This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 913

2007 Today's Calculation Of Integral, 179
Tags: calculus , integration , calculus computations , logarithm
Evaluate the following integrals. (1) Meiji University $\int_{\frac{1}{e}}^{e}\frac{(\log x)^{2}}{x}dx.$ (2) Tokyo University of Science $\int_{0}^{1}\frac{7x^{3}+23x^{2}+21x+15}{(x^{2}+1)(x+1)^{2}}dx.$

1995 Israel Mathematical Olympiad, 1
Tags: system of equations , algebra , logarithm
Solve the system $$\begin{cases} x+\log\left(x+\sqrt{x^2+1}\right)=y \\ y+\log\left(y+\sqrt{y^2+1}\right)=z \\ z+\log\left(z+\sqrt{z^2+1}\right)=x \end{cases}$$

2009 Today's Calculation Of Integral, 446
Tags: calculus , integration , calculus computations , logarithm
Evaluate $ \int_0^1 \frac{(1\minus{}2x)e^{x}\plus{}(1\plus{}2x)e^{\minus{}x}}{(e^x\plus{}e^{\minus{}x})^3}\ dx.$

2006 AMC 12/AHSME, 21
Tags: geometry , ratio , inequalities , function , logarithm
Let \[ S_1 \equal{} \{ (x,y)\ | \ \log_{10} (1 \plus{} x^2 \plus{} y^2)\le 1 \plus{} \log_{10}(x \plus{} y)\} \]and \[ S_2 \equal{} \{ (x,y)\ | \ \log_{10} (2 \plus{} x^2 \plus{} y^2)\le 2 \plus{} \log_{10}(x \plus{} y)\}. \]What is the ratio of the area of $ S_2$ to the area of $ S_1$? $ \textbf{(A) } 98\qquad \textbf{(B) } 99\qquad \textbf{(C) } 100\qquad \textbf{(D) } 101\qquad \textbf{(E) } 102$

2012 Balkan MO Shortlist, C1
Tags: induction , inequalities , floor function , combinatorics , logarithm
Let $n$ be a positive integer. Let $P_n=\{2^n,2^{n-1}\cdot 3, 2^{n-2}\cdot 3^2, \dots, 3^n \}.$ For each subset $X$ of $P_n$, we write $S_X$ for the sum of all elements of $X$, with the convention that $S_{\emptyset}=0$ where $\emptyset$ is the empty set. Suppose that $y$ is a real number with $0 \leq y \leq 3^{n+1}-2^{n+1}.$ Prove that there is a subset $Y$ of $P_n$ such that $0 \leq y-S_Y < 2^n$

1997 National High School Mathematics League, 12
Tags: logarithm
Let $a=\lg z+\lg\left[x(yz)^{-1}+1\right],b=\lg x^{-1}+\lg(xyz+1),c=\lg y+\lg\left[(xyz)^{-1}+1\right]$, if $M=\max\{a,b,c\}$, then the minumum value of $M$ is________.

2014 District Olympiad, 3
Tags: floor function , algebra , logarithm
Let $p$ and $n$ be positive integers, with $p\geq2$, and let $a$ be a real number such that $1\leq a<a+n\leq p$. Prove that the set \[ \mathcal {S}=\left\{\left\lfloor \log_{2}x\right\rfloor +\left\lfloor \log_{3}x\right\rfloor +\cdots+\left\lfloor \log_{p}x\right\rfloor\mid x\in\mathbb{R},a\leq x\leq a+n\right\} \] has exactly $n+1$ elements.

1966 AMC 12/AHSME, 24
Tags: logarithm
If $\log_MN=\log_NM$, $M\ne N$, $MN>0$, $M\ne 1$, $N\ne 1$, then $MN$ equals: $\text{(A)} \ \frac12 \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ 10 \qquad \text{(E)} \ \text{a number greater than 2 and less than 10}$

2006 Iran Team Selection Test, 2
Tags: ceiling function , algebra , logarithm
Let $n$ be a fixed natural number. [b]a)[/b] Find all solutions to the following equation : \[ \sum_{k=1}^n [\frac x{2^k}]=x-1 \] [b]b)[/b] Find the number of solutions to the following equation ($m$ is a fixed natural) : \[ \sum_{k=1}^n [\frac x{2^k}]=x-m \]

1979 AMC 12/AHSME, 18
Tags: logarithm
To the nearest thousandth, $\log_{10}2$ is $.301$ and $\log_{10}3$ is $.477$. Which of the following is the best approximation of $\log_5 10$? $\textbf{(A) }\frac{8}{7}\qquad\textbf{(B) }\frac{9}{7}\qquad\textbf{(C) }\frac{10}{7}\qquad\textbf{(D) }\frac{11}{7}\qquad\textbf{(E) }\frac{12}{7}$

