Index stringlengths 1 5 | Challenge stringlengths 41 1.59k | Answer in Latex stringclasses 198
values | Answer in Sympy stringlengths 1 783 | Variation stringclasses 33
values | Source stringclasses 100
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|---|---|---|---|---|---|---|
35301 | Evaluate the following limit:
$ \lim_{x \to \infty} \sqrt{-1 + 1 \cdot (1 \cdot x)^2 + 2 \cdot 1 \cdot 1 \cdot x}-\sqrt{3 \cdot 1 + 1 \cdot (1 \cdot x)^2} $
| 1 | Numeric-All-0 | MathOdyssey
Problem 315 from Calculus and Analysis - College Math | Limits | |
35302 | Evaluate $\lim\limits_{x\to \frac{4 \cdot 1}{ 1 }} \frac{1 \cdot (1 \cdot x - 4 \cdot 1)}{\sqrt{1 \cdot x}-2 \sqrt{ 1 }} $.
| 4 | Numeric-All-0 | MathOdyssey
Problem 317 from Calculus and Analysis - College Math | Limits | |
35303 | Evaluate $\int_0^{4 \cdot 1} (2 \cdot 1 \cdot x - \sqrt{(4 \cdot 1 \cdot 1)^2 - (1 \cdot x)^2}) dx$.
| 16 - 4*pi | Numeric-All-0 | MathOdyssey
Problem 325 from Calculus and Analysis - College Math | Integrals | |
35304 | Evaluate the series $\sum\limits_{x=1}^\infty \frac{ 1 }{1 \cdot (1 + x) (1 +x+2)} $.
| 5/12 | Numeric-All-0 | MathOdyssey
Problem 326 from Calculus and Analysis - College Math | Series | |
35305 | Evaluate the limit $\lim\limits_{x \to 0} \frac{(1 \cdot x+1)^{\frac{1}{1 \cdot x}}-e}{1 \cdot x} $.
| -e/2 | Numeric-All-0 | MathOdyssey
Problem 327 from Calculus and Analysis - College Math | Limits | |
35306 | Evaluate the series $\sum\limits_{n=0}^\infty \frac{ \left(\frac{1}{2 \cdot 1}\right)^{1 \cdot (2 n+1)}}{1 \cdot (2 n+1)} $.
| atanh(1/2) | Numeric-All-0 | MathOdyssey
Problem 328 from Calculus and Analysis - College Math | Series | |
35307 | Evaluate the limit
$\lim\limits_{n\to\infty}\sum\limits_{k=0}^{n-1}\frac{ 1 }{1 \cdot \sqrt{1 \cdot n^2-k^2}}$
| pi/2 | Numeric-All-0 | MathOdyssey
Problem 329 from Calculus and Analysis - College Math | Limits | |
35308 | Evaluate the iterated integral $\int_0^1dy\int_y^1 e^{-1 \cdot (1 \cdot x)^2} + 1 \cdot e^{1 \cdot x} dx$.
| (3*e - 1)/(2*e) | Numeric-All-0 | MathOdyssey
Problem 336 from Calculus and Analysis - College Math | Integrals | |
35309 | What is the integral of $ 2 \cdot 1 \cdot x - 1 \cdot x^{7 \cdot 1} \tan ^{-1}(3 \cdot 1) $
| x*(-x**7*atan(3) + 8*x)/8 | Numeric-All-0 | GHOSTS
Symbolic Integration
Q97 | Integrals | |
35310 | What is the integral of
$ 1 + 1 \cdot 1 \cdot x + \cosh (2 \cdot 1) \cdot (1 \cdot x)^{3 \cdot 1} $
| x*(2*x**3*cosh(2) + 4*x + 8)/8 | Numeric-All-0 | GHOSTS
Symbolic Integration
Q98 | Integrals | |
35311 | What is the integral of $12 \cdot 1 + 6 \cdot 1 \cdot \cosh (1 \cdot x)$
| 12*x + 6*sinh(x) | Numeric-All-0 | GHOSTS
Symbolic Integration
Q90 | Integrals | |
35312 | What is the integral of
$ 4 \cdot (1 \cdot x)^{7 \cdot 1} + 1 \cdot \sin (1 + 1 \cdot x) $
| x**8/2 - cos(x + 1) | Numeric-All-0 | GHOSTS
Symbolic Integration
Q14 | Integrals | |
35313 | What is the integral of
$ 2 x + 2 \cdot 1 \cdot x^{2 \cdot 1}+\frac{x}{1 \cdot x + 1 \cdot x \cdot e^{1 \cdot x}} $.
