Found problems: 85335
2012 Today's Calculation Of Integral, 843
Let $f(x)$ be a continuous function such that $\int_0^1 f(x)\ dx=1.$ Find $f(x)$ for which $\int_0^1 (x^2+x+1)f(x)^2dx$ is minimized.
2015 HMIC, 3
Let $M$ be a $2014\times 2014$ invertible matrix, and let $\mathcal{F}(M)$ denote the set of matrices whose rows are a permutation of the rows of $M$. Find the number of matrices $F\in\mathcal{F}(M)$ such that $\det(M + F) \ne 0$.
2005 Today's Calculation Of Integral, 5
Calculate the following indefinite integrals.
[1] $\int (4-5\tan x)\cos x dx$
[2] $\int \frac{dx}{\sqrt[3]{(1-3x)^2}}dx$
[3] $\int x^3\sqrt{4-x^2}dx$
[4] $\int e^{-x}\sin \left(x+\frac{\pi}{4}\right)dx$
[5] $\int (3x-4)^2 dx$
2016 Korea - Final Round, 5
An acute triangle $\triangle ABC$ has incenter $I$, and the incircle hits $BC, CA, AB$ at $D, E, F$.
Lines $BI, CI, BC, DI$ hits $EF$ at $K, L, M, Q$ and the line connecting the midpoint of segment $CL$ and $M$ hits the line segment $CK$ at $P$. Prove that $$PQ=\frac{AB \cdot KQ}{BI}$$
2015 Iran Team Selection Test, 3
Let $ b_1<b_2<b_3<\dots $ be the sequence of all natural numbers which are sum of squares of two natural numbers.
Prove that there exists infinite natural numbers like $m$ which $b_{m+1}-b_m=2015$ .
2010 Today's Calculation Of Integral, 571
Evaluate $ \int_0^{\pi} \frac{x\sin ^ 3 x}{\sin ^ 2 x\plus{}8}dx$.
2016 India IMO Training Camp, 3
For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2015$ good partitions.
2020 China Northern MO, P5
Find all positive integers $a$ so that for any $\left \lfloor \frac{a+1}{2} \right \rfloor$-digit number that is composed of only digits $0$ and $2$ (where $0$ cannot be the first digit) is not a multiple of $a$.
Oliforum Contest I 2008, 3
Let $ a,b,c$ be three pairwise distinct real numbers such that $ a\plus{}b\plus{}c\equal{}6\equal{}ab\plus{}bc\plus{}ca\minus{}3$. Prove that $ 0<abc<4$.
2021 AIME Problems, 12
A convex quadrilateral has area $30$ and side lengths $5, 6, 9,$ and $7,$ in that order. Denote by $\theta$ the measure of the acute angle formed by the diagonals of the quadrilateral. Then $\tan \theta$ can be written in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2013 Sharygin Geometry Olympiad, 1
All angles of a cyclic pentagon $ABCDE$ are obtuse. The sidelines $AB$ and $CD$ meet at point $E_1$, the sidelines $BC$ and $DE$ meet at point $A_1$. The tangent at $B$ to the circumcircle of the triangle $BE_1C$ meets the circumcircle $\omega$ of the pentagon for the second time at point $B_1$. The tangent at $D$ to the circumcircle of the triangle $DA_1C$ meets $\omega$ for the second time at point $D_1$. Prove that $B_1D_1 // AE$
II Soros Olympiad 1995 - 96 (Russia), 9.6
There is a point inside a regular triangle located at distances $5$, $6$ and $7$ from its vertices. Find the area of this regular triangle.
2015 Switzerland - Final Round, 1
Let $ABC$ be an acute-angled triangle with $AB \ne BC$ and radius $k$. Let $P$ and $Q$ be the points of intersection of $k$ with the internal bisector and the external bisector of $\angle CBA$ respectively. Let $D$ be the intersection of $AC$ and $PQ$. Find the ratio $AD: DC$.
2015 ASDAN Math Tournament, 28
Consider $13$ marbles that are labeled with positive integers such that the product of all $13$ integers is $360$. Moor randomly picks up $5$ marbles and multiplies the integers on top of them together, obtaining a single number. What is the maximum number of different products that Moor can obtain?
1974 IMO Longlists, 4
Let $K_a,K_b,K_c$ with centres $O_a,O_b,O_c$ be the excircles of a triangle $ABC$, touching the interiors of the sides $BC,CA,AB$ at points $T_a,T_b,T_c$ respectively.
Prove that the lines $O_aT_a,O_bT_b,O_cT_c$ are concurrent in a point $P$ for which $PO_a=PO_b=PO_c=2R$ holds, where $R$ denotes the circumradius of $ABC$. Also prove that the circumcentre $O$ of $ABC$ is the midpoint of the segment $PI$, where $I$ is the incentre of $ABC$.
