Found problems: 85335
2005 Today's Calculation Of Integral, 78
Let $\alpha,\beta$ be the distinct positive roots of the equation of $2x=\tan x$.
Evaluate
\[\int_0^1 \sin \alpha x\sin \beta x\ dx\]
2012 AMC 10, 8
The sums of three whole numbers taken in pairs are $12$, $17$, and $19$. What is the middle number?
$ \textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 8 $
2023 Brazil Team Selection Test, 3
Find all positive integers $n \geqslant 2$ for which there exist $n$ real numbers $a_1<\cdots<a_n$ and a real number $r>0$ such that the $\tfrac{1}{2}n(n-1)$ differences $a_j-a_i$ for $1 \leqslant i<j \leqslant n$ are equal, in some order, to the numbers $r^1,r^2,\ldots,r^{\frac{1}{2}n(n-1)}$.
2008 IMO Shortlist, 7
Let $ ABCD$ be a convex quadrilateral with $ BA\neq BC$. Denote the incircles of triangles $ ABC$ and $ ADC$ by $ \omega_{1}$ and $ \omega_{2}$ respectively. Suppose that there exists a circle $ \omega$ tangent to ray $ BA$ beyond $ A$ and to the ray $ BC$ beyond $ C$, which is also tangent to the lines $ AD$ and $ CD$. Prove that the common external tangents to $ \omega_{1}$ and $\omega_{2}$ intersect on $ \omega$.
[i]Author: Vladimir Shmarov, Russia[/i]
2015 IMO Shortlist, C6
Let $S$ be a nonempty set of positive integers. We say that a positive integer $n$ is [i]clean[/i] if it has a unique representation as a sum of an odd number of distinct elements from $S$. Prove that there exist infinitely many positive integers that are not clean.
2007 Peru Iberoamerican Team Selection Test, P4
Each of the squares on a $15$×$15$ board has a zero. At every step you choose a row or a column, we delete all the numbers from it and then we write the numbers from $1$ to $15$ in the empty cells, in an arbitrary order. find the sum
possible maximum of the numbers on the board that can be achieved after a number finite number of steps.
1965 AMC 12/AHSME, 4
Line $ l_2$ intersects line $ l_1$ and line $ l_3$ is parallel to $ l_1$. The three lines are distinct and lie in a plane. The number of points equidistant from all three lines is:
$ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 8$
2017 Germany, Landesrunde - Grade 11/12, 3
Find the smallest prime number that can not be written in the form $\left| 2^a-3^b \right|$ with non-negative integers $a,b$.
2008 BAMO, 5
A positive integer $N$ is called stable if it is possible to split the set of all positive divisors of $N$ (including $1$ and $N$) into two subsets that have no elements in common, which have the same sum. For example, 6 is stable, because $1+2+3=6$, but 10 is not stable. Is $2^{2008}\cdot2008$ stable?
2022 New Zealand MO, 2
Find all triples $(a, b, c) $ of real numbers such that $a^2 + b^2 + c^2 = 1$ and $a(2b - 2a - c) \ge \frac12$.
2016 NIMO Problems, 5
The equation $x^3 - 3x^2 - 7x - 1 = 0$ has three distinct real roots $a$, $b$, and $c$. If \[\left( \dfrac{1}{\sqrt[3]{a}-\sqrt[3]{b}} + \dfrac{1}{\sqrt[3]{b}-\sqrt[3]{c}} + \dfrac{1}{\sqrt[3]{c}-\sqrt[3]{a}} \right)^2 = \dfrac{p\sqrt[3]{q}}{r}\] where $p$, $q$, $r$ are positive integers such that $\gcd(p, r) = 1$ and $q$ is not divisible by the cube of a prime, find $100p + 10q + r$.
[i]Proposed by Michael Tang and David Altizio[/i]
1958 AMC 12/AHSME, 1
The value of $ [2 \minus{} 3(2 \minus{} 3)^{\minus{}1}]^{\minus{}1}$ is:
$ \textbf{(A)}\ 5\qquad
\textbf{(B)}\ \minus{}5\qquad
\textbf{(C)}\ \frac{1}{5}\qquad
\textbf{(D)}\ \minus{}\frac{1}{5}\qquad
\textbf{(E)}\ \frac{5}{3}$
2010 Contests, 2
Let $ABC$ be a triangle with $ \widehat{BAC}\neq 90^\circ $. Let $M$ be the midpoint of $BC$. We choose a variable point $D$ on $AM$. Let $(O_1)$ and $(O_2)$ be two circle pass through $ D$ and tangent to $BC$ at $B$ and $C$. The line $BA$ and $CA$ intersect $(O_1),(O_2)$ at $ P,Q$ respectively.
[b]a)[/b] Prove that tangent line at $P$ on $(O_1)$ and $Q$ on $(O_2)$ must intersect at $S$.
