Found problems: 85335
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.
1952 Putnam, A1
Let \[ f(x) = \sum_{i=0}^{i=n} a_i x^{n - i}\] be a polynomial of degree $n$ with integral coefficients. If $a_0, a_n,$ and $f(1)$ are odd, prove that $f(x) = 0$ has no rational roots.
1971 Miklós Schweitzer, 2
Prove that there exists an ordered set in which every uncountable subset contains an uncountable, well-ordered subset and that cannot be represented as a union of a countable family of well-ordered subsets.
[i]A. Hajnal[/i]
1987 All Soviet Union Mathematical Olympiad, 453
Each field of the $1987\times 1987$ board is filled with numbers, which absolute value is not greater than one. The sum of all the numbers in every $2\times 2$ square equals $0$. Prove that the sum of all the numbers is not greater than $1987$.
2021 Estonia Team Selection Test, 1
a) There are $2n$ rays marked in a plane, with $n$ being a natural number. Given that no two marked rays have the same direction and no two marked rays have a common initial point, prove that there exists a line that passes through none of the initial points of the marked rays and intersects with exactly $n$ marked rays.
(b) Would the claim still hold if the assumption that no two marked rays have a common initial point was dropped?
2007 F = Ma, 20
A point-like mass moves horizontally between two walls on a frictionless surface with initial kinetic energy $E$. With every collision with the walls, the mass loses $1/2$ its kinetic energy to thermal energy. How many collisions with the walls are necessary before the speed of the mass is reduced by a factor of $8$?
$ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 16 $
2019 India PRMO, 6
Let $\overline{abc}$ be a three digit number with nonzero digits such that $a^2 + b^2 = c^2$. What is the largest possible prime factor of $\overline{abc}$
2014 Postal Coaching, 4
Let $m$ and $n$ be odd positive integers. Each square of an $m$ by $n$ board is coloured red or blue. A row is said to be red-dominated if there are more red squares than blue squares in the row. A column is said to be blue-dominated if there are more blue squares than red squares in the column. Determine the maximum possible value of the number of red-dominated rows plus the number of blue-dominated columns. Express your answer in terms of $m$ and $n$.
1962 AMC 12/AHSME, 39
Two medians of a triangle with unequal sides are $ 3$ inches and $ 6$ inches. Its area is $ 3 \sqrt{15}$ square inches. The length of the third median in inches, is:
$ \textbf{(A)}\ 4 \qquad
\textbf{(B)}\ 3 \sqrt{3} \qquad
\textbf{(C)}\ 3 \sqrt{6} \qquad
\textbf{(D)}\ 6 \sqrt{3} \qquad
\textbf{(E)}\ 6 \sqrt{6}$
2018 USA TSTST, 9
Show that there is an absolute constant $c < 1$ with the following property: whenever $\mathcal P$ is a polygon with area $1$ in the plane, one can translate it by a distance of $\frac{1}{100}$ in some direction to obtain a polygon $\mathcal Q$, for which the intersection of the interiors of $\mathcal P$ and $\mathcal Q$ has total area at most $c$.
[i]Linus Hamilton[/i]
2019 IFYM, Sozopol, 6
Does there exist a function $f: \mathbb N \to \mathbb N$ such that for all integers $n \geq 2$,
\[ f(f(n-1)) = f (n+1) - f(n)\, ?\]