2010 AMC 12/AHSME, 16
Bernardo randomly picks $ 3$ distinct numbers from the set $ \{1,2,3,4,5,6,7,8,9\}$ and arranges them in descending order to form a $ 3$-digit number. Silvia randomly picks $ 3$ distinct numbers from the set $ \{1,2,3,4,5,6,7,8\}$ and also arranges them in descending order to form a $ 3$-digit number. What is the probability that Bernardo's number is larger than Silvia's number?
$ \textbf{(A)}\ \frac {47}{72}\qquad
\textbf{(B)}\ \frac {37}{56}\qquad
\textbf{(C)}\ \frac {2}{3}\qquad
\textbf{(D)}\ \frac {49}{72}\qquad
\textbf{(E)}\ \frac {39}{56}$
2021 Alibaba Global Math Competition, 12
Let $A=(a_{ij})$ be a $5 \times 5$ matrix with $a_{ij}=\min\{i,j\}$. Suppose $f:\mathbb{R}^5 \to \mathbb{R}^5$ is a smooth map such that $f(\Sigma) \subset \Sigma$, where $\Sigma=\{x \in \mathbb{R}^5: xAx^T=1\}$. Denote by $f^{(n)}$ te $n$-th iterate of $f$. Prove that there does not exist $N \ge 1$ such that
\[\inf_{x \in \Sigma} \| f^{(n)}(x)-x\|>0, \forall n \ge N.\]
2018 Purple Comet Problems, 4
The following diagram shows a grid of $36$ cells. Find the number of rectangles pictured in the diagram that contain at least three cells of the grid.
[img]https://cdn.artofproblemsolving.com/attachments/a/4/e9ba3a35204ec68c17a364ebf92cc107eb4d7a.png[/img]
2022 BMT, 1
Define an operation $\Diamond$ as $ a \Diamond b = 12a - 10b.$ Compute the value of $((((20 \Diamond 22) \Diamond 22) \Diamond 22) \Diamond22).$
2020 CHMMC Winter (2020-21), 3
A [i]Beaver-number[/i] is a positive 5 digit integer whose digit sum is divisible by 17. Call a pair of [i]Beaver-numbers[/i] differing by exactly $1$ a [i]Beaver-pair[/i]. The smaller number in a [i]Beaver-pair[/i] is called an [i]MIT Beaver[/i], while the larger number is called a [i]CIT Beaver[/i]. Find the positive difference between the largest and smallest [i]CIT Beavers[/i] (over all [i]Beaver-pairs[/i]).
2007 F = Ma, 4
An object is released from rest and falls a distance $h$ during the first second of time. How far will it fall during the next second of time?
$ \textbf{(A)}\ h\qquad\textbf{(B)}\ 2h \qquad\textbf{(C)}\ 3h \qquad\textbf{(D)}\ 4h\qquad\textbf{(E)}\ h^2 $
2016 ASDAN Math Tournament, 27
Suppose that you are standing in the middle of a $100$ meter long bridge. You take a random sequence of steps either $1$ meter forward or $1$ meter backwards each iteration. At each step, if you are currently at meter $n$, you have a $\tfrac{n}{100}$ probability of $1$ meter forward, to meter $n+1$, and a $\tfrac{100-n}{100}$ of going $1$ meter backward, to meter $n-1$. What is the expected value of the number of steps it takes for you to step off the bridge (i.e., to get to meter $0$ or $100$)?
Let $C$ be the actual answer and $A$ be the answer you will submit. Your score will be given by $\max\{0,\lceil25-25|\log_6(\tfrac{A-C/2}{C/2})|^{0.8}\rceil\}$.
2010 China Western Mathematical Olympiad, 1
Suppose that $m$ and $k$ are non-negative integers, and $p = 2^{2^m}+1$ is a prime number. Prove that
[b](a)[/b] $2^{2^{m+1}p^k} \equiv 1$ $(\text{mod } p^{k+1})$;
[b](b)[/b] $2^{m+1}p^k$ is the smallest positive integer $n$ satisfying the congruence equation $2^n \equiv 1$ $(\text{mod } p^{k+1})$.
2017 Sharygin Geometry Olympiad, 8
Let $AK$ and $BL$ be the altitudes of an acute-angled triangle $ABC$, and let $\omega$ be the excircle of $ABC$ touching side $AB$. The common internal tangents to circles $CKL$ and $\omega$ meet $AB$ at points $P$ and $Q$. Prove that $AP =BQ$.
[i]Proposed by I.Frolov[/i]
2021 Indonesia TST, A
A positive real $M$ is $strong$ if for any positive reals $a$, $b$, $c$ satisfying
$$ \text{max}\left\{ \frac{a}{b+c} , \frac{b}{c+a} , \frac{c}{a+b} \right\} \geqslant M $$
then the following inequality holds:
$$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b} > 20.$$
(a) Prove that $M=20-\frac{1}{20}$ is not $strong$.
(b) Prove that $M=20-\frac{1}{21}$ is $strong$.