Found problems: 85335
Prove or disprove that there exists a positive real number $u$ such that $\lfloor u^n \rfloor -n$ is an even integer for all positive integer $n$.
Five positive integers from $1$ to $15$ are chosen without replacement. What is the probability that their sum is divisible by $3$?
In the figure below, how many ways are there to select two squares which do not share an edge?
[asy]
size(3cm);
for (int t = -2; t <= 2; t=t+1) { draw( shift((t,0))*unitsquare ) ; if (t!=0) draw( shift((0,t))*unitsquare ); }
[/asy]
[i]Proposed by Evan Chen[/i]
Suppose $x$ is a positive real number such that $\{x\}, [x]$ and $x$ are in a geometric progression. Find the least positive integer $n$ such that $x^n > 100$. (Here $[x]$ denotes the integer part of $x$ and $\{x\} = x - [x]$.)
In the acute triangle $ABC$, $\angle BAC$ is less than $\angle ACB $. Let $AD$ be a diameter of $\omega$, the circle circumscribed to said triangle. Let $E$ be the point of intersection of the ray $AC$ and the tangent to $\omega$ passing through $B$. The perpendicular to $AD$ that passes through $E$ intersects the circle circumscribed to the triangle $BCE$, again, at the point $F$. Show that $CD$ is an angle bisector of $\angle BCF$.
Let $n$ be a positive integer, and consider a sequence $a_1 , a_2 , \dotsc , a_n $ of positive integers. Extend it periodically to an infinite sequence $a_1 , a_2 , \dotsc $ by defining $a_{n+i} = a_i $ for all $i \ge 1$. If \[a_1 \le a_2 \le \dots \le a_n \le a_1 +n \] and \[a_{a_i } \le n+i-1 \quad\text{for}\quad i=1,2,\dotsc, n, \] prove that \[a_1 + \dots +a_n \le n^2. \]
Let $ x$, $ y$, $ z$ be real numbers such that $ 0 < x,y,z < 1$ and $ xyz \equal{} (1 \minus{} x)(1 \minus{} y)(1 \minus{} z)$. Show that at least one of the numbers $ (1 \minus{} x)y,(1 \minus{} y)z,(1 \minus{} z)x$ is greater than or equal to $ \frac {1}{4}$
Find the locus of the intersection points of the medians all triangles inscribed in a given circle.
For positive integers $ n$, define $ A(n)$ to be $ \frac {(2n)!}{(n!)^{2}}$. Determine the sets of positive integers $ n$ for which
(a) $ A(n)$ is an even number,
(b) $ A(n)$ is a multiple of $ 4$.
For a positive integer $n$, let $f(n)$ be the number of (not necessarily distinct) primes in the prime factorization of $k$. For example, $f(1) = 0, f(2) = 1, $ and $f(4) = f(6) = 2$. let $g(n)$ be the number of positive integers $k \leq n$ such that $f(k) \geq f(j)$ for all $j \leq n$. Find $g(1) + g(2) + \ldots + g(100)$.
Let be two $ 2\times 2 $ real matrices $ A,B, $ such that $ AB=\begin{pmatrix} 1&1\\1&2 \end{pmatrix} . $
Calculate $ \left((BA)^{-1} +BA\right)^{2019 } . $
[i]Dan Nedeianu[/i]
For every positive integer $ n$ let
\[ a_n\equal{}\sum_{k\equal{}n}^{2n}\frac{(2k\plus{}1)^n}{k}\]
Show that there exists no $ n$, for which $ a_n$ is a non-negative integer.
There are many $a\times b$ rectangular cardboard pieces ($a,b\in\mathbb{N}$ such that $a<b$). It is given that by putting such pieces together without overlapping one can make $49\times 51$ rectangle, and $99\times 101$ rectangle. Can one uniquely determine $a,b$ from this?
Find the smallest constant $ C$ such that for all real $ x,y$
\[ 1\plus{}(x\plus{}y)^2 \leq C \cdot (1\plus{}x^2) \cdot (1\plus{}y^2)\]
holds.
