 14$.
The final answer is $\boxed{x < 10,\ x = 11,\ x > 14}$. | x < 10,\ x = 11,\ x > 14 | \boxed{x < 10,\ x = 11,\ x > 14} | unverifiable | no valid sample points | 127 | ||
falin_algebra.1023 | 1023 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Olympiad "Pokori Vorobyovy Gory-2017", No. 3) Solve the inequality
$$\frac{8^{-|x-4|}}{\sqrt{x^{2}-5x+3}} \leq \frac{3\cdot 2^{-|x+2|}}{\sqrt{x^{2}+13x-14}+9}.$$
In the answer, write the sum of the lengths of the solution intervals that belong to the set $[-2020;2018]$. If the inequality has no solutions, write $-1$. | $4017$.
The final answer is $\boxed{4017}$. | 4017 | \boxed{4017} | suspected | integer 4014 satisfies the inequality but is not listed | 127 | ||
falin_algebra.1026 | 1026 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (ДВИ, 2019, No. 2) Find $a + b + c$, given that $a + 3b = 2$, $b + 3c = 4$, $c + 3a = 6$. | 3 | \boxed{3} | unverifiable | multi-block | 129 | |||
falin_algebra.1027 | 1027 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (ФНМ, 2000, April, No. 1) Solve the system
$$\begin{cases} 3x + y - z = 4, \\ x - 2y + 3z = 0, \\ x^2 + 2y + z^2 = 6x. \end{cases}$$ | (1; 2; 1); \left(\frac{44}{25}; -\frac{28}{5}; -\frac{108}{25}\right) | \boxed{(1; 2; 1); \left(\frac{44}{25}; -\frac{28}{5}; -\frac{108}{25}\right)} | verified | the printed tuple satisfies all 3 equations | 129 | |||
falin_algebra.1028 | 1028 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Soil Science, 1994, No. 2) Solve the system
$$\begin{cases} 3 \cdot 2^x + y = 13, \\ 2^{2x+1} + 3y = 35. \end{cases}$$ | \left(-1; \frac{23}{2}\right), (2; 1) | \boxed{\left(-1; \frac{23}{2}\right), (2; 1)} | verified | the printed tuple satisfies all 2 equations | 129 | |||
falin_algebra.1029 | 1029 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Economics, management dept., 2007, No. 2) Solve the system of equations
$$\begin{cases} x^2 = 81, \\ x + 1 = -2\sqrt{y^2 + 10y + 41}. \end{cases}$$ | \{(-9; -5)\} | \boxed{\{(-9; -5)\}} | verified | the printed tuple satisfies all 2 equations | 130 | |||
falin_algebra.1030 | 1030 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Economics (management), 1996, No. 2) Solve the system
$$\begin{cases} x + 3^y = 2, \\ x^3 + 27^y = 26. \end{cases}$$ | \{(-1; 1)\} | \boxed{\{(-1; 1)\}} | verified | the printed tuple satisfies all 2 equations | 130 | |||
falin_algebra.1031 | 1031 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Psychology, 1989, No. 3) Solve the system
$$\begin{cases} \log_x 25 + 2y = 2, \\ -(\log_x 0.2)^3 + y = 1. \end{cases}$$ | \{(5; 0),\ \left(\tfrac{1}{5}; 2\right)\} | \boxed{\{(5; 0),\ \left(\tfrac{1}{5}; 2\right)\}} | unverifiable | not a genuine system | 130 | |||
falin_algebra.1032 | 1032 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Mech.-Math., 1994, No. 2) Solve the system
$$\begin{cases} 2^x + 2y = 1, \\ 3y - 6y^2 = 2^{x-1}. \end{cases}$$ | \{\left(\log_2 \tfrac{2}{3}; \tfrac{1}{6}\right)\} | \boxed{\{\left(\log_2 \tfrac{2}{3}; \tfrac{1}{6}\right)\}} | verified | the printed tuple satisfies all 2 equations | 130 | |||
falin_algebra.1033 | 1033 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Mech.-Math., 2010, No. 1) Solve the system of equations
$$\begin{cases} 3x^y = 4^x + 8, \\ y = \dfrac{x+1}{\log_2 x}. \end{cases}$$ | \{(2; 3)\} | \boxed{\{(2; 3)\}} | verified | the printed tuple satisfies all 2 equations | 130 | |||
falin_algebra.1034 | 1034 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Geology, 1987, No. 4) Solve the system
$$\begin{cases} 2x + 3y = 16, \\ \log_2 x + \log_2 y = 3. \end{cases}$$ | \{(2; 4),\ (6; 4/3)\} | \boxed{\{(2; 4),\ (6; 4/3)\}} | verified | the printed tuple satisfies all 2 equations | 130 | |||
falin_algebra.1036 | 1036 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Psychology, 1991, No. 3) Solve the system
$$\begin{cases} \log_3 x - 2^y + y = 3, \\ y \cdot 2^y + 2^y \cdot \log_3 x = 4. \end{cases}$$ | \{(81; 0)\} | \boxed{\{(81; 0)\}} | verified | the printed tuple satisfies all 2 equations | 130 | |||