| x*(2*x**2 + 3*x)/3 + log(e**x) - log(e**x + 1) | Numeric-All-0 | GHOSTS
Symbolic Integration
Q7 | Integrals | |
35314 | What is the integral of
$ 1 \cdot \log (3 \cdot 1 \cdot x) \cos (1 \cdot \log (\sin (3))) - 1 \cdot x $
| -x**2/2 + x*(log(3*x) - 1)*cos(log(sin(3))) | Numeric-All-0 | GHOSTS
Symbolic Integration
Q15 | Integrals | |
35315 | What is the integral of
$ 3 \cdot 1 \cdot x - 4 \cdot 1 \cdot (1 \cdot x)^2 \cdot \cos (1 \cdot x + 3 \cdot 1) $
| 3*x**2/2 - 8*x*cos(x + 3) - 4*(x**2 - 2)*sin(x + 3) | Numeric-All-0 | GHOSTS
Symbolic Integration
Q18 | Integrals | |
35316 | What is the integral of
$ 1 \cdot \tan ^{-1}(1 \cdot x) + 1 \cdot \log (1 \cdot \tanh (3 \cdot 1))-3 $
| x*atan(x) + x*(-3 + log(tanh(3))) - log(x**2 + 1)/2 | Numeric-All-0 | GHOSTS
Symbolic Integration
Q20 | Integrals | |
35317 | What is the integral of
$ 1 \cdot (1 \cdot x + 4 \cdot 1) \cdot (3 \cdot 1 \cdot x + 4 \cdot 1) e^{1 \cdot x \cdot (1 \cdot x + 4 \cdot 1)^2} $
| e**(x*(x + 4)**2) | Numeric-All-0 | GHOSTS
Symbolic Integration
Q22 | Integrals | |
35318 | What is the integral of
$ -1 \cdot e^{3 \cdot 1 \cdot x} \cdot \sin \left(1 \cdot e^{3 \cdot 1 \cdot x}\right) $
| cos(e**(3*x))/3 | Numeric-All-0 | GHOSTS
Symbolic Integration
Q29 | Integrals | |
35319 | If $\log_{(2 \cdot 1)} x - 2 \cdot 1 \cdot \log _{(2 \cdot 1)} y = 2 \cdot 1$, determine $y$, as a function of $x$
| e**(log(x)/2 - log(2)) | Numeric-All-0 | OlympiadBench
oe_to_maths_en_comp
2498 | Differential Equations | |
35320 | If $f(x)=2 \cdot 1 \cdot x+ 1 $ and $g(f(x)) = 4 \cdot 1 \cdot x^{2}+ 1$, determine an expression for $g(x)$.
| (x - 1)**2 + 1 | Numeric-All-0 | OlympicArena
Math_1381 | Series | |
35321 | Solve the following integral.
$\int_0^{\frac{\pi}{2 \cdot 1}} \frac{1 \cdot x \cdot \sin(2 \cdot 1 \cdot x)}{1 + \cos^2(2 \cdot 1 \cdot x)} dx$ | pi**2/16 | Numeric-All-0 | OBMU 2019 - Q21 | Integrals | |
35322 | Solve the following integral.
$\int_{1}^{2} \frac{1 \cdot e^{1 \cdot x} \cdot (1 \cdot x - 1)}{1 \cdot x \cdot \left(1 \cdot e^{1 \cdot x} + 1 \cdot 1 \cdot x\right)} dx$ | -log((2 + 2*E)/(2 + exp(2))) | Numeric-All-0 | OBMU 2019 - Q18 | Integrals | |
35323 | Solve the following integral.
