Found problems: 85335
Eight people are sitting at a circular table, it is known that any three consecutive people at the table have an odd number of coins (among the three people), show that each person has at least one coin.
Determine the smallest prime that does not divide any five-digit number whose digits are in a strictly increasing order.
Find all functions $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$ such that for all $x, y\in\mathbb{R}^+$,
\[ \frac{f(x)}{y^2} - \frac{f(y)}{x^2} \le \left(\frac{1}{x}-\frac{1}{y}\right)^2\]
($\mathbb{R}^+$ denotes the set of positive real numbers.)
[i](Proposed by Ivan Chan Guan Yu)[/i]
Given a non-right triangle $ABC$ with $BC>AC>AB$. Two points $P_1 \neq P_2$ on the plane satisfy that, for $i=1,2$, if $AP_i, BP_i$ and $CP_i$ intersect the circumcircle of the triangle $ABC$ at $D_i, E_i$, and $F_i$, respectively, then $D_iE_i \perp D_iF_i$ and $D_iE_i = D_iF_i \neq 0$. Let the line $P_1P_2$ intersects the circumcircle of $ABC$ at $Q_1$ and $Q_2$. The Simson lines of $Q_1$, $Q_2$ with respect to $ABC$ intersect at $W$.
Prove that $W$ lies on the nine-point circle of $ABC$.
Let $ABC$ be a triangle with $\angle{BAC} > 90$. Let $D$ be a point on the segment $BC$ and $E$ be a point on line $AD$ such that $AB$ is tangent to the circumcircle of triangle $ACD$ at $A$ and $BE$ is perpendicular to $AD$. Given that $CA=CD$ and $AE=CE$. Determine $\angle{BCA}$ in degrees.
Rectangular sheet of paper $ABCD$ is folded as shown in the figure. Find the rato $DK: AB$, given that $C_1$ is the midpoint of $AD$.
[img]https://3.bp.blogspot.com/-9EkSdxpGnPU/W6dWD82CxwI/AAAAAAAAJHw/iTkEOejlm9U6Dbu427vUJwKMfEOOVn0WwCK4BGAYYCw/s400/Yasinsky%2B2017%2BVIII-IX%2Bp1.png[/img]
Prove for every integer $n>0$, there exists an integer $k>0$ such that $2^nk$ can be written in decimal notation using only digits 1 and 2.
Right triangle $ABC$ has a right angle at $C$. Point $D$ on side $\overline{AB}$ is the base of the altitude of $\triangle ABC$ from $C$. Point $E$ on side $\overline{BC}$ is the base of the altitude of $\triangle CBD$ from $D$. Given that $\triangle ACD$ has area $48$ and $\triangle CDE$ has area $40$, fi nd the area of $\triangle DBE$.
Let $P(x)$ be a polynomial with positive integer coefficients and degree 2015. Given that there exists some $\omega \in \mathbb{C}$ satisfying
$$\omega^{73} = 1\quad \text{and}$$
$$P(\omega^{2015}) + P(\omega^{2015^2}) + P(\omega^{2015^3}) + \ldots + P(\omega^{2015^{72}}) = 0,$$
what is the minimum possible value of $P(1)$?
Let $P(x_1,\dots,x_n)$ denote a polynomial with real coefficients in the variables $x_1,\dots,x_n,$ and suppose that
(a) $\left(\frac{\partial^2}{\partial x_1^2}+\cdots+\frac{\partial^2}{\partial x_n^2} \right)P(x_1,\dots,x_n)=0$ (identically)
and that
(b) $x_1^2+\cdots+x_n^2$ divides $P(x_1,\dots,x_n).$
Show that $P=0$ identically.
For any natural number $n= \overline{a_k...a_1a_0}$ $(a_k \ne 0)$ in decimal system write $p(n)=a_0 \cdot a_1 \cdot ... \cdot a_k$, $s(n)=a_0+ a_1+ ... + a_k$, $n^*= \overline{a_0a_1...a_k}$. Consider $P=\{n | n=n^*, \frac{1}{3} p(n)= s(n)-1\}$ and let $Q$ be the set of numbers in $P$ with all digits greater than $1$.
(a) Show that $P$ is infinite.
(b) Show that $Q$ is finite.
(c) Write down all the elements of $Q$.
Do there exist a positive integer $k$ and a non-constant sequence $a_1, a_2, a_3, ...$ of positive integers such that $a_n = gcd(a_{n+k}, a_{n+k+1})$ for all positive integers $n$?
