Found problems: 85335
Let $m$ be a positive integer not divisible by 3. Prove that there are infinitely many positive integers $n$ such that $s(n)$ and $s(n+1)$ are divisible by $m$, where $s(x)$ is the sum of digits of $x$.
[i]Dorel Miheț[/i]
For how many integer values of $x$ is $|2x|\leq 7\pi?$
$\textbf{(A) }16 \qquad\textbf{(B) }17\qquad\textbf{(C) }19\qquad\textbf{(D) }20\qquad\textbf{(E) }21$
Trapezoid $ABCD$ has $\overline{AB} \parallel \overline{CD}$, $BC = CD = 43$, and $\overline{AD} \perp \overline{BD}$. Let $O$ be the intersection of the diagonals $\overline{AC}$ and $\overline{BD}$, and let $P$ be the midpoint of $\overline{BD}$. GIven that $OP = 11$, the length $AD$ can be written in the form $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. What is $m + n$?
$\textbf{(A)}\: 65\qquad\textbf{(B)}\: 132\qquad\textbf{(C)}\: 157\qquad\textbf{(D)}\: 194\qquad\textbf{(E)}\: 215$
$ABC$ is acute-angled triangle. $AA_1,BB_1,CC_1$ are altitudes. $X,Y$ - midpoints of $AC_1,A_1C$. $XY=BB_1$.
Prove that one side of $ABC$ in $\sqrt{2}$ greater than other side.
Does there exist an increasing sequence of positive integers for which any large enough integer can be expressed uniquely as the sum of two (possibly equal) terms of the sequence?
[i]Proposed by Vlad Spătaru and David Anghel[/i]
Find all positive integers $n\geqslant 2$ such that $d_{i+1}/d_i$ is an integer for all $1\leqslant i < k$, where $1=d_1<d_2<\dots <d_k=n$ are all the positive divisors of $n$.
[i]Proposed by Pavel Ciurea[/i]
Determine all sequences of non-negative integers $a_1, \ldots, a_{2016}$ all less than or equal to $2016$ satisfying $i+j\mid ia_i+ja_j$ for all $i, j\in \{ 1,2,\ldots, 2016\}$.
Let \( S \) be a set consisting of \( 2024 \) points on a plane, such that no three points in \( S \) are collinear. A line \( \ell \) passing through two points in \( S \) is called a "weakly balanced line" if it satisfies the following condition:
(Condition) The line \( \ell \) divides the plane into two regions, one containing exactly \( 1010 \) points of \( S \), and the other containing exactly \( 1012 \) points of \( S \) (where each region contains no points lying on \( \ell \)).
Let \( \omega(S) \) denote the number of weakly balanced lines among the lines passing through two points in \( S \). Find the smallest possible value of \( \omega(S) \).
Suppose $x,y,z\in \mathbb R^+$. Prove that \[\frac {x}{\sqrt{yz+4xy+4xz}}+\frac {y}{\sqrt{zx+4yz+4yx}}+\frac {z}{\sqrt{xy+4zx+4zy}}\geq 1\].
Let $P(X)$ be a polynomial with real coefficients such that $\{P(n)\}\leq\frac{1}{n}, \forall n\in\mathbb{N}$, where $\{a\}$ is the fractional part of the number $a$. Show that $P(n)\in\mathbb{Z}, \forall n\in\mathbb{N}$.
We say a finite set $S$ of points in the plane is [i]very[/i] if for every point $X$ in $S$, there exists an inversion with center $X$ mapping every point in $S$ other than $X$ to another point in $S$ (possibly the same point).
(a) Fix an integer $n$. Prove that if $n \ge 2$, then any line segment $\overline{AB}$ contains a unique very set $S$ of size $n$ such that $A, B \in S$.
(b) Find the largest possible size of a very set not contained in any line.
(Here, an [i]inversion[/i] with center $O$ and radius $r$ sends every point $P$ other than $O$ to the point $P'$ along ray $OP$ such that $OP\cdot OP' = r^2$.)
[i]Proposed by Sammy Luo[/i]
What is the $1992^\text{nd}$ letter in this sequence?
\[\text{ABCDEDCBAABCDEDCBAABCDEDCBAABCDEDC}\cdots \]
$\text{(A)}\ \text{A} \qquad \text{(B)}\ \text{B} \qquad \text{(C)}\ \text{C} \qquad \text{(D)}\ \text{D} \qquad \text{(E)}\ \text{E}$
A uniform disk, a thin hoop, and a uniform sphere, all with the same mass and same outer radius, are each free to rotate about a fixed axis through its center. Assume the hoop is connected to the rotation axis by light spokes. With the objects starting from rest, identical forces are simultaneously applied to the rims, as shown. Rank the objects according to their kinetic energies after a given time $t$, from least to greatest.