1969 German National Olympiad, 4
Tags: system of equations , system , logarithm , algebra
Solve the system of equations: $$|\log_2(x + y)| + | \log_2(x - y)| = 3$$ $$xy = 3$$

2007 Moldova National Olympiad, 12.3
Tags: inequalities , function , algebra , logarithm
For $a,b \in [1;\infty)$ show that \[ab\leq e^{a-1}+b\ln b\]

2019 AMC 12/AHSME, 12
Tags: algebra , quadratic formula , logarithm
Positive real numbers $x \neq 1$ and $y \neq 1$ satisfy $\log_2{x} = \log_y{16}$ and $xy = 64$. What is $(\log_2{\tfrac{x}{y}})^2$? $\textbf{(A) } \frac{25}{2} \qquad\textbf{(B) } 20 \qquad\textbf{(C) } \frac{45}{2} \qquad\textbf{(D) } 25 \qquad\textbf{(E) } 32$

1985 Traian Lălescu, 1.4
Tags: algebra , floor , logarithm , equation
Let $ a $ be a non-negative real number distinct from $ 1. $ [b]a)[/b] For which positive values $ x $ the equation $$ \left\lfloor\log_a x\right\rfloor +\left\lfloor \frac{1}{3} +\log_a x\right\rfloor =\left\lfloor 2\cdot\log_a x\right\rfloor $$ is true? [b]b)[/b] Solve $ \left\lfloor\log_3 x\right\rfloor +\left\lfloor \frac{1}{3} +\log_3 x\right\rfloor =3. $

2014 Contests, 3
Tags: geometry , trapezoid , parallelogram , trigonometry , geometric transformation , dilation , logarithm
Let $ABCD$ be a trapezoid (quadrilateral with one pair of parallel sides) such that $AB < CD$. Suppose that $AC$ and $BD$ meet at $E$ and $AD$ and $BC$ meet at $F$. Construct the parallelograms $AEDK$ and $BECL$. Prove that $EF$ passes through the midpoint of the segment $KL$.

2004 IMC, 5
Tags: integration , inequalities , calculus , logarithm
Prove that \[ \int^1_0 \int^1_0 \frac { dx \ dy }{ \frac 1x + |\log y| -1 } \leq 1 . \]

2011 Today's Calculation Of Integral, 710
Tags: calculus , integration , trigonometry , calculus computations , logarithm
Evaluate $\int_0^{\frac{\pi}{4}} \frac{\sin \theta (\sin \theta \cos \theta +2)}{\cos ^ 4 \theta}\ d\theta$.

2007 Today's Calculation Of Integral, 181
Tags: calculus , integration , trigonometry , function , derivative , analytic geometry , logarithm
For real number $a,$ find the minimum value of $\int_{0}^{\frac{\pi}{2}}\left|\frac{\sin 2x}{1+\sin^{2}x}-a\cos x\right| dx.$

2010 Today's Calculation Of Integral, 574
Tags: calculus , integration , calculus computations , logarithm
Let $ n$ be a positive integer. Prove that $ x^ne^{1\minus{}x}\leq n!$ for $ x\geq 0$,

2010 Today's Calculation Of Integral, 647
Tags: calculus , integration , trigonometry , calculus computations , logarithm
Evaluate \[\int_0^{\pi} xp^x\cos qx\ dx,\ \int_0^{\pi} xp^x\sin qx\ dx\ (p>0,\ p\neq 1,\ q\in{\mathbb{N^{+}}})\] Own

PEN G Problems, 15
Tags: inequalities , logarithm , irrational number
Prove that for any $ p, q\in\mathbb{N}$ with $ q > 1$ the following inequality holds: \[ \left\vert\pi\minus{}\frac{p}{q}\right\vert\ge q^{\minus{}42}.\]

2007 Princeton University Math Competition, 6
Tags: probability , calculus , integration , geometry , trapezoid , logarithm
Take the square with vertices $(0,0)$, $(1,0)$, $(0,1)$, and $(1,1)$. Choose a random point in this square and draw the line segment from it to $(0,0)$. Choose a second random point in this square and draw the line segment from it to $(1,0)$. What is the probability that the two line segments intersect?

2010 Today's Calculation Of Integral, 547
Tags: calculus , integration , function , derivative , real analysis , calculus computations , logarithm
Find the minimum value of $ \int_0^1 |e^{ \minus{} x} \minus{} a|dx\ ( \minus{} \infty < a < \infty)$.

II Soros Olympiad 1995 - 96 (Russia), 11.1
Tags: algebra , logarithm
Solve the equation $$\log_{10} (x^3+x)=\log_2 x.$$

1995 AIME Problems, 2
Tags: quadratic , modular arithmetic , algebra , polynomial , vieta , logarithm , search
Find the last three digits of the product of the positive roots of \[ \sqrt{1995}x^{\log_{1995}x}=x^2. \]