Found problems: 85335
MMPC Part II 1996 - 2019, 2016.5
Consider four real numbers $x$, $y$, $a$, and $b$, satisfying $x + y = a + b$ and $x^2 + y^2 = a^2 + b^2$. Prove that $x^n + y^n = a^n + b^n$, for all $n \in \mathbb{N}$.
2018 PUMaC Live Round, 7.3
Kite $ABCD$ has right angles at $B$ and $D$, and $AB<BC$. Points $E\in AB$ and $F\in AD$ satisfy $AE=4$, $EF=7$, and $FA=5$. If $AB=8$ and points $X$ lies outside $ABCD$ while satisfying $XE-XF=1$ and $XE+XF+2XA=23$, then $XA$ may be written as $\tfrac{a-b\sqrt{c}}{d}$ for $a,b,c,d$ positive integers with $\gcd(a^2,b^2,c,d^2)=1$ and $c$ squarefree. Find $a+b+c+d$.
2016 USA Team Selection Test, 1
Let $S = \{1, \dots, n\}$. Given a bijection $f : S \to S$ an [i]orbit[/i] of $f$ is a set of the form $\{x, f(x), f(f(x)), \dots \}$ for some $x \in S$. We denote by $c(f)$ the number of distinct orbits of $f$. For example, if $n=3$ and $f(1)=2$, $f(2)=1$, $f(3)=3$, the two orbits are $\{1,2\}$ and $\{3\}$, hence $c(f)=2$.
Given $k$ bijections $f_1$, $\ldots$, $f_k$ from $S$ to itself, prove that \[ c(f_1) + \dots + c(f_k) \le n(k-1) + c(f) \] where $f : S \to S$ is the composed function $f_1 \circ \dots \circ f_k$.
[i]Proposed by Maria Monks Gillespie[/i]
2013 239 Open Mathematical Olympiad, 8
The product of the positive numbers $a, b, c, d$ and $e$ is equal to $1$. Prove that
$$ \frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{d^2}+\frac{d^2}{e^2}+\frac{e^2}{a^2} \geq a+b+c+d+e .$$
2014 Harvard-MIT Mathematics Tournament, 14
Let $ABCD$ be a trapezoid with $AB\parallel CD$ and $\angle D=90^\circ$. Suppose that there is a point $E$ on $CD$ such that $AE=BE$ and that triangles $AED$ and $CEB$ are similar, but not congruent. Given that $\tfrac{CD}{AB}=2014$, find $\tfrac{BC}{AD}$.
2022 Princeton University Math Competition, A2 / B4
Compute the sum of all positive integers whose positive divisors sum to $186.$
2009 Korea National Olympiad, 2
Let $ABC$ be a triangle and $ P, Q ( \ne A, B, C ) $ are the points lying on segments $ BC , CA $. Let $ I, J, K $ be the incenters of triangle $ ABP, APQ, CPQ $. Prove that $ PIJK $ is a convex quadrilateral.
2007 Harvard-MIT Mathematics Tournament, 31
A sequence $\{a_n\}_{n\geq 0}$ of real numbers satisfies the recursion $a_{n+1}=a_n^3-3a_n^2+3$ for all positive integers $n$. For how many values of $a_0$ does $a_{2007}=a_0$?
1952 Putnam, B5
If the terms of a sequence $a_{1}, a_{2}, \ldots$ are monotonic, and if $\sum_{n=1}^{\infty} a_n$ converges, show that $\sum_{n=1}^{\infty} n(a_{n} -a_{n+1 })$ converges.
2002 Romania National Olympiad, 3
Let $[ABCDEF]$ be a frustum of a regular pyramid. Let $G$ and $G'$ be the centroids of bases $ABC$ and $DEF$ respectively. It is known that $AB=36,DE=12$ and $GG'=35$.
$a)$ Prove that the planes $(ABF),(BCD),(CAE)$ have a common point $P$, and the planes $(DEC),(EFA),(FDB)$ have a common point $P'$, both situated on $GG'$.
$b)$ Find the length of the segment $[PP']$.
2017 Polish MO Finals, 5
Point $M$ is the midpoint of $BC$ of a triangle $ABC$, in which $AB=AC$. Point $D$ is the orthogonal projection of $M$ on $AB$. Circle $\omega$ is inscribed in triangle $ACD$ and tangent to segments $AD$ and $AC$ at $K$ and $L$ respectively. Lines tangent to $\omega$ which pass through $M$ cross line $KL$ at $X$ and $Y$, where points $X$, $K$, $L$ and $Y$ lie on $KL$ in this specific order. Prove that points $M$, $D$, $X$ and $Y$ are concyclic.
JOM 2015 Shortlist, A9
Let \(2n\) positive reals \(a_1, a_2, \cdots, a_n, b_1, b_2, \cdots, b_n\) satisfy \(a_{i+1}\ge 2a_i\) and \(b_{i+1} \le b_i\) for \(1\le i\le n-1\). Find the least constant \(C\) that satisfy: \[\displaystyle \sum^{n}_{i=1}{\frac{a_i}{b_i}} \ge \displaystyle \frac{C(a_1+a_2+\cdots+a_n)}{b_1+b_2+\cdots+b_n}\] and determine all equality case with that constant \(C\).
