Found problems: 85335
The figure below shows a dotted grid $8$ cells wide and $3$ cells tall consisting of $1''\times1''$ squares. Carl places $1$-inch toothpicks along some of the sides of the squares to create a closed loop that does not intersect itself. The numbers in the cells indicate the number of sides of that square that are to be covered by toothpicks, and any number of toothpicks are allowed if no number is written. In how many ways can Carl place the toothpicks? [asy]
size(6cm);
for (int i=0; i<9; ++i) {
draw((i,0)--(i,3),dotted);
}
for (int i=0; i<4; ++i){
draw((0,i)--(8,i),dotted);
}
for (int i=0; i<8; ++i) {
for (int j=0; j<3; ++j) {
if (j==1) {
label("1",(i+0.5,1.5));
}}}
[/asy] $\textbf{(A) }130\qquad\textbf{(B) }144\qquad\textbf{(C) }146\qquad\textbf{(D) }162\qquad\textbf{(E) }196$
Consider a plane $\epsilon$ and three non-collinear points $A,B,C$ on the same side of $\epsilon$; suppose the plane determined by these three points is not parallel to $\epsilon$. In plane $\epsilon$ take three arbitrary points $A',B',C'$. Let $L,M,N$ be the midpoints of segments $AA', BB', CC'$; Let $G$ be the centroid of the triangle $LMN$. (We will not consider positions of the points $A', B', C'$ such that the points $L,M,N$ do not form a triangle.) What is the locus of point $G$ as $A', B', C'$ range independently over the plane $\epsilon$?
Determine all the pairs $ (p , n )$ of a prime number $ p$ and a positive integer $ n$ for which $ \frac{ n^p + 1 }{p^n + 1} $ is an integer.
Let $k$ be an integer $\geq 3$. Sequence $\{a_n\}$ satisfies that $a_k = 2k$ and for all $n > k$, we have
\[a_n =
\begin{cases}
a_{n-1}+1 & \text{if } (a_{n-1},n) = 1 \\
2n & \text{if } (a_{n-1},n) > 1
\end{cases}
\]
Prove that there are infinitely many primes in the sequence $\{a_n - a_{n-1}\}$.
Let $ABCD$ be a rhombus, and let $K$ and $L$ be points such that $K$ lies inside the rhombus, $L$ lies outside the rhombus, and $KA = KB = LC = LD$. Prove that there exist points $X$ and $Y$ on lines $AC$ and $BD$ such that $KXLY$ is also a rhombus.
[i]Proposed by Ankan Bhattacharya[/i]
For a positive real number $ a > 1$, prove the following inequality.
$ \frac {1}{a \minus{} 1}\left(1 \minus{} \frac {\ln a}{a\minus{}1}\right) < \int_0^1 \frac {x}{a^x}\ dx < \frac {1}{\ln a}\left\{1 \minus{} \frac {\ln (\ln a \plus{} 1)}{\ln a}\right\}$
Find all integer solutions of the equation $x^2y^5 - 2^x5^y = 2015 + 4xy$.
(Malik Talbi)
If
$(x^2-x-1)^n=a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+...+a_1x+a_0$,
where $a_i,i\in\{0,1,2,..,2n\}$, find $a_1+a_3+...+a_{2n-1}$ and $a_0+a_2+a_4+...+a_{2n}$.
A regular triangle has side length $n{}$. We divided its sides into $n{}$ equal parts and drew a line segment parallel with each side through the dividing points. A lattice of $1+2+\ldots+(n+1)$ intersection points is thus formed. For which positive integers $n{}$ can this lattice be partitioned into triplets of points which are the vertices of a regular triangle of side length $1$?
[i]Proposed by Alexander Gunning, Cambridge, UK[/i]
Let $z=x+yi$ be a complex number with $x$ and $y$ rational and with $|z|=1.$ Prove that the number $|z^{2n} -1|$ is rational for every integer $n$.
Let there be 50 natural numbers $ a_i$ such that $ 0 < a_1 < a_2 < ... < a_{50} < 150$. What is the greatest possible sum of the differences $ d_j$ where each $ d_j \equal{} a_{j \plus{} 1} \minus{} a_j$?
For an integer $m\geq 1$, we consider partitions of a $2^m\times 2^m$ chessboard into rectangles consisting of cells of chessboard, in which each of the $2^m$ cells along one diagonal forms a separate rectangle of side length $1$. Determine the smallest possible sum of rectangle perimeters in such a partition.