falin_algebra.1038 | 1038 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Economics, 2010, No. 2) Solve the system of equations
$$\begin{cases} (x-y)\sqrt{y+1}=0, \\ \sqrt{x^2+6x+y+10}=-y^2-8y+x+2. \end{cases}$$ | \left\{(-6;-1),\;\left(\dfrac{-7+3\sqrt{5}}{2};\dfrac{-7+3\sqrt{5}}{2}\right)\right\}. | \boxed{\left\{(-6;-1),\;\left(\dfrac{-7+3\sqrt{5}}{2};\dfrac{-7+3\sqrt{5}}{2}\right)\right\}.} | verified | the printed tuple satisfies all 2 equations | 131 | |||
falin_algebra.1039 | 1039 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (ДВИ, 2012, August, No. 2) Solve the system of equations:
$$\begin{cases} x^4 \cdot \sqrt{x-y}=0, \\ y^2+5y=24+xy. \end{cases}$$ | \left\{(0;-8),\;\left(\dfrac{24}{5};\dfrac{24}{5}\right)\right\}. | \boxed{\left\{(0;-8),\;\left(\dfrac{24}{5};\dfrac{24}{5}\right)\right\}.} | verified | the printed tuple satisfies all 2 equations | 131 | |||
falin_algebra.1040 | 1040 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Sevastopol, 2008, No. 4) Solve the system of equations
$$\begin{cases} 2x=2-y^2, \\ 2y=2-x^2. \end{cases}$$ | (-1-\sqrt{3};-1-\sqrt{3}),\;(-1+\sqrt{3};-1+\sqrt{3}). | \boxed{(-1-\sqrt{3};-1-\sqrt{3}),\;(-1+\sqrt{3};-1+\sqrt{3}).} | verified | the printed tuple satisfies all 2 equations | 131 | |||
falin_algebra.1041 | 1041 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Geology, 1995, July, No. 5) Solve the system
$$\begin{cases} y^2+xy=42x, \\ 6x^2+6xy=7y. \end{cases}$$ | \left\{(0;0),\;(1;6),\;\left(\dfrac{7}{5};-\dfrac{42}{5}\right)\right\}. | \boxed{\left\{(0;0),\;(1;6),\;\left(\dfrac{7}{5};-\dfrac{42}{5}\right)\right\}.} | verified | the printed tuple satisfies all 2 equations | 131 | |||
falin_algebra.1042 | 1042 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Soc., 2007, July, No. 4) Solve the system
$$\begin{cases} x^2-2y-3=0, \\ y^2+2x-3=0. \end{cases}$$ | \{(1;-1),\;(-3;3)\}. | \boxed{\{(1;-1),\;(-3;3)\}.} | verified | the printed tuple satisfies all 2 equations | 131 | |||
falin_algebra.1046 | 1046 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (BMC, oral, 2005) Solve the system of equations
$$\begin{cases} y + 2x + z = 2\cdot(x+y)(z+x), \\ z + 2y + x = 4\cdot(y+z)(x+y), \\ x + 2z + y = (z+x)(y+z). \end{cases}$$ | (0,0,0), \left(-\dfrac{17}{15};\dfrac{23}{15};-\dfrac{13}{15}\right). | \boxed{(0,0,0), \left(-\dfrac{17}{15};\dfrac{23}{15};-\dfrac{13}{15}\right).} | unverifiable | tuple has 1 values for 3 unknowns | 132 | |||
falin_algebra.1047 | 1047 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Tashkent, 2006, №10) Solve the system of equations
$$\begin{cases} x^3 + y^3 = xyz + z^3 - 11, \\ x^3 + z^3 = xyz + y^3 + 21, \\ y^3 + z^3 = xyz + x^3 + 3. \end{cases}$$ | \left\{(1;-2;2),\;\left(\dfrac{5}{\sqrt[3]{13}};\dfrac{2}{\sqrt[3]{13}};\dfrac{6}{\sqrt[3]{13}}\right)\right\right\}. | \boxed{\left\{(1;-2;2),\;\left(\dfrac{5}{\sqrt[3]{13}};\dfrac{2}{\sqrt[3]{13}};\dfrac{6}{\sqrt[3]{13}}\right)\right\right\}.} | verified | the printed tuple satisfies all 3 equations | 132 | |||
falin_algebra.1048 | 1048 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (BMC, oral, 2002) Solve the system of equations
$$\begin{cases} x^2 + 5y^2 + 6z^2 + 8(yz + xz + xy) = 36, \\ 6x^2 + y^2 + 5z^2 + 8(yz + xz + xy) = 36, \\ 5x^2 + 6y^2 + z^2 + 8(yz + xz + xy) = 36. \end{cases}$$ | \{(1;1;1),(-1;-1;-1),(3;-3;-3),(-3;3;3),(3;-3;3),(-3;3;-3),(3;3;-3),(-3;-3;3)\} | \boxed{\{(1;1;1),(-1;-1;-1),(3;-3;-3),(-3;3;3),(3;-3;3),(-3;3;-3),(3;3;-3),(-3;-3;3)\}} | verified | the printed tuple satisfies all 3 equations | 132 | |||
falin_algebra.1049 | 1049 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (BMC, 2007, oral) Solve the system of equations
$$\begin{cases} \log_2 x + \log_4 y + \log_4 z = 2, \\ \log_3 y + \log_9 z + \log_9 x = 2, \\ \log_4 z + \log_{16} x + \log_{16} y = 2. \end{cases}$$ | \left(\dfrac{2}{3};\dfrac{27}{8};\dfrac{32}{3}\right). | \boxed{\left(\dfrac{2}{3};\dfrac{27}{8};\dfrac{32}{3}\right).} | verified | the printed tuple satisfies all 3 equations | 132 | |||
falin_algebra.105 | 105 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (DCE, 2014, No. 1) Write in explicit form the natural number given by the expression $\sqrt{7 - 4\sqrt{3}}\,(8 + 4\sqrt{3})$. | 4. | \boxed{4.} | unverifiable | unclassified | 21 | |||
