Found problems: 85335
LMT Accuracy Rounds, 2021 F2
A random rectangle (not necessarily a square) with positive integer dimensions is selected from the $2\times4$ grid below. The probability that the selected rectangle contains only white squares can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a+b$.
[asy]
fill((2,0)--(3,0)--(3,1)--(2,1)--cycle,blue);
fill((2,1)--(3,1)--(3,2)--(2,2)--cycle,blue);
draw((0,0)--(4,0),black);
draw((0,0)--(0,2),black);
draw((4,0)--(4,2),black);
draw((4,2)--(0,2),black);
draw((0,1)--(4,1),black);
draw((1,0)--(1,2),black);
draw((2,0)--(2,2),black);
draw((3,0)--(3,2),black);
[/asy]
2016 LMT, 7
Let $R(x)$ be a function that takes a natural number as input and returns a rectangle. $R(1)$ is known to have integer side lengths. Let $p(x)$ be the perimeter of $R(x)$ and let $a(x)$ be the area of $R(x)$. Suppose that $p(x+5)=6 p(x)$ for all $x$ in the domain of $R$ and that $a(x+2)=12a(x)$ for all $x> 6$ in the domain of $R$. For $x \leq 6$, $a(x+1)=a(x)+2$. Suppose $p(16)=1296$, and let the side lengths of $R(11)$ be $a$ and $b$ with $a\leq b$. Find the ordered pair $(a,b)$.
[i]Proposed by Matthew Weiss
2001 Miklós Schweitzer, 7
Let $e_1,\ldots, e_n$ be semilines on the plane starting from a common point. Prove that if there is no $u\not\equiv 0$ harmonic function on the whole plane that vanishes on the set $e_1\cup \cdots \cup e_n$, then there exists a pair $i,j$ of indices such that no $u\not\equiv 0$ harmonic function on the whole plane exists that vanishes on $e_i\cup e_j$.
2016 CCA Math Bonanza, I2
Rectangle $ABCD$ has perimeter $178$ and area $1848$. What is the length of the diagonal of the rectangle?
[i]2016 CCA Math Bonanza Individual Round #2[/i]
2015 Kazakhstan National Olympiad, 1
Prove that $$\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{(n+1)^2} < n \cdot \left(1-\frac{1}{\sqrt[n]{2}}\right).$$
2020 MBMT, 33
Circle $\omega_1$ with center $K$ of radius $4$ and circle $\omega_2$ of radius $6$ intersect at points $W$ and $U$. If the incenter of $\triangle KWU$ lies on circle $\omega_2$, find the length of $\overline{WU}$. (Note: The incenter of a triangle is the intersection of the angle bisectors of the angles of the triangle)
[i]Proposed by Bradley Guo[/i]
2018 BMT Spring, Tie 1
Compute the least positive $x$ such that $25x - 6$ is divisible by $1001$.
2018 Harvard-MIT Mathematics Tournament, 7
Let $[n]$ denote the set of integers $\left\{ 1, 2, \ldots, n \right\}$. We randomly choose a function $f:[n] \to [n]$, out of the $n^n$ possible functions. We also choose an integer $a$ uniformly at random from $[n]$. Find the probability that there exist positive integers $b, c \geq 1$ such that $f^b(1) = a$ and $f^c(a) = 1$. ($f^k(x)$ denotes the result of applying $f$ to $x$ $k$ times.)
MathLinks Contest 6th, 7.2
Let $ABCD$ be a cyclic quadrilateral. Let $M, N$ be the midpoints of the diagonals $AC$ and $BD$ and let $P$ be the midpoint of $MN$. Let $A',B',C',D'$ be the intersections of the rays $AP$, $BP$, $CP$ and $DP$ respectively with the circumcircle of the quadrilateral $ABCD$.
Find, with proof, the value of the sum
\[ \sigma = \frac{ AP}{PA'} + \frac{BP}{PB'} + \frac{CP}{PC'} + \frac{DP}{PD'} . \]
2012 IMAC Arhimede, 5
On the circumference of a circle, there are $3n$ colored points that divide the circle on $3n$ arches, $n$ of which have lenght $1$, $n$ of which have length $2$ and the rest of them have length $3$ . Prove that there are two colored points on the same diameter of the circle.
2015 All-Russian Olympiad, 1
Real numbers $a$ and $b$ are chosen so that each of two quadratic trinomials $x^2+ax+b$ and $x^2+bx+a$ has two distinct real roots,and the product of these trinomials has exactly three distinct real roots.Determine all possible values of the sum of these three roots. [i](S.Berlov)[/i]
2002 Moldova National Olympiad, 12.8
\[\bf{\sum_{n=1}^{\infty}3^n.sin^3(\frac{\pi}{3^n})=?}\]
1997 China Team Selection Test, 2
Let $n$ be a natural number greater than 6. $X$ is a set such that $|X| = n$. $A_1, A_2, \ldots, A_m$ are distinct 5-element subsets of $X$. If $m > \frac{n(n - 1)(n - 2)(n - 3)(4n - 15)}{600}$, prove that there exists $A_{i_1}, A_{i_2}, \ldots, A_{i_6}$ $(1 \leq i_1 < i_2 < \cdots, i_6 \leq m)$, such that $\bigcup_{k = 1}^6 A_{i_k} = 6$.