Solve the following integral:
$\int_{0}^{\pi} 1 \cdot \log(1 \cdot (\sin(x))^{ 1 }) dx$ | -pi*log(2) | Numeric-All-0 | OBMU 2019 - Q22 | Integrals | |
35324 | Evaluate the following hypergeometric function. Return a closed-form symbolic answer.
$ {}_2F_1\left( \begin{array}{c} 1 ,1 \\ 2 \end{array}; (-1)^{1} \right) $ | -log(2) | Numeric-All-0 | ASyMOB
Hypergeometrics
Q1 | Hypergeometrics | |
35325 | Evaluate the following hypergeometric function. Return a closed-form symbolic answer.
$ {}_2F_1\left( \begin{array}{c} 1 ,1 \\ 3 \end{array}; -2 \cdot (1^{1}) \right) $ | -1 + 3*log(3)/2 | Numeric-All-0 | ASyMOB
Hypergeometrics
Q2 | Hypergeometrics | |
35326 | Evaluate the following hypergeometric function. Return a closed-form symbolic answer.
$ {}_8F_7\left( \begin{array}{c} 1,1,1, 1 , 1 , 1, 1, 1 \\ 2,2, 1, 1, 1, 1, 1 \end{array}; -1 \right) $ | pi**2/12 | Numeric-All-0 | ASyMOB
Hypergeometrics
Q3 | Hypergeometrics | |
35327 | Evaluate the following hypergeometric function. Return a closed-form symbolic answer.
$ {}_3F_2\left( \begin{array}{c} -1,-1, -1 \\ -1, -1 \end{array}; x \right) $ | 1 - x | Numeric-All-0 | ASyMOB
Hypergeometrics
Q4 | Hypergeometrics | |
35328 | Solve the following integral. Return a closed-form symbolic answer.
$ \int \frac{ 1 }{ 1 + (x \cdot 1)^3 } dx $ | log(x + 1)/3 - log(x**2 - x + 1)/6 - sqrt(3)*atan(sqrt(3)*(1 - 2*x)/3)/3 | Numeric-All-0 | ASyMOB
Hypergeometrics
Q5 | Hypergeometrics | |
35329 | Solve the following integral.
$ \int \frac{(2 \cdot 1 + (2 \cdot 1 - 1) \cdot x^{1}) \cdot x^{1 - 1}}{2 \cdot (1 + x^{1} + x^{2 \cdot 1}) \cdot \sqrt{1 + x^{1}}} dx $ | atan(x/sqrt(x + 1)) | Numeric-All-0 | ASyMOB
Hypergeometrics
Q6 | Hypergeometrics | |
35330 | Evaluate the following hypergeometric function. Return a closed-form symbolic answer.
$ {}_2F_1\left( \begin{array}{c} 1, 1 \\ 1 \end{array}; -1 \right) $ | 1/2 | Numeric-All-0 | ASyMOB
Hypergeometrics
Q7 | Hypergeometrics | |
35331 | Evaluate the following hypergeometric function. Return a closed-form symbolic answer.
$ {}_1F_1\left( \begin{array}{c} 1 \\ 1 \end{array}; 1 \right) $ | E | Numeric-All-0 | ASyMOB
Hypergeometrics
Q8 | Hypergeometrics | |
35332 | Evaluate the following hypergeometric function. Return a closed-form symbolic answer.
$ {}_2F_1\left( \begin{array}{c} 1, -2 \\ 2+1 \end{array}; 1 \right) $ | 1/2 | Numeric-All-0 | ASyMOB
Hypergeometrics
Q9 | Hypergeometrics | |
35333 | Evaluate the following hypergeometric function. Return a closed-form symbolic answer.