There are 7 pieces of paper in the hat. On the $ n $th piece of paper there is written the number $ 2^n-1 $ ($ n = 1, 2, \ldots, 7 $). We draw cards randomly until the sum exceeds 124. What is the most probable value of this sum?
Find all functions $f : \mathbb N \to \mathbb N$ satisfying, for all $x \in \mathbb N$, \[ f(2f(x)) = x + 1998 . \]
We consider a function $f:\mathbb R\to\mathbb R$ such that
$$f(x+y)+f(xy-1)=f(x)f(y)+f(x)+f(y)+1$$
for each $x,y\in\mathbb R$.
i) Calculate $f(0)$ and $f(-1)$.
ii) Prove that $f$ is an even function.
iii) Give an example of such a function.
iv) Find all monotone functions with the above property.
[i]Proposed by Mihály Bencze and Marius Drăgan[/i]
Let $D$ and $E$ be points on sides $AB$ and $AC$ of a triangle $ABC$ such that $DB = BC = CE$. The segments $BE$ and $CD$ intersect at point $P$. Prove that the incenter of triangle $ABC$ lies on the circles circumscribed around the triangles $BDP$ and $CEP$.
Let $\triangle ABC$ and $A'$,$B'$,$C'$ the symmetrics of vertex over opposite sides.The intersection of the circumcircles of $\triangle ABB'$ and $\triangle ACC'$ is $A_1$.$B_1$ and $C_1$ are defined similarly.Prove that lines $AA_1$,$BB_1$ and $CC_1$ are concurent.
For some positive integer \( n \), Katya wrote the numbers from \( 1 \) to \( 2^n \) in a row in increasing order. Oleksii rearranged Katya's numbers and wrote the new sequence directly below the first row. Then, they calculated the sum of the two numbers in each column.
Katya calculated \( N \), the number of powers of two among the results, while Oleksii calculated \( K \), the number of distinct powers of two among the results. What is the maximum possible value of \( N + K \)?
[i]Proposed by Oleksii Masalitin[/i]
Let $ABC$ be a triangle with side lengths $AB = 2011$, $BC = 2012$, $AC = 2013$. Create squares $S_1 =ABB'A''$, $S_2 = ACC''A'$ , and $S_3 = CBB''C'$ using the sides $AB$, $AC$, $BC$ respectively, so that the side $B'A''$ is on the opposite side of $AB$ from $C$, and so forth. Let square $S_4$ have side length $A''A' $, square $S_5$ have side length $C''C'$, and square $S_6$ have side length $B''B'$. Let $A(S_i)$ be the area of square $S_i$ . Compute $\frac{A(S_4)+A(S_5)+A(S_6)}{A(S_1)+A(S_2)+A(S_3)}$?
Consider functions $f$ from the whole numbers (non-negative integers) to the whole numbers that have the following properties:
$\bullet$ For all $x$ and $y$, $f(xy) = f(x)f(y)$,
$\bullet$ $f(30) = 1$, and
$\bullet$ for any $n$ whose last digit is $7$, $f(n) = 1$.
Obviously, the function whose value at $n$ is $ 1$ for all $n$ is one such function. Are there any others? If not, why not, and if so, what are they?
How many ordered four-tuples of integers $(a,b,c,d)$ with $0 < a < b < c < d < 500$ satisfy $a + d = b + c$ and $bc - ad = 93$?
Which integers can be represented as \[\frac{(x+y+z)^{2}}{xyz}\] where $x$, $y$, and $z$ are positive integers?
Let $ ABC$ be a triangle in which $ AB\equal{}AC$. Let $ D$ be the midpoint of $ BC$ and $ P$ be a point on $ AD$. Suppose $ E$ is the foot of perpendicular from $ P$ on $ AC$. Define
\[ \frac{AP}{PD}\equal{}\frac{BP}{PE}\equal{}\lambda , \ \ \ \frac{BD}{AD}\equal{}m , \ \ \ z\equal{}m^2(1\plus{}\lambda)\]
Prove that
\[ z^2 \minus{} (\lambda^3 \minus{} \lambda^2 \minus{} 2)z \plus{} 1 \equal{} 0\]
Hence show that $ \lambda \ge 2$ and $ \lambda \equal{} 2$ if and only if $ ABC$ is equilateral.
Find all polynomials $P(x)$ with real coefficients that satisfy \[P(x\sqrt{2})=P(x+\sqrt{1-x^2})\]for all real $x$ with $|x|\le 1$.
During the $2005$ iTest, you will be introduced to Joe and Kathryn, two high school seniors. If $J$ is the number of distinct permutations of $JOE$, and $K$ is the number of distinct permutations of $KATHRYN$, find $K -J$.