1950 AMC 12/AHSME, 38
If the expression $ \begin{pmatrix}a & c \\
d & b \end{pmatrix}$ has the value $ ab\minus{}cd$ for all values of $a, b, c$ and $d$, then the equation $ \begin{pmatrix}2x & 1 \\
x & x \end{pmatrix} = 3$:
$\textbf{(A)}\ \text{Is satisfied for only 1 value of }x \qquad\\
\textbf{(B)}\ \text{Is satisified for only 2 values of }x \qquad\\
\textbf{(C)}\ \text{Is satisified for no values of }x \qquad\\
\textbf{(D)}\ \text{Is satisfied for an infinite number of values of }x \qquad\\
\textbf{(E)}\ \text{None of these.}$
2007 Germany Team Selection Test, 1
The sequence of real numbers $a_0,a_1,a_2,\ldots$ is defined recursively by \[a_0=-1,\qquad\sum_{k=0}^n\dfrac{a_{n-k}}{k+1}=0\quad\text{for}\quad n\geq 1.\]Show that $ a_{n} > 0$ for all $ n\geq 1$.
[i]Proposed by Mariusz Skalba, Poland[/i]
Ukrainian From Tasks to Tasks - geometry, 2012.2
The triangle $ABC$ is equilateral. Find the locus of the points $M$ such that the triangles $ABM$ and $ACM$ are both isosceles.
2000 Mexico National Olympiad, 4
Let $a$ and $b$ be positive integers not divisible by $5$. A sequence of integers is constructed as follows: the first term is $5$, and every consequent term is obtained by multiplying its precedent by $a$ and adding $b$. (For example, if $a = 2$ and $b = 4$, the first three terms are $5,14,32$.) What is the maximum possible number of primes that can occur before encoutering the first composite term?
2018 AIME Problems, 3
Kathy has $5$ red cards and $5$ green cards. She shuffles the $10$ cards and lays out $5$ of the cards in a row in a random order. She will be happy if and only if all the red cards laid out are adjacent and all the green cards laid out are adjacent. For example, card orders RRGGG, GGGGR, or RRRRR will make Kathy happy, but RRRGR will not. The probability that Kathy will be happy is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2014 PUMaC Combinatorics A, 1
What is the largest $n$ such that a square cannot be partitioned into $n$ smaller, non-overlapping squares?
2009 Bulgarian Spring Mathematical Competition, Problem 10.3
On the side $BC$ of the triangle $\Delta ABC$ is choosen point $K$,such that $2\angle BAK=3\angle KAC$.Prove that $AB^2AC^3>AK^5$
2002 Manhattan Mathematical Olympiad, 2
Let us consider the sequence $1,2, 3, \ldots , 2002$. Somebody choses $1002$ numbers from the sequence. Prove that there are two of the chosen numbers which are relatively prime (i.e. do not have any common divisors except $1$).
2006 Vietnam National Olympiad, 2
Let $ABCD$ be a convex quadrilateral. Take an arbitrary point $M$ on the line $AB$, and let $N$ be the point of intersection of the circumcircles of triangles $MAC$ and $MBC$ (different from $M$). Prove that:
a) The point $N$ lies on a fixed circle;
b) The line $MN$ passes though a fixed point.
2007 Estonia Team Selection Test, 2
Let $D$ be the foot of the altitude of triangle $ABC$ drawn from vertex $A$. Let $E$ and $F$ be points symmetric to $D$ w.r.t. lines $AB$ and $AC$, respectively. Let $R_1$ and $R_2$ be the circumradii of triangles $BDE$ and $CDF$, respectively, and let $r_1$ and $r_2$ be the inradii of the same triangles. Prove that $|S_{ABD} - S_{ACD}| > |R_1r_1 - R_2r_2|$
2014 Bulgaria JBMO TST, 5
From the foot $D$ of the height $CD$ in the triangle $ABC,$ perpendiculars to $BC$ and $AC$ are drawn, which they intersect at points $M$ and $N.$ Let $\angle CAB = 60^{\circ} , \angle CBA = 45^{\circ} ,$ and $H$ be the orthocentre of $MNC.$ If $O$ is the midpoint of $CD,$ find $\angle COH.$
2006 National Olympiad First Round, 30
How many integer triples $(x,y,z)$ are there such that \[\begin{array}{rcl} x - yz^2&\equiv & 1 \pmod {13} \\ xz+y&\equiv& 4 \pmod {13} \end{array}\] where $0\leq x < 13$, $0\leq y <13$, and $0\leq z< 13$?
$
\textbf{(A)}\ 10
\qquad\textbf{(B)}\ 23
\qquad\textbf{(C)}\ 36
\qquad\textbf{(D)}\ 49
\qquad\textbf{(E)}\ \text{None of above}
$