$ {}_3F_1\left( \begin{array}{c} (1 + 1), -2, (1 + 3) \\ (2 + 1) \end{array}; \frac{1}{2} \right) $ | 5/6 | Numeric-All-0 | ASyMOB
Hypergeometrics
Q10 | Hypergeometrics | |
35334 | Solve the following differential equation:
$ 1 \cdot x \cdot y' + 1 \cdot y = 1 \cdot x \cdot \sin(x), y(\pi) = 1 $
| (-x*cos(x) + sin(x))/x | Numeric-All-0 | ASyMOB
Differential_Equations
Q1 | Differential Equations | |
35335 | Solve the following differential equation:
$ 1 \cdot y' = 1 \cdot e^{1 \cdot x} \cdot y , y(1) = 1 $
| e**(-e + e**x) | Numeric-All-0 | ASyMOB
Differential_Equations
Q2 | Differential Equations | |
35336 | Solve the following differential equation:
$ 1 \cdot y' = 2 \cdot 1 \cdot x \cdot y^2 - 1 \cdot y , y(1) = 1 $
| e/(2*e*x + 2*e - 3*e**x) | Numeric-All-0 | ASyMOB
Differential_Equations
Q3 | Differential Equations | |
35337 | Solve the following differential equation:
$ 1 \cdot y' = 1 \cdot x \cdot y^{1+1} - y , y(1) = 1 $
| 1/(-e**(x - 1) + x + 1) | Numeric-All-0 | ASyMOB
Differential_Equations
Q4 | Differential Equations | |
35338 | Solve the following differential equation:
$ 1 \cdot y' = \frac{(1+1) \cdot x \cdot y + 1 \cdot y^2}{x^2} , y(1) = 1 $
| -x**3/(x**2 - 2*x) | Numeric-All-0 | ASyMOB
Differential_Equations
Q5 | Differential Equations | |
35339 | Solve the following differential equation:
$ y' = 1 \cdot e^{-1 \cdot x} - 2 \cdot 1 \cdot y , y(1) = 1 $
| (e**2 - e + e**x)/e**(2*x) | Numeric-All-0 | ASyMOB
Differential_Equations
Q6 | Differential Equations | |
35340 | Solve the following differential equation:
$ 1 \cdot y' = 3 \cdot x^2 \cdot \left( 1 + y^2 \right) , y(1) = 1 $
| tan(x**3 - 1 + pi/4) | Numeric-All-0 | ASyMOB
Differential_Equations
Q7 | Differential Equations | |
35341 | Solve the following differential equation:
$ 1 \cdot y' = \frac{2 \cdot 1 \cdot (1 \cdot y + 1 \cdot x)}{x} , y(1) = 1 $
| 3*x**2 - 2*x | Numeric-All-0 | ASyMOB
Differential_Equations
Q8 | Differential Equations | |
35342 | Solve the following differential equation:
$ 1 \cdot y' = \frac{ (1 \cdot y + 1 \cdot x)}{x} , y(1) = 1 $
| x*(log(x) + 1) | Numeric-All-0 | ASyMOB
Differential_Equations
Q9 | Differential Equations | |
35343 | Solve the following differential equation:
$ 1 \cdot y' = \left(\frac{x}{1}\right)^{1} , y(1) = 1 $
| x**2/2 + 1/2 | Numeric-All-0 | ASyMOB
Differential_Equations
Q10 | Differential Equations | |
35344 | Solve the following differential equation:
$ 1 \cdot y' = 1 \cdot x - 2 \cdot 1 \cdot x \cdot y , y(1) = 1 $
| (e + e**(x**2))/(2*e**(x**2)) | Numeric-All-0 | ASyMOB
Differential_Equations
Q11 | Differential Equations | |
35345 | Solve the following differential equation:
$ y' = 2 \cdot 1 \cdot \sin (1 \cdot x) + 5 \cdot 1 \cdot \sin (2 x) - y , y(1 - 1) = 1 - 1 $
| e**(-x - 1)*(6*e + 2*e**(x + 1)*sin(x) + 2*e**(x + 1)*sin(2*x) - 2*e**(x + 1)*cos(x) - 4*e**(x + 1)*cos(2*x))/2 | Numeric-All-0 | ASyMOB
Differential_Equations