2015 Turkmenistan National Math Olympiad, 4
Find the max and minimum without using dervivate:
$\sqrt{x} +4 \cdot \sqrt{\frac{1}{2} - x}$
2008 Moldova National Olympiad, 9.3
From the vertex $ A$ of the equilateral triangle $ ABC$ a line is drown that intercepts the segment $ [BC]$ in the point $ E$. The point $ M \in (AE$ is such that $ M$ external to $ ABC$, $ \angle AMB \equal{} 20 ^\circ$ and $ \angle AMC \equal{} 30 ^ \circ$. What is the measure of the angle $ \angle MAB$?
2009 Harvard-MIT Mathematics Tournament, 8
Triangle $ABC$ has side lengths $AB=231$, $BC=160$, and $AC=281$. Point $D$ is constructed on the opposite side of line $AC$ as point $B$ such that $AD=178$ and $CD=153$. Compute the distance from $B$ to the midpoint of segment $AD$.
2007 Tournament Of Towns, 7
Nancy shuffles a deck of $52$ cards and spreads the cards out in a circle face up, leaving one spot empty. Andy, who is in another room and does not see the cards, names a card. If this card is adjacent to the empty spot, Nancy moves the card to the empty spot, without telling Andy; otherwise nothing happens. Then Andy names another card and so on, as many times as he likes, until he says "stop."
[list][b](a)[/b] Can Andy guarantee that after he says "stop," no card is in its initial spot?
[b](b)[/b] Can Andy guarantee that after he says "stop," the Queen of Spades is not adjacent to
the empty spot?[/list]
2014 Sharygin Geometry Olympiad, 1
Let $ABCD$ be a cyclic quadrilateral. Prove that $AC > BD$ if and only if $(AD-BC)(AB- CD) > 0$.
(V. Yasinsky)
1956 AMC 12/AHSME, 50
In triangle $ ABC$, $ \overline{CA} \equal{} \overline{CB}$. On $ CB$ square $ BCDE$ is constructed away from the triangle. If $ x$ is the number of degrees in angle $ DAB$, then
$ \textbf{(A)}\ x\text{ depends upon triangle }ABC \qquad\textbf{(B)}\ x\text{ is independent of the triangle}$
$ \textbf{(C)}\ x\text{ may equal angle }CAD \qquad\textbf{(D)}\ x\text{ can never equal angle }CAB$
$ \textbf{(E)}\ x\text{ is greater than }45^{\circ}\text{ but less than }90^{\circ}$
2018 Azerbaijan BMO TST, 1
Find all positive integers $(x,y)$ such that
$x^2+y^2=2017(x-y)$
1952 AMC 12/AHSME, 35
With a rational denominator, the expression $ \frac {\sqrt {2}}{\sqrt {2} \plus{} \sqrt {3} \minus{} \sqrt {5}}$ is equivalent to:
$ \textbf{(A)}\ \frac {3 \plus{} \sqrt {6} \plus{} \sqrt {15}}{6} \qquad\textbf{(B)}\ \frac {\sqrt {6} \minus{} 2 \plus{} \sqrt {10}}{6} \qquad\textbf{(C)}\ \frac {2 \plus{} \sqrt {6} \plus{} \sqrt {10}}{10}$
$ \textbf{(D)}\ \frac {2 \plus{} \sqrt {6} \minus{} \sqrt {10}}{6} \qquad\textbf{(E)}\ \text{none of these}$
2010 Today's Calculation Of Integral, 557
Find the folllowing limit.
\[ \lim_{n\to\infty} \frac{(2n\plus{}1)\int_0^1 x^{n\minus{}1}\sin \left(\frac{\pi}{2}x\right)dx}{(n\plus{}1)^2\int_0^1 x^{n\minus{}1}\cos \left(\frac{\pi}{2}x\right)dx}\ \ (n\equal{}1,\ 2,\ \cdots).\]
2012 AMC 10, 2
A circle of radius $5$ is inscribed in a rectangle as shown. The ratio of the the length of the rectangle to its width is $2\ :\ 1$. What is the area of the rectangle?
[asy]
draw((0,0)--(0,10)--(20,10)--(20,0)--cycle);
draw(circle((10,5),5));
[/asy]
$ \textbf{(A)}\ 50\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 125\qquad\textbf{(D)}\ 150\qquad\textbf{(E)}\ 200 $
2006 Baltic Way, 5
An occasionally unreliable professor has devoted his last book to a certain binary operation $*$. When this operation is applied to any two integers, the result is again an integer. The operation is known to satisfy the following axioms:
$\text{a})\ x*(x*y)=y$ for all $x,y\in\mathbb{Z}$;
$\text{b})\ (x*y)*y=x$ for all $x,y\in\mathbb{Z}$.
The professor claims in his book that
$1.$ The operation $*$ is commutative: $x*y=y*x$ for all $x,y\in\mathbb{Z}$.
$2.$ The operation $*$ is associative: $(x*y)*z=x*(y*z)$ for all $x,y,z\in\mathbb{Z}$.
Which of these claims follow from the stated axioms?
2005 Bulgaria Team Selection Test, 3
Let $\mathbb{R}^{*}$ be the set of non-zero real numbers. Find all functions $f : \mathbb{R}^{*} \to \mathbb{R}^{*}$ such that $f(x^{2}+y) = (f(x))^{2} + \frac{f(xy)}{f(x)}$, for all $x,y \in \mathbb{R}^{*}$ and $-x^{2} \not= y$.
2011 Sharygin Geometry Olympiad, 14
In triangle $ABC$, the altitude and the median from vertex $A$ form (together with line $BC$) a triangle such that the bisectrix of angle $A$ is the median; the altitude and the median from vertex $B$ form (together with line AC) a triangle such that the bisectrix of angle $B$ is the bisectrix. Find the ratio of sides for triangle $ABC$.