[i]Proposed by Gerhard Woeginger, Netherlands[/i]
What is remainder when $2014^{2015}$ is divided by $121$?
$
\textbf{(A)}\ 45
\qquad\textbf{(B)}\ 34
\qquad\textbf{(C)}\ 23
\qquad\textbf{(D)}\ 12
\qquad\textbf{(E)}\ 1
$
Three circles $C_1,C_2,C_3$, with radii $1, 2, 3$ respectively, are externally tangent.
In the area enclosed by these circles, there is a circle $C_4$ which is externally tangent to all three circles.
Find the radius of $C_4$.
[asy]
unitsize(0.4 cm);
pair[] O;
real[] r;
O[1] = (-12/5,16/5);
r[1] = 1;
O[2] = (0,5);
r[2] = 2;
O[3] = (0,0);
r[3] = 3;
O[4] = (-132/115, 351/115);
r[4] = 6/23;
draw(Circle(O[1],r[1]));
draw(Circle(O[2],r[2]));
draw(Circle(O[3],r[3]));
draw(Circle(O[4],r[4]));
label("$C_1$", O[1]);
label("$C_2$", O[2]);
label("$C_3$", O[3]);
[/asy]
The monic quadratic trinomial $f(x)$ has $2$ different roots. Could it be that the equation $f(f(x)) = 0$ has $3$ different root, and the equation $f(f(f(x))) = 0$ has $7$ different roots?
Determine all positive integers $n$, $n\ge2$, such that the following statement is true:
If $(a_1,a_2,...,a_n)$ is a sequence of positive integers with $a_1+a_2+\cdots+a_n=2n-1$, then there is block of (at least two) consecutive terms in the sequence with their (arithmetic) mean being an integer.
A collector offers to buy state quarters for $2000\%$ of their face value. At that rate how much will Bryden get for his four state quarters?
$ \text{(A)}\ 20\text{ dollars}\qquad\text{(B)}\ 50\text{ dollars}\qquad\text{(C)}\ 200\text{ dollars}\qquad\text{(D)}\ 500\text{ dollars}\qquad\text{(E)}\ 2000\text{ dollars} $
Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f(0)+1=f(1)$ and for any real numbers $x$ and $y$,
$$ f(xy-x)+f(x+f(y))=yf(x)+3 $$
* A shipment of $13.5$ tons is packed in a number of weightless containers. Each loaded container weighs not more than $350$ kg. Prove that $11$ trucks each of which is capable of carrying · $1.5$ ton can carry this load.
Show that there exists a positive constant $C$ such that, for all positive reals $a$ and $b$ with $a + b$ being an integer, we have
$$\left\{a^3\right\} + \left\{b^3\right\} + \frac{C}{(a+b)^6} \le 2. $$
Here $\{x\} = x - \lfloor x\rfloor$ is the fractional part of $x$.
[i]Proposed by Li4 and Untro368.[/i]
Let $P(x)$ be a polynomial of odd degree. Prove that the equation $P(P(x)) = 0$ has at least as many different real roots as the equation $P(x) = 0$
[hide=original wording]Пусть P(x) — многочлен нечетной степени. Докажите, что уравнение P(P(x)) = 0 имеет не меньше различных действительных корней, чем уравнение P(x) = 0[/hide]
Given a triangle $ ABC$ with $ \angle A \equal{} 120^{\circ}$. The points $ K$ and $ L$ lie on the sides $ AB$ and $ AC$, respectively. Let $ BKP$ and $ CLQ$ be equilateral triangles constructed outside the triangle $ ABC$. Prove that $ PQ \ge\frac{\sqrt 3}{2}\left(AB \plus{} AC\right)$.
Show that the determinant: \[ \begin{vmatrix} 0 & a & b & c \\ -a & 0 & d & e \\ -b & -d & 0 & f \\ -c & -e & -f & 0 \end{vmatrix} \] is non-negative, if its elements $a, b, c,$ etc., are real.
Shaq sees the numbers $1$ through $2017$ written on a chalkboard. He repeatedly chooses three numbers, erases them, and writes one plus their median. (For instance, if he erased $-2, -1, 0$ he would replace them with $0$.) If $M$ is the maximum possible final value remaining on the board, and if m is the minimum, compute $M - m$.
A complex number $z$ whose real and imaginary parts are integers satis fies $\left(Re(z) \right)^4 +\left(Re(z^2)\right)^2 + |z|^4 =(2018)(81)$, where $Re(w)$ and $Im(w)$ are the real and imaginary parts of $w$, respectively. Find $\left(Im(z) \right)^2$
.