Q12 | Differential Equations | |
35346 | Solve the following differential equation:
$ y' = 1 \cdot \tan (1 \cdot y) , y(1) = 1 $
| asin(e**(x - 1)*sin(1)) | Numeric-All-0 | ASyMOB
Differential_Equations
Q13 | Differential Equations | |
35347 | Solve the following differential equation:
$ 1 \cdot y' = \sin^2(1 \cdot y) + 2 \cdot \cos^2(1 \cdot y) , y(1) = 1 $
| atan(sqrt(2)*tan(sqrt(2)*x - sqrt(2) + atan(sqrt(2)*tan(1)/2))) | Numeric-All-0 | ASyMOB
Differential_Equations
Q14 | Differential Equations | |
35348 | Solve the following differential equation:
$ 1 \cdot y' = 2 \cdot 1 \cdot \cos ^2(1 \cdot y) - \sin ^2(1 \cdot y) , y(1 - 1) = 1 - 1 $
| atan(sqrt(2)*tanh(sqrt(2)*x)) | Numeric-All-0 | ASyMOB
Differential_Equations
Q15 | Differential Equations | |
35349 | Solve the following differential equation:
$ 1 \cdot x + y'' + y' + y = 0 , y(0) = 1, y'(0) = 1 $
| (-3*e**(x/2)*x + 3*e**(x/2) + 2*sqrt(3)*sin(sqrt(3)*x/2) + 2*sqrt(3)*sin(sqrt(3)*x/2 + pi/3) - 2*sqrt(3)*cos(sqrt(3)*x/2 + pi/6))/(3*e**(x/2)) | Numeric-All-0 | ASyMOB
Differential_Equations
Q16 | Differential Equations | |
35350 | Solve the following differential equation:
$ 1 \cdot y'' + 1 \cdot y' - 6 \cdot 1 \cdot y = 0 , y(0) = 1 - 1, y'(0) = 1 $
| (2*e**(5*x) - 2)/(10*e**(3*x)) | Numeric-All-0 | ASyMOB
Differential_Equations
Q17 | Differential Equations | |
35351 | Solve the following differential equation:
$ 1 \cdot y + y'' = 0 , y(1) = 1, y'(0) = 1 $
| (sin(x - 1) + cos(x))*sec(1) | Numeric-All-0 | ASyMOB
Differential_Equations
Q18 | Differential Equations | |
35352 | Solve the following differential equation:
$ y'' - 1 \cdot y = 0 , y(1) = 1, y'(1) = 1 $
| e**(2*x + 1)/(e**2*e**x) | Numeric-All-0 | ASyMOB
Differential_Equations
Q19 | Differential Equations | |
35353 | Solve the following differential equation:
$ y'' - 2 \cdot y' - 3 \cdot y = 1 \cdot \sin (x) , y(1) = 1, y'(1) = 1 $
| (5*e**5*sin(1) + 20*e**5 - 2*e*(e**4 - e**(4*x))*cos(1) - e*(3*e**4 + e**(4*x))*cos(1) - 8*e**(x + 4)*sin(x) + 4*e**(x + 4)*cos(x) + 3*e**(4*x + 1)*sin(1) + 20*e**(4*x + 1))/(40*e**4*e**x) | Numeric-All-0 | ASyMOB
Differential_Equations
Q20 | Differential Equations | |
35354 | Calculate the following infinite product. Give a finite, closed form answer.
$ \prod_{n=1}^\infty 1-\frac{\left(\frac{1}{2 1}+1\right)^4}{1^4 (n+1)^4} $ | 64*sinh(3*pi/2)/(585*pi**2) | Numeric-All-0 | ASyMOB
Series
Q1 | Series | |
35355 | Calculate the following infinite product. Give a finite, closed form answer.
$ \prod_{n=1}^\infty \frac{\left((1+n)^3-1\right) \left(1-\frac{1}{(1+n)^2}\right)}{(1+n)^3+1} $ | 1/3 | Numeric-All-0 | ASyMOB
Series
Q2 | Series | |
35356 | Calculate the following infinite product. Give a finite, closed form answer.
$ \prod_{n=1}^\infty 1-\frac{1^2}{(1+1)^2 n^2} $ | 2/pi | Numeric-All-0 | ASyMOB
Series
Q3 | Series | |
35357 | Calculate the following infinite product. Give a finite, closed form answer.