[asy]
size(225);
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
filldraw(circle((0,0),1),gray(.7));
draw((0,-1)--(2,-1),EndArrow);
label("$\vec{F}$",(1, -1),S);
label("Disk",(-1,0),W);
filldraw(circle((5,0),1),gray(.7));
filldraw(circle((5,0),0.75),white);
draw((5,-1)--(7,-1),EndArrow);
label("$\vec{F}$",(6, -1),S);
label("Hoop",(6,0),E);
filldraw(circle((10,0),1),gray(.5));
draw((10,-1)--(12,-1),EndArrow);
label("$\vec{F}$",(11, -1),S);
label("Sphere",(11,0),E);
[/asy]
$ \textbf{(A)} \ \text{disk, hoop, sphere}$
$\textbf{(B)}\ \text{sphere, disk, hoop}$
$\textbf{(C)}\ \text{hoop, sphere, disk}$
$\textbf{(D)}\ \text{disk, sphere, hoop}$
$\textbf{(E)}\ \text{hoop, disk, sphere} $
Decide whether there exists a function $f : Z \rightarrow Z$ such that for each $k =0,1, ...,1996$ and for any integer $m$ the equation $f (x)+kx = m$ has at least one integral solution $x$.
Let ${\omega}$ be the circumcircle of an acute-angled triangle ${ABC}$. Point ${D}$ lies on the arc ${BC}$ of ${\omega}$ not containing point ${A}$. Point ${E}$ lies in the interior of the triangle ${ABC}$, does not lie on the line ${AD}$, and satis fies ${\angle DBE =\angle ACB}$ and ${\angle DCE = \angle ABC}$. Let ${F}$ be a point on the line ${AD}$ such that lines ${EF}$ and ${BC}$ are parallel, and let ${G}$ be a point on ${\omega}$ different from ${A}$ such that ${AF = FG}$. Prove that points ${D,E, F,G}$ lie on one circle.
(Slovakia)
A pentagon can be divided into equilateral triangles. Find all the possibilities that the sizes of the angles of this pentagon can be.
Let P be a finite, partially ordered set with one largest element, which is the only upper bound of the set of minimal elements. Prove that any monotonic function $f : P^n\to P$ can be written in the form $g( x_1 , x_2 , ..., x_n , c_1 , ..., c_m )$, where $c_i\in P$ and g is a monotonic, idempotent function. (g is idempotent iff $g(x , x , ..., x) = x\,\forall x\in P$)
For which $n$ can we find positive integers $a_1,a_2,\dots,a_n$ such that
\[
a_1^2+a_2^2+\cdots+a_n^2
\]
is a square?
Let $ABC$ be a triangle and let $P$ be a point in its interior. Denote by $D$, $E$, $F$ the feet of the perpendiculars from $P$ to the lines $BC$, $CA$, $AB$, respectively. Suppose that \[AP^2 + PD^2 = BP^2 + PE^2 = CP^2 + PF^2.\] Denote by $I_A$, $I_B$, $I_C$ the excenters of the triangle $ABC$. Prove that $P$ is the circumcenter of the triangle $I_AI_BI_C$.
[i]Proposed by C.R. Pranesachar, India [/i]
For positive integers $a$ and $b$, an $(a,b)$-shuffle of a deck of $a+b$ cards is any shuffle that preserves the relative order of the top $a$ cards and the relative order of the bottom $b$ cards. Let $n$, $k$, $a_1$, $a_2$, $\dots$, $a_k$, $b_1$, $b_2$, $\dots$, $b_k$ be fixed positive integers such that $a_i+b_i=n$ for all $1\leq i\leq k$. Big Bird has a deck of $n$ cards and will perform an $(a_i,b_i)$-shuffle for each $1\leq i\leq k$, in ascending order of $i$. Suppose that Big Bird can reverse the order of the deck. Prove that Big Bird can also achieve any of the $n!$ permutations of the cards.
[i]Linus Tang[/i]
In the triangle $ABC$, the median $BD$ is drawn and through its midpoint and vertex $A$ the line $\ell$. Thus the triangle $ABC$ is divided into three triangles and one quadrilateral. Determine the areas of these figures if the area of triangle $ABC$ is equal to $S$.
Determine all triplets of integers $(x, y, z)$ that validate the inequality $x^2 + y^2 + z^2 <xy + 3y + 2z$.
[b](a)[/b] The integers $x$, $x^2$ and $x^3$ begin with the same digit. Does it imply that this digit is $1$?
[i]($2$ points) [/i]
[b](b)[/b] The same question for the integers $x, x^2, x^3, \cdots, x^{2015}$
[i]($3$ points)[/i]
.
Let $ \alpha(n)$ be the number of digits equal to one in the binary representation of a positive integer $ n.$ Prove that:
(a) the inequality $ \alpha(n) (n^2 ) \leq \frac{1}{2} \alpha(n)(\alpha(n) + 1)$ holds;
(b) the above inequality is an equality for infinitely many positive integers, and
(c) there exists a sequence $ (n_i )^{\infty}_1$ such that $ \frac{\alpha ( n^2_i )}{\alpha (n_i }$
goes to zero as $ i$ goes to $ \infty.$
[i]Alternative problem:[/i] Prove that there exists a sequence a sequence $ (n_i )^{\infty}_1$ such that $ \frac{\alpha ( n^2_i )}{\alpha (n_i )}$
(d) $ \infty;$
(e) an arbitrary real number $ \gamma \in (0,1)$;
(f) an arbitrary real number $ \gamma \geq 0$;
as $ i$ goes to $ \infty.$
Points on a square with side length $ c$ are either painted blue or red. Find the smallest possible value of $ c$ such that how the points are painted, there exist two points with same color having a distance not less than $ \sqrt {5}$.
$\textbf{(A)}\ \frac {\sqrt {10} }{2} \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ \sqrt {5} \qquad\textbf{(D)}\ 2\sqrt {2} \qquad\textbf{(E)}\ \text{None}$