falin_algebra.1051 | 1051 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | **1051.** (HSE, 2007, No. 2) Solve the system of equations
$$\begin{cases} \dfrac{x^2}{y} + x = 6y, \\ x^2 - 2y = xy. \end{cases}$$ | ** $\left\{(2;\,1),\ \left(-\tfrac{1}{2};\,\tfrac{1}{6}\right)\right\}.$
The final answer is $\boxed{\{(2;1),\ (-\tfrac{1}{2};\tfrac{1}{6})\}}$. | \{(2;1),\ (-\tfrac{1}{2};\tfrac{1}{6})\} | \boxed{\{(2;1),\ (-\tfrac{1}{2};\tfrac{1}{6})\}} | verified | the printed tuple satisfies all 2 equations | 133 | ||
falin_algebra.1052 | 1052 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | **1052*.** (Geography, 1995, May, No. 1) Solve the system
$$\begin{cases} y^2 + 2xy = 15, \\ 2x^2 + xy = 5. \end{cases}$$ | $\textbf{Solution to problem 1052}$. Notice that the left-hand sides of the equations contain homogeneous second-degree polynomials. To obtain a homogeneous equation one must arrange for the right-hand side to be $0$. To do this, multiply the second equation by $3$ and subtract from the first equation:
$$y^2 - xy - 6x... | \{(1;3),\ (-1;-3)\} | \boxed{\{(1;3),\ (-1;-3)\}} | verified | the printed tuple satisfies all 2 equations | 133 | ||
falin_algebra.1054 | 1054 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | **1054.** (Geography, 2002, July, No. 6) Solve the system
$$\begin{cases} x^3 = 4x + y, \\ y^3 = 4y + x. \end{cases}$$ | ** $(0;\,0),\ (\pm\sqrt{5};\,\pm\sqrt{5}),\ (\pm\sqrt{3};\,\mp\sqrt{3}),\ \left(\tfrac{\sqrt{6}+\sqrt{2}}{2};\,\tfrac{-\sqrt{6}+\sqrt{2}}{2}\right),\ \left(\tfrac{-\sqrt{6}-\sqrt{2}}{2};\,\tfrac{\sqrt{6}-\sqrt{2}}{2}\right),\ \left(\tfrac{\sqrt{6}-\sqrt{2}}{2};\,\tfrac{-\sqrt{6}-\sqrt{2}}{2}\right),\ \left(\tfrac{-\sqr... | (0;0),\ (\pm\sqrt{5};\pm\sqrt{5}),\ (\pm\sqrt{3};\mp\sqrt{3}),\ (\tfrac{\sqrt{6}+\sqrt{2}}{2};\tfrac{-\sqrt{6}+\sqrt{2}}{2}),\ (\tfrac{-\sqrt{6}-\sqrt{2}}{2};\tfrac{\sqrt{6}-\sqrt{2}}{2}),\ (\tfrac{\sqrt{6}-\sqrt{2}}{2};\tfrac{-\sqrt{6}-\sqrt{2}}{2}),\ (\tfrac{-\sqrt{6}+\sqrt{2}}{2};\tfrac{\sqrt{6}+\sqrt{2}}{2}) | \boxed{(0;0),\ (\pm\sqrt{5};\pm\sqrt{5}),\ (\pm\sqrt{3};\mp\sqrt{3}),\ (\tfrac{\sqrt{6}+\sqrt{2}}{2};\tfrac{-\sqrt{6}+\sqrt{2}}{2}),\ (\tfrac{-\sqrt{6}-\sqrt{2}}{2};\tfrac{\sqrt{6}-\sqrt{2}}{2}),\ (\tfrac{\sqrt{6}-\sqrt{2}}{2};\tfrac{-\sqrt{6}-\sqrt{2}}{2}),\ (\tfrac{-\sqrt{6}+\sqrt{2}}{2};\tfrac{\sqrt{6}+\sqrt{2}}{2})... | verified | the printed tuple satisfies all 2 equations | 133 | ||
falin_algebra.1058 | 1058 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (ISAA, 2008, No. 3) Find all solutions of the system of equations
$$\begin{cases} x^2 + xy - 2y^2 = 12 + x - 10y, \\ x^2 + 3xy + 2y^2 = 16 + 6x + 4y. \end{cases}$$ | (-8; 6), \left(-\tfrac{5}{2}; \tfrac{1}{2}\right), \left(\tfrac{2}{3}; \tfrac{11}{3}\right) | \boxed{(-8; 6), \left(-\tfrac{5}{2}; \tfrac{1}{2}\right), \left(\tfrac{2}{3}; \tfrac{11}{3}\right)} | verified | the printed tuple satisfies all 2 equations | 134 | |||
falin_algebra.1059 | 1059 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (VShB, 2008, No. 4) Find all pairs of numbers $x$ and $y$ that are solutions of the system of equations
$$\begin{cases} x^2 + y^2 = 2xy + 1, \\ y(2 - x) = 3xy - 2x - 2. \end{cases}$$ | \{(2; 1), (0; -1), (-1; 0), (1; 2)\} | \boxed{\{(2; 1), (0; -1), (-1; 0), (1; 2)\}} | verified | the printed tuple satisfies all 2 equations | 134 | |||
falin_algebra.1060 | 1060 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Physics, 2007, March, No. 5) Solve the system of equations
$$\begin{cases} z^2 + 2y\sqrt{z} - 1 = 0, \\ y^2 + z^4 = (1 + z^3)y\sqrt{z}. \end{cases}$$ | \left\{ \left(\sqrt{\sqrt{2} - 1}, \sqrt{2} - 1\right), \left(\tfrac{1}{4}\sqrt[4]{2}, \tfrac{1}{2}\sqrt{2}\right) \right\} | \boxed{\left\{ \left(\sqrt{\sqrt{2} - 1}, \sqrt{2} - 1\right), \left(\tfrac{1}{4}\sqrt[4]{2}, \tfrac{1}{2}\sqrt{2}\right) \right\}} | unverifiable | tuple has 1 values for 2 unknowns | 134 | |||
falin_algebra.1061 | 1061 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Physics, 1997, July, No. 5) Solve the system of equations