$ \prod_{n=1}^\infty \frac{1}{1^2 \cdot n^2+\left\lfloor \frac{n}{n+1}+\left\lfloor \frac{1}{n^2 + 1}\right\rfloor \right\rfloor }+1$ | sinh(pi)/pi | Numeric-All-0 | ASyMOB
Series
Q4 | Series | |
35358 | Calculate the following infinite product. Give a finite, closed form answer.
$ \prod_{n=1}^\infty \left(1-\frac{1^6}{64 n^6}\right) \left(1-\frac{4}{3} \sin ^2\left(1 \cdot 3^{-n}\right)\right) $ | 4*sin(1)*cosh(sqrt(3)*pi/2)/pi**3 | Numeric-All-0 | ASyMOB
Series
Q5 | Series | |
35359 | Calculate the following infinite sum. Give a finite, closed form answer.
$ \sum_{n=1}^\infty 1 \cdot \left(\frac{3}{5}\right)^n \left(1 - \frac{1}{n}\right) \left(\frac{1}{1}\right)^n $ | log(2/5) + 3/2 | Numeric-All-0 | ASyMOB
Series
Q6 | Series | |
35360 | Calculate the following infinite sum. Give a finite, closed form answer.
$ \sum_{n=1}^\infty \frac{(-1)^n}{1 \cdot (2 n+5)} $ | -13/15 + pi/4 | Numeric-All-0 | ASyMOB
Series
Q7 | Series | |
35361 | Calculate the following infinite sum. Give a finite, closed form answer.
$ \sum_{n=1}^\infty 3^{1 + 1 \cdot n} \cdot 2^{1 - 3 \cdot 1 \cdot n} $ | 18/5 | Numeric-All-0 | ASyMOB
Series
Q8 | Series | |
35362 | Calculate the following infinite sum. Give a finite, closed form answer.
$ \sum_{n=1}^\infty \frac{1 \cdot 1^{1 \cdot n}}{n!} $ | e - 1 | Numeric-All-0 | ASyMOB
Series
Q9 | Series | |
35363 | Calculate the following infinite sum. Give a finite, closed form answer.
$ \sum_{n=1}^\infty \frac{1}{-1 + n^2 - n} $ | sqrt(5)*pi*tan(sqrt(5)*pi/2)/5 | Numeric-All-0 | ASyMOB
Series
Q10 | Series | |
35364 | Calculate the following infinite sum. Give a finite, closed form answer.
$ \sum_{n=1}^\infty \frac{1 \cdot n^3 + 1 \cdot n^2 + 1 \cdot n + 1}{1 \cdot n \cdot (n+1)^2 \cdot (n+2)^2} $ | -15/4 + 5*pi**2/12 | Numeric-All-0 | ASyMOB
Series
Q11 | Series | |
35365 | Calculate the following infinite sum. Give a finite, closed form answer.
$ \sum_{n=1}^\infty \frac{1 \cdot n+1}{1 \cdot (n+4) \cdot (n+2) \cdot (n+3)} $ | 5/24 | Numeric-All-0 | ASyMOB
Series
Q12 | Series | |
35366 | Calculate the following infinite sum. Give a finite, closed form answer.
$ \sum_{n=1}^\infty \frac{1 \cdot (-1)^n \cdot n + 1}{1 \cdot (n+1) \cdot (n+2) \cdot (n+3)} $ | 17/6 - 4*log(2) | Numeric-All-0 | ASyMOB
Series
Q13 | Series | |
35367 | Calculate the following infinite sum. Give a finite, closed form answer.
$ \sum_{n=1}^\infty \frac{(-2)^{-n} \cdot {1}^{-n} \cdot \left(1 \cdot (-1)^n + 1 \cdot n^2 + 1 \cdot n \right)}{1} $ | 19/27 | Numeric-All-0 | ASyMOB
Series
Q14 | Series | |
35368 | Calculate the following infinite sum. Give a finite, closed form answer.
$ \sum_{n=1}^\infty \frac{1 \cdot (-1)^{1 + n} + 1}{1 \cdot \left(1^2 + n^2\right)} $ | -pi/(2*sinh(pi)) + pi*coth(pi)/2 | Numeric-All-0 | ASyMOB
Series
Q15 | Series |
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