Function $f(n), n \in \mathbb N$, is defined as follows:
Let $\frac{(2n)!}{n!(n+1000)!} = \frac{A(n)}{B(n)}$ , where $A(n), B(n)$ are coprime positive integers; if $B(n) = 1$, then $f(n) = 1$; if $B(n) \neq 1$, then $f(n)$ is the largest prime factor of $B(n)$. Prove that the values of $f(n)$ are finite, and find the maximum value of $f(n).$
Find the greatest positive integer $n$ for which there exist $n$ nonnegative integers $x_1, x_2,\ldots , x_n$, not all zero, such that for any $\varepsilon_1, \varepsilon_2, \ldots, \varepsilon_n$ from the set $\{-1, 0, 1\}$, not all zero, $\varepsilon_1 x_1 + \varepsilon_2 x_2 + \cdots + \varepsilon_n x_n$ is not divisible by $n^3$.
Let $f(x)$ be a real-valued function defined on the positive reals such that
(1) if $x < y$, then $f(x) < f(y)$,
(2) $f\left(2xy\over x+y\right) \geq {f(x) + f(y)\over2}$ for all $x$.
Show that $f(x) < 0$ for some value of $x$.
Solve the inequation
$\sqrt {3-x}-\sqrt {x+1}>\frac {1}{2}$.
What figure can the central projection of a triangle be? (The center of the projection does not lie on the plane of the triangle.)
Express $\sqrt{2 + \sqrt{3}}$ in the form $\frac{a + \sqrt{b}}{\sqrt{c}}$, where $a$ is a positive integer and $b$ and $c$ are square-free positive integers.
Let $ABC$ be a non-right-angled triangle, with $AC\ne BC$. Let $F$ be the midpoint of side $BC$. Let $D$ be a point on line $AB$ satisfying$CA=CD$,and let $E$ be a point on line $BC$ satisfying $EB = ED$. The line passing through $A$ and parallel to $ED$ meets line $FD$ at point $I$. Line $AF$ meets line $ED$ at point $J$. Prove that points $C$, $I$ and $J$ are collinear.
A square of area $2$ is inscribed in a square of area $3$, creating four congruent triangles, as shown below. What is the ratio of the shorter leg to the longer leg in the shaded right triangle?
[asy]
size(200);
defaultpen(linewidth(0.6pt)+fontsize(10pt));
real y = sqrt(3);
pair A,B,C,D,E,F,G,H;
A = (0,0);
B = (0,y);
C = (y,y);
D = (y,0);
E = ((y + 1)/2,y);
F = (y, (y - 1)/2);
G = ((y - 1)/2, 0);
H = (0,(y + 1)/2);
fill(H--B--E--cycle, gray);
draw(A--B--C--D--cycle);
draw(E--F--G--H--cycle);
[/asy]
$\textbf{(A) }\frac15\qquad\textbf{(B) }\frac14\qquad\textbf{(C) }2-\sqrt3\qquad\textbf{(D) }\sqrt3-\sqrt2\qquad\textbf{(E) }\sqrt2-1$
Let $a_1,a_2,...,a_{2007}$ be real number such that $ a_1+a_2+...+a_{2007}\geq 2007^{2}$ and $a_1^{2}+a_2^{2}+...+a_{2007}^{2}\leq 2007^{3}-1 $. Prove that $ a_k\in[2006;2008]$ for all $k\in\left \{ 1,2,...,2007 \right \}$
Charles has $ 5q \plus{} 1$ quarters and Richard has $ q \plus{} 5$ quarters. The difference in their money in dimes is:
$ \textbf{(A)}\ 10(q \minus{} 1) \qquad\textbf{(B)}\ \frac {2}{5}(4q \minus{} 4) \qquad\textbf{(C)}\ \frac {2}{5}(q \minus{} 1) \\
\textbf{(D)}\ \frac {5}{2}(q \minus{} 1) \qquad\textbf{(E)}\ \text{none of these}$
Consider integers $ a_i,i\equal{}\overline{1,2002}$ such that
$ a_1^{ \minus{} 3} \plus{} a_2^{ \minus{} 3} \plus{} \ldots \plus{} a_{2002}^{ \minus{} 3} \equal{} \frac {1}{2}$
Prove that at least 3 of the numbers are equal.