$$\begin{cases} y - |x - 3| = 1, \\ |x - y| = 3. \end{cases}$$ | \left\{ \left(\tfrac{1}{2}; \tfrac{7}{2}\right) \right\} | \boxed{\left\{ \left(\tfrac{1}{2}; \tfrac{7}{2}\right) \right\}} | verified | the printed tuple satisfies all 2 equations | 134 | |||
falin_algebra.1062 | 1062 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Mech-Math, 1970, No. 2) Solve the system
$$\begin{cases} |xy - 2| = 6 - x^2, \\ 2 + 3y^2 = 2xy. \end{cases}$$ | \left\{ \left(\sqrt{6}; \tfrac{\sqrt{6}}{3}\right), \left(-\sqrt{6}; -\tfrac{\sqrt{6}}{3}\right) \right\} | \boxed{\left\{ \left(\sqrt{6}; \tfrac{\sqrt{6}}{3}\right), \left(-\sqrt{6}; -\tfrac{\sqrt{6}}{3}\right) \right\}} | verified | the printed tuple satisfies all 2 equations | 134 | |||
falin_algebra.1063 | 1063 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Physics, 1998, July, No. 5) Solve the system of equations
$$\begin{cases} x + |x + y| = 0, \\ y - 2 + \sqrt{x - y + 5} = 0. \end{cases}$$ | \{(-1; 0)\} | \boxed{\{(-1; 0)\}} | verified | the printed tuple satisfies all 2 equations | 134 | |||
falin_algebra.1065 | 1065 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (FNM, 2002, April, No. 3) Solve the equation
$$\sqrt{x^2-3xy+y^2+1} + |2x^2+5xy-3y^2| = 0.$$ | (1;2),\,(-1;-2) | \boxed{(1;2),\,(-1;-2)} | unverifiable | no solutions parsed | 135 | |||
falin_algebra.1066 | 1066 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Mechanics-Mathematics, 1984, No. 4) Solve the system
$$\begin{cases} 2^{3x+1}+2^{y-2} = 3\cdot 2^{y+3x}, \\ \sqrt{3x^2+1+xy} = \sqrt{x+1}. \end{cases}$$ | (0; 3-\log_2 11),\ \left(\dfrac{\log_2(3+2\sqrt{2})-1}{3};\, 2-\log_2(3+2\sqrt{2})\right) | \boxed{(0; 3-\log_2 11),\ \left(\dfrac{\log_2(3+2\sqrt{2})-1}{3};\, 2-\log_2(3+2\sqrt{2})\right)} | verified | the printed tuple satisfies all 2 equations | 135 | |||
falin_algebra.1067 | 1067 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Geology, 2000, No. 5) Solve the system
$$\begin{cases} x+y+3\sqrt{x+y} = 18, \\ x^2+y^2 = 125. \end{cases}$$ | (11;-2),\,(-2;11) | \boxed{(11;-2),\,(-2;11)} | verified | the printed tuple satisfies all 2 equations | 135 | |||
falin_algebra.1068 | 1068 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Physics, 2006, No. 5) Find all solutions of the system of equations
$$\begin{cases} 3x+y+\sqrt{9x^2-y^2} = 6, \\ y\sqrt{(3x+y)(3x-y)} = 2 \end{cases}$$
where $x$ and $y$ are taken as coordinates of points, and indicate which of these points is closer to the origin. | \left(\dfrac{8}{9};\dfrac{5-\sqrt{7}}{3}\right),\ \left(\dfrac{8}{9};\dfrac{5+\sqrt{7}}{3}\right). The first point is closer to the origin. | \boxed{\left(\dfrac{8}{9};\dfrac{5-\sqrt{7}}{3}\right),\ \left(\dfrac{8}{9};\dfrac{5+\sqrt{7}}{3}\right). The first point is closer to the origin.} | verified | the printed tuple satisfies all 2 equations | 135 | |||
falin_algebra.1069 | 1069 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | Solve the system $$\begin{cases} x+4\sqrt{x-y}=y+12, \\ |2(x+1)+y|+2|2x+(y-1)|=3. \end{cases}$$ | \left(\frac{5}{3};-\frac{7}{3}\right) | \boxed{\left(\frac{5}{3};-\frac{7}{3}\right)} | refuted | no assignment of the printed tuple (\frac{5}{3}; -\frac{7}{3}\right) satisfies the system | 135 | |||
falin_algebra.107 | 107 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (VMK, oral, 1995) Prove the identity
$$\sqrt[3]{20 + 14\sqrt{2}} + \sqrt[3]{20 - 14\sqrt{2}} = 4.$$
**Hint:** $20 \pm 14\sqrt{2} = (2 \pm \sqrt{2})^3$. | no-gold | 21 | ||||||
falin_algebra.1070 | 1070 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Physics, 2002, March, No. 5) Solve the system of equations
$$\begin{cases} 3\sqrt{3x^2-y^4} = 2x-7y, \\ 6\sqrt{3x^2-y^4} = x-8y. \end{cases}$$ | (0;0),\,(-2\sqrt{11}; \sqrt{11}) | \boxed{(0;0),\,(-2\sqrt{11}; \sqrt{11})} | verified | the printed tuple satisfies all 2 equations | 135 | |||
falin_algebra.1071 | 1071 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | Solve the system:
$$\left\{
\begin{array}{l}
25\cdot\left(\dfrac{1}{\sqrt{5}}\right)^{2y-2}=\dfrac{1}{5^{2x}},\\
\sqrt{x+2}+\sqrt{2x+y+14}=\sqrt{x+y+12}.