[b]b)[/b] Prove that $S$ lies on a fix line.
1969 IMO Longlists, 51
$(NET 6)$ A curve determined by $y =\sqrt{x^2 - 10x+ 52}, 0\le x \le 100,$ is constructed in a rectangular grid. Determine the number of squares cut by the curve.
2014 Harvard-MIT Mathematics Tournament, 8
Let $ABC$ be a triangle with sides $AB = 6$, $BC = 10$, and $CA = 8$. Let $M$ and $N$ be the midpoints of $BA$ and $BC$, respectively. Choose the point $Y$ on ray $CM$ so that the circumcircle of triangle $AMY$ is tangent to $AN$. Find the area of triangle $NAY$.
2016 Japan MO Preliminary, 4
There is a $11\times 11$ square grid. We divided this in $5$ rectangles along unit squares. How many ways that one of the rectangles doesn’t have a edge on basic circumference.
Note that we count as different ways that one way coincides with another way by rotating or reversing.
LMT Guts Rounds, 2020 F35
Estimate the number of ordered pairs $(p,q)$ of positive integers at most $2020$ such that the cubic equation $x^3-px-q=0$ has three distinct real roots. If your estimate is $E$ and the answer is $A$, your score for this problem will be \[\Big\lfloor15\min\Big(\frac{A}{E},\frac{E}{A}\Big)\Big\rfloor.\]
[i]Proposed by Alex Li[/i]
1955 AMC 12/AHSME, 32
If the discriminant of $ ax^2\plus{}2bx\plus{}c\equal{}0$ is zero, then another true statement about $ a$, $ b$, and $ c$ is that:
$ \textbf{(A)}\ \text{they form an arithmetic progression} \\
\textbf{(B)}\ \text{they form a geometric progression} \\
\textbf{(C)}\ \text{they are unequal} \\
\textbf{(D)}\ \text{they are all negative numbers} \\
\textbf{(E)}\ \text{only b is negative and a and c are positive}$
LMT Speed Rounds, 2016.1
Find the ordered triple of natural numbers $(x,y,z)$ such that $x \le y \le z$ and $x^x+y^y+z^z = 3382.$
[i]Proposed by Evan Fang
2015 BMT Spring, 2
Let $g(x)=1+2x+3x^2+4x^3+\ldots$. Find the coefficient of $x^{2015}$ of $f(x)=\frac{g(x)}{1-x}$.
1959 AMC 12/AHSME, 37
When simplified the product $\left(1-\frac13\right)\left(1-\frac14\right)\left(1-\frac15\right)\cdots\left(1-\frac1n\right)$ becomes:
$ \textbf{(A)}\ \frac1n \qquad\textbf{(B)}\ \frac2n\qquad\textbf{(C)}\ \frac{2(n-1)}{n}\qquad\textbf{(D)}\ \frac{2}{n(n+1)}\qquad\textbf{(E)}\ \frac{3}{n(n+1)} $
2024 Princeton University Math Competition, 6
Ben has a square of side length $2.$ He wants to put a circle and an equilateral triangle inside the square such that the circle and equilateral triangle do not overlap. The maximum possible sum of the areas of the circle and triangle is $\tfrac{a\pi+b\sqrt{c}+d\sqrt{e}}{f},$ where $a,c,e,f$ are positive integers, $b$ and $d$ are integers, $c$ and $e$ are square-free, and $\gcd(a,b,d,f)=1.$ Find $a+b+c+d+e+f.$
2015 Azerbaijan IMO TST, 1
Let $\omega$ be the circumcircle of an acute-angled triangle $ABC$. The lines tangent to $\omega$ at the points $A$ and $B$ meet at $K$. The line passing through $K$ and parallel to $BC$ intersects the side $AC$ at $S$. Prove that $BS=CS$
Geometry Mathley 2011-12, 15.2
Let $O$ be the centre of the circumcircle of triangle $ABC$. Point $D$ is on the side $BC$. Let $(K)$ be the circumcircle of $ABD$. $(K)$ meets $AO$ at $E$ that is distinct from $A$.
(a) Prove that $B,K,O,E$ are on the same circle that is called $(L)$.
(b) $(L)$ intersects $AB$ at $F$ distinct $B$. Point $G$ is on $(L)$ such that $EG \parallel OF$. $GK$ meets $AD$ at $S, SO$ meets $BC$ at $T$ . Prove that $O,E, T,C$ are on the same circle.
Trần Quang Hùng
2011 Purple Comet Problems, 17
Find the number of ordered quadruples $(a, b, c, d)$ where each of $a, b, c,$ and $d$ are (not necessarily distinct) elements of $\{1, 2, 3, 4, 5, 6, 7\}$ and $3abc + 4abd + 5bcd$ is even. For example, $(2, 2, 5, 1)$ and $(3, 1, 4, 6)$ satisfy the conditions.