\end{array}
\right.$$ | ((-2;-1)) | \boxed{((-2;-1))} | disputed | not a single relation | 136 | |||
falin_algebra.1072 | 1072 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Physics, 1999, May, No. 5) Solve the system of equations $$\begin{cases} 2^{\left(\frac{y}{x}+\frac{8x}{y}\right)} = 64, \\ \sqrt{y} - \sqrt{3x} = \frac{1}{2+\sqrt{3}}. \end{cases}$$ | (1; 4) | \boxed{(1; 4)} | verified | the printed tuple satisfies all 2 equations | 136 | |||
falin_algebra.1074 | 1074 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Physics, 1998, March, No. 5) Solve the system of equations $$\begin{cases} 5^{\log_4(2x-y)} = 1, \\ 9^{3x-2y} - 6 \cdot 3^{3x-2y} = 27. \end{cases}$$ | (0; -1) | \boxed{(0; -1)} | verified | the printed tuple satisfies all 2 equations | 136 | |||
falin_algebra.1075 | 1075 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (HSE, 2003, July, No. 4) Solve the system of equations $$\begin{cases} 5\log_{32}(x+y) + \log_{\frac{1}{2}}(3y-8) = 0, \\ x^2 + 2x + y^2 + y = 12. \end{cases}$$ | (-2; 3) | \boxed{(-2; 3)} | verified | the printed tuple satisfies all 2 equations | 136 | |||
falin_algebra.1076 | 1076 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Physics, 2000, March, No. 5) Solve the system of equations $$\begin{cases} \log_{\sqrt{6}} x + 2\log_6 y = 2, \\ \log_{27}(3y-3x-1)^3 + \log_3(3y-3x+1) = \log_3 8. \end{cases}$$ | (2; 3) | \boxed{(2; 3)} | verified | the printed tuple satisfies all 2 equations | 136 | |||
falin_algebra.1077 | 1077 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Physics, 1996, July, No. 5) Solve the system of equations $$\begin{cases} \log_2(3y^2 - x^2) = 3, \\ 8\log_{16}(-x) + \log_2 y^2 = 4. \end{cases}$$ | (-2; -2), (-2; 2) | \boxed{(-2; -2), (-2; 2)} | verified | the printed tuple satisfies all 2 equations | 136 | |||
falin_algebra.1078 | 1078 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Geology, 1980, No. 3) Solve the system
$$\begin{cases} \log_2(x^2 + y^2) = 5, \\ 2\log_4 x + \log_2 y = 4. \end{cases}$$ | (4; 4) | \boxed{(4; 4)} | verified | the printed tuple satisfies all 2 equations | 137 | |||
falin_algebra.1079 | 1079 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Faculty of Mechanics and Mathematics, "Applicant-2008", 2008, April, No. 1) Find all solutions of the system of equations
$$\begin{cases} \log_{\sqrt{3}}(4x + y - 7) = \log_3(2x + y - 5), \\ (0.125)^{x-1} = 2^{y-6}. \end{cases}$$ | (0; 9) | \boxed{(0; 9)} | verified | the printed tuple satisfies all 2 equations | 137 | |||
falin_algebra.108 | 108 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (VMK, oral, 2000) Is the number
$$\sqrt[3]{20 + 14\sqrt{2}} + \sqrt[3]{20 - 14\sqrt{2}}$$
an integer? | no-gold | 21 | ||||||
falin_algebra.1081 | 1081 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Faculty of Philology, 1967, No. 3) Solve the system of equations
$$\begin{cases} \log_x 3 = \log_y 9, \\ x^2 - 7y^2 + 26 = 0. \end{cases}$$ | \left(\sqrt{2}; 2\right) | \boxed{\left(\sqrt{2}; 2\right)} | verified | the printed tuple satisfies all 2 equations | 137 | |||
falin_algebra.1084 | 1084 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Economics (Cybernetics), 1977, No. 3) Solve the system
$$\begin{cases} y^{1 - \frac{2}{5}\log_x y} = x^{\frac{2}{5}}, \\ 1 + \log_x\left(1 - \frac{3y}{x}\right) = \log_x 4. \end{cases}$$ | (16; 4) | \boxed{(16; 4)} | verified | the printed tuple satisfies all 2 equations | 137 | |||
falin_algebra.1085 | 1085 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Sevastopol, 2003, May, No. 6) Solve the system of equations
$$\begin{cases} (\log_y 2) \cdot \log_2(1 - x) = -1, \\ x^{\log_y x} = \sqrt{xy}. \end{cases}$$ | no-gold | 137 | ||||||
falin_algebra.1086 | 1086 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (почвовед., 1998, №4) Solve the system
$$\begin{cases} y^x = 3y, \\ 2\log_3 y + \log_y 3 = 3x. \end{cases}$$ | \left\{\left(\dfrac{3}{2}; 9\right), \left(-1; \dfrac{1}{\sqrt{3}}\right)\right\} | \boxed{\left\{\left(\dfrac{3}{2}; 9\right), \left(-1; \dfrac{1}{\sqrt{3}}\right)\right\}} | verified | the printed tuple satisfies all 2 equations | 138 | |||
falin_algebra.1087 | 1087 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (ВМК, устный, 2003) Find all positive solutions of the system of equations
$$\begin{cases} x^{x-y} = y^2, \\ y^{x-y} = x^6 y^4. \end{cases}$$ | \{(1; 1), (\sqrt{2} - 1; \sqrt{2} + 1)\} | \boxed{\{(1; 1), (\sqrt{2} - 1; \sqrt{2} + 1)\}} | verified | the printed tuple satisfies all 2 equations | 138 | |||
falin_algebra.1088 | 1088 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (мех-мат, 2008, №3) Solve the system
$$\begin{cases} \log_2(2x^3 + 4x^2 y - 3x^2) = \log_{11}(4xy^2 + 24y^3 - 12y^2), \\ \log_{11}(x^3 + 6x^2 y - 3x^2) = \log_2(8xy^2 + 16y^3 - 12y^2). \end{cases}$$ | (1; \dfrac{1}{2}) | \boxed{(1; \dfrac{1}{2})} | verified | the printed tuple satisfies all 2 equations | 138 | |||
falin_algebra.1089 | 1089 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Олимпиада "Покори Воробьёвы горы-2012", очный тур) Solve the equation
$$|1 - x - y - xy| + |2x^2 y^2 - 2x^2 y - 2xy^2 + 2xy - 9| + \dfrac{|xy|}{xy} = -1.$$ | two solutions: x = -\dfrac{1}{2}, y = 3 and x = 3, y = -\dfrac{1}{2} | \boxed{two solutions: x = -\dfrac{1}{2}, y = 3 and x = 3, y = -\dfrac{1}{2}} | unverifiable | no solutions parsed | 138 | |||
falin_algebra.109 | 109 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (VMK, oral, 1995) Prove the identity $$\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}=1.$$ Hint: $\sqrt{5}\pm 2=\left(\dfrac{\sqrt{5}}{2}\pm \dfrac{1}{2}\right)^3$. | no-gold | 22 | ||||||
falin_algebra.1091 | 1091 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (ВМК, устный, 2008, июль) Solve the system
$$\begin{cases} \log_8 \dfrac{y}{x} = 3^x - 3^y, \\ 2^{1-x} + 8^{-2x} = 65^x \cdot 4^{-3y}, \\ x, y > 0. \end{cases}$$ | \{(1; 1)\} | \boxed{\{(1; 1)\}} | unverifiable | no valid sample points | 138 | |||
falin_algebra.1092 | 1092 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (ВМК, устный, 2003) Solve the system of equations
$$\begin{cases} x^4 + y^{-4} = y^4 + x^{-4}, \\ 3x^2 + 6xy + 7y^2 = 16. \end{cases}$$ | \{(1; 1), (-1; -1), (2; -2), (-2; 2)\} | \boxed{\{(1; 1), (-1; -1), (2; -2), (-2; 2)\}} | verified | the printed tuple satisfies all 2 equations | 138 | |||
falin_algebra.1093 | 1093 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (VMK, oral, 2005) Solve the system of equations
$$\begin{cases} x^2 + \frac{1}{y^6} = y^2 + \frac{1}{x^6}, \\ (x^2 - y^2) \cdot 2^{2x - 3y + 6} + 3x = 12. \end{cases}$$ | $\{(4; 4), (4; -4)\}$.
The final answer is $\boxed{\{(4; 4), (4; -4)\}}$. | \{(4; 4), (4; -4)\} | \boxed{\{(4; 4), (4; -4)\}} | verified | the printed tuple satisfies all 2 equations | 139 | ||
falin_algebra.1094 | 1094 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (VMK, 2007, oral) Solve the system of equations
$$\begin{cases} \sqrt{x} - \sqrt{y} = \log_3 \frac{y}{x}, \\ 2^{x+2} + 8^x = 5 \cdot 4^y. \end{cases}$$ | $(2; 2)$.
The final answer is $\boxed{(2; 2)}$. | (2; 2) | \boxed{(2; 2)} | verified | the printed tuple satisfies all 2 equations | 139 | ||
falin_algebra.1095 | 1095 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Higher School of Public Audit, "Applicant-2008", No. 5) Solve the system of equations
$$\begin{cases} \log_2 x + \log_{\frac{1}{2}} y = 1, \\ x^{\log_4 x} + y^{\log_{\frac{1}{4}} x} = 2. \end{cases}$$ | $(1; \frac{1}{2}), (2; 1)$.
The final answer is $\boxed{(1; \tfrac{1}{2}),\ (2; 1)}$. | (1; \tfrac{1}{2}),\ (2; 1) | \boxed{(1; \tfrac{1}{2}),\ (2; 1)} | verified | the printed tuple satisfies all 2 equations | 139 | ||
falin_algebra.1096 | 1096 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Soil Science, 2007, July, No. 6) Solve the system
$$\begin{cases} x + y + xy = 7, \\ x^2 + y^2 + xy = 13. \end{cases}$$ | $\{(1; 3), (3; 1)\}$.
The final answer is $\boxed{\{(1; 3),\ (3; 1)\}}$. | \{(1; 3),\ (3; 1)\} | \boxed{\{(1; 3),\ (3; 1)\}} | verified | the printed tuple satisfies all 2 equations | 139 | ||
falin_algebra.1097 | 1097 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Geology, 1997, July, No. 6) Find all solutions of the equation
$$|x + y - 3xy + 13| + |x^2 y + xy^2 - 30| = 0.$$ | $(2; 3),\ (3; 2),\ \left(\frac{-27 + \sqrt{744}}{3},\ \frac{-27 - \sqrt{744}}{3}\right),\ \left(\frac{-27 - \sqrt{744}}{3},\ \frac{-27 + \sqrt{744}}{3}\right)$.
The final answer is $\boxed{(2;\,3),\ (3;\,2),\ \left(\tfrac{-27+\sqrt{744}}{3},\ \tfrac{-27-\sqrt{744}}{3}\right),\ \left(\tfrac{-27-\sqrt{744}}{3},\ \tfrac{... | (2;\,3),\ (3;\,2),\ \left(\tfrac{-27+\sqrt{744}}{3},\ \tfrac{-27-\sqrt{744}}{3}\right),\ \left(\tfrac{-27-\sqrt{744}}{3},\ \tfrac{-27+\sqrt{744}}{3}\right) | \boxed{(2;\,3),\ (3;\,2),\ \left(\tfrac{-27+\sqrt{744}}{3},\ \tfrac{-27-\sqrt{744}}{3}\right),\ \left(\tfrac{-27-\sqrt{744}}{3},\ \tfrac{-27+\sqrt{744}}{3}\right)} | unverifiable | no solutions parsed | 139 | ||
falin_algebra.11.36 | 11.36 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | The figure shows the graph of a function (piecewise linear, with vertices approximately at $(-5, 26)$, $(-4, 16)$, $(-3, 11)$, $(-1, 11)$, $(0, 11)$, $(3, 16)$, $(3.5, 26)$).
держит "almost reciprocal" numbers $2x^2$ and $\frac{1}{x^2+1}$. This naturally suggests using the arithmetic–geometric mean inequality. To do t... | no-gold | 542 | ||||||
falin_algebra.11.60 | 11.60 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | Justify the validity of the following theorem:
$$\sqrt{a} > b \Updownarrow \begin{cases} b \geq 0 \\ a > b^2 \end{cases} \text{ or } \begin{cases} b < 0 \\ a \geq 0 \end{cases}$$ | no-gold | 383 | ||||||
falin_algebra.11.85 | 11.85 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | Applying formula (11.83) to the polynomial $p_{kn+m}(x)$ first, and then formula (11.84), we obtain the more general formula:
$$p_{kn+m}(x) = p_{kn}(x) + x^{kn} \cdot p_m(x) = p_n(x) \cdot p_k\left(x^n\right) + x^{kn} \cdot p_m(x). \qquad (11.85)$$
If we define $p_0(x)$ as $0$ (a sum containing $0$ summands equals $0$ ... | ** the number $2^n - 1$ is divisible by $7$ if and only if $n$ is divisible by $3$. $\square$
The final answer is $\boxed{the number 2^n - 1 is divisible by 7 if and only if n is divisible by 3}$. | the number 2^n - 1 is divisible by 7 if and only if n is divisible by 3 | \boxed{the number 2^n - 1 is divisible by 7 if and only if n is divisible by 3} | unverifiable | error: ValueError | 422 | ||
falin_algebra.110 | 110 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Tashkent, 2007, No. 1) Compare the numbers $\left(\sqrt{5}+\sqrt{10}\right)^2$ and $14+\sqrt{200}$. | the first number is greater than the second. | \boxed{the first number is greater than the second.} | unverifiable | unclassified | 22 | |||
falin_algebra.1101 | 1101 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Geology, 1998, July, No. 7) Solve the system of equations
$$\begin{cases} x - 5 = y(1 - x), \\ xy^2 - x^2 y = 6. \end{cases}$$ | (2;3),\,(-3;-2),\,(3+2\sqrt{2}; -3+2\sqrt{2}),\,(3-2\sqrt{2}; -3-2\sqrt{2}) | \boxed{(2;3),\,(-3;-2),\,(3+2\sqrt{2}; -3+2\sqrt{2}),\,(3-2\sqrt{2}; -3-2\sqrt{2})} | verified | the printed tuple satisfies all 2 equations | 140 | |||
falin_algebra.1102 | 1102 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (VMK, oral, 2004) Solve the system of equations
$$\begin{cases} x^2 - y^2 + x - y = 5, \\ x^3 - x^2 y - xy^2 + y^3 = 6. \end{cases}$$ | \left\{\left(\frac{11}{6}; -\frac{7}{6}\right),\, \left(\frac{7}{4}; -\frac{1}{4}\right)\right\} | \boxed{\left\{\left(\frac{11}{6}; -\frac{7}{6}\right),\, \left(\frac{7}{4}; -\frac{1}{4}\right)\right\}} | verified | the printed tuple satisfies all 2 equations | 140 | |||
falin_algebra.1103 | 1103 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Geology, 2001, No. 5) Find all solutions of the system of equations
$$\begin{cases} \dfrac{xy}{2} + \dfrac{5}{2x+y-xy} = 5, \\ 2x + y + \dfrac{10}{xy} = 4 + xy. \end{cases}$$ | \{(1;5),\,(\tfrac{5}{2}; 2)\} | \boxed{\{(1;5),\,(\tfrac{5}{2}; 2)\}} | verified | the printed tuple satisfies all 2 equations | 140 | |||
falin_algebra.1104 | 1104 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Physics, 2003, March, No. 5) Solve the system
$$\begin{cases} \dfrac{17}{2x^2+3y} + \dfrac{12}{3x^2-2y} = 3, \\ \dfrac{6}{3x^2-2y} + \dfrac{34}{2x^2+3y} = 3 \end{cases}$$
and plot its solutions on the coordinate plane $Oxy$. | (-2;3),\,(2;3) | \boxed{(-2;3),\,(2;3)} | verified | the printed tuple satisfies all 2 equations | 140 | |||
falin_algebra.1105 | 1105 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Mech.-Math., 1979, No. 3C) Find all solutions of the system of equations
$$\begin{cases} \dfrac{2}{2x-y} + \dfrac{3}{x-2y} = \dfrac{1}{2} \\ \dfrac{2}{2x-y} - \dfrac{1}{x-2y} = \dfrac{1}{18}. \end{cases}$$ | (5;-2) | \boxed{(5;-2)} | verified | the printed tuple satisfies all 2 equations | 140 | |||
falin_algebra.1107 | 1107 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Mech.-Math., 1980, No. 4C) Solve the system of equations
$$\begin{cases} x^3 - \sqrt{y} = 1, \\ 5x^6 - 8x^3 \sqrt{y} + 2y = 2. \end{cases}$$ | no-gold | 140 | ||||||
falin_algebra.1108 | 1108 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Physics, 2003, July, No. 3) Solve the system of equations
$$\begin{cases} \sqrt{x-y} = 9 - |x + 2y|, \\ x(x + 4y - 2) + y(4y + 2) = 41. \end{cases}$$ | (5; 1), \left(\frac{1}{3}; -\frac{11}{3}\right) | \boxed{(5; 1), \left(\frac{1}{3}; -\frac{11}{3}\right)} | verified | the printed tuple satisfies all 2 equations | 141 | |||
falin_algebra.1109 | 1109 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Chemistry, 1991, No. 3) Solve the system
$$\begin{cases} \sqrt{2x-1} + \sqrt{y+3} = 3, \\ 2xy - y + 6x - 3 = 4. \end{cases}$$ | \left\{(1; 1), \left(\frac{5}{2}; -2\right)\right\} | \boxed{\left\{(1; 1), \left(\frac{5}{2}; -2\right)\right\}} | verified | the printed tuple satisfies all 2 equations | 141 | |||
falin_algebra.111 | 111 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (VMK, oral, 1999) Compare $\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}$ and $1$. | the numbers are equal. | \boxed{the numbers are equal.} | unverifiable | unclassified | 22 | |||
falin_algebra.1111 | 1111 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (LSS, 2014, No. 6) Find all positive $x$, $y$ satisfying the system of equations
$$\begin{cases} x^{3/2} + y = 16, \\ x + y^{2/3} = 8. \end{cases}$$ | (4; 8) | \boxed{(4; 8)} | verified | the printed tuple satisfies all 2 equations | 141 | |||
falin_algebra.1112 | 1112 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (MSU CMC, oral, 2004) Solve the system of equations
$$\begin{cases} (x-y)^4 = 13x - 4, \\ \sqrt{x+y} + \sqrt{3x-y} = \sqrt{2}. \end{cases}$$ | \left\{\left(\frac{5}{16}; -\frac{3}{16}\right), \left(\frac{5}{16}; \frac{13}{16}\right)\right\} | \boxed{\left\{\left(\frac{5}{16}; -\frac{3}{16}\right), \left(\frac{5}{16}; \frac{13}{16}\right)\right\}} | verified | the printed tuple satisfies all 2 equations | 141 | |||
falin_algebra.1113 | 1113 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (MSU CMC, oral, 2002) Solve the system of equations
$$\begin{cases} \sqrt{x} + \sqrt{y} = 3, \\ \sqrt{x+5} + \sqrt{y+3} = 5. \end{cases}$$ | \left\{(4; 1), \left(\frac{121}{64}; \frac{169}{64}\right)\right\} | \boxed{\left\{(4; 1), \left(\frac{121}{64}; \frac{169}{64}\right)\right\}} | verified | the printed tuple satisfies all 2 equations | 141 | |||
falin_algebra.1115 | 1115 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Physics, 1976, No. 3) Find all solutions of the system of equations
$$\begin{cases} 3^{\frac{x-y}{2}} + 3^x \cdot 3^{-y} = 12, \\ 3^x + 3^{-y} = 10. \end{cases}$$ | \{(0; -2), (2; 0)\} | \boxed{\{(0; -2), (2; 0)\}} | verified | the printed tuple satisfies all 2 equations | 141 | |||
falin_algebra.1116 | 1116 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Physics, 2004, March, No. 5) Solve the system of equations
$$\begin{cases} |2^x - 2^y| + 2^x + 3 \cdot 2^y = 12\sqrt{2}, \\ |2^x + 2^{-y}| + 2^x - 33 \cdot 2^{-y} = 0. \end{cases}$$ | \left(\frac{5}{2}; \frac{3}{2}\right), \left(\frac{7}{2} - \log_2 3; \frac{1}{2} + \log_2 3\right) | \boxed{\left(\frac{5}{2}; \frac{3}{2}\right), \left(\frac{7}{2} - \log_2 3; \frac{1}{2} + \log_2 3\right)} | verified | the printed tuple satisfies all 2 equations | 142 | |||
falin_algebra.1117 | 1117 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Physics, 1999, July, No. 4) Find all solutions of the system of equations
$$\begin{cases} 3 \cdot 5^{x+1} - \frac{18}{3^{2-y}} = 35, \\ \frac{10}{5^{1-x}} + 2 \cdot 3^{y+1} = 36. \end{cases}$$ | (\log_5 3; \log_3 5) | \boxed{(\log_5 3; \log_3 5)} | verified | the printed tuple satisfies all 2 equations | 142 | |||
falin_algebra.1118 | 1118 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Faculty of Physics, 2008, No. 6) Solve the system of equations
$$\begin{cases} 7 + 9^x = 2^y + 4 \cdot 3^x, \\ 11 - 3 \cdot 2^{y+1} = 3^x - 4^y. \end{cases}$$ | (1; 2), (\log_3 2; \log_3 23) | \boxed{(1; 2), (\log_3 2; \log_3 23)} | verified | the printed tuple satisfies all 2 equations | 142 | |||
falin_algebra.1119 | 1119 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Economics (management), 2003, July, No. 3) Find all solutions of the system of equations
$$\begin{cases} 3 \cdot 4^x + 2^{x+1} \cdot 3^y - 9^y = 0, \\ 2 \cdot 4^x - 5 \cdot 2^x \cdot 3^y + 9^y = -8. \end{cases}$$ | (1/2; \log_3 3\sqrt{2}) | \boxed{(1/2; \log_3 3\sqrt{2})} | verified | the printed tuple satisfies all 2 equations | 142 | |||
falin_algebra.112 | 112 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (VMK, 1992, July, No. 1) Which of the two numbers $\sqrt[3]{\dfrac{1990}{1991}}$ or $\sqrt[3]{\dfrac{1991}{1992}}$ is larger? | the second. | \boxed{the second.} | unverifiable | unclassified | 22 | |||
falin_algebra.1120 | 1120 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Economics, 2003, July, No. 3) Find all solutions of the system of equations
$$\begin{cases} 4^x - 19 \cdot 3^{x \log_3 2 + y} + 4 \cdot 9^y = -10, \\ 4^x + 6 \cdot 2^{x+y \log_2 3} - 9^y = 5. \end{cases}$$ | \left(-\frac{1}{2}; \log_3 \frac{3}{\sqrt{2}}\right), \left(\log_2 \frac{2}{\sqrt{3}}; -\frac{1}{2}\right) | \boxed{\left(-\frac{1}{2}; \log_3 \frac{3}{\sqrt{2}}\right), \left(\log_2 \frac{2}{\sqrt{3}}; -\frac{1}{2}\right)} | verified | the printed tuple satisfies all 2 equations | 142 | |||
falin_algebra.1122 | 1122 | falin_algebra | Entrance exam algebra | 12 | University entrance algebra: polynomials, equations, inequalities, logarithms | (Geography, 2001, May, No. 1) Find all solutions of the system of equations
$$\begin{cases} 9 \cdot 2^x \cdot 5^y - 5 \cdot 3^{y+x} = 3^x \cdot 5^y, \\ 2^{x-2} \cdot 3^{y-x+1} \cdot 5^{1-y} = 1. \end{cases}$$ | (2; 1) | \boxed{(2; 1)} | verified | the printed tuple satisfies all 2 equations | 142 |
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