Found problems: 85335
A cube side $5$ is made up of unit cubes. Two small cubes are [i]adjacent [/i] if they have a common face. Can we start at a cube adjacent to a corner cube and move through all the cubes just once? (The path must always move from a cube to an adjacent cube).
Let a real number $k$ and points $S,A,SA=1$ in plane be given. Denote $A'$ the image of $A$ under rotation by an oriented angle $\varphi$ with respect to center $S$. Similarly, let $A''$ be the image of $A'$ under homothety with the factor $\frac{1}{\cos\varphi-k\sin\varphi}$ with respect to center $S.$ Denote the locus \[\ell=\bigl\{A''\mid\varphi\in(-\pi,\pi],\cos\varphi-k\sin\varphi\neq0\bigr\}.\] Show that $\ell$ is a line containing $A.$
Let $a$ and $b$ be two positive real numbers such that $a+b=ab$.
Prove that $\frac{a}{b^{2}+4}+\frac{b}{a^{2}+4}\geq \frac{1}{2}$.
Let $H$ be the orthocenter of an acute-angled triangle A$BC$. The perpendicular bisector to segment $BH$ meets $BA$ and $BC$ at points $A_0, C_0$ respectively. Prove that the perimeter of triangle $A_0OC_0$ ($O$ is the circumcenter of triangle $ABC$) is equal to $AC$.
Given a triangle $ABC$ with $AB>BC$, $ \Omega $ is circumcircle. Let $M$, $N$ are lie on the sides $AB$, $BC$ respectively, such that $AM=CN$. $K(.)=MN\cap AC$ and $P$ is incenter of the triangle $AMK$, $Q$ is K-excenter of the triangle $CNK$ (opposite to $K$ and tangents to $CN$). If $R$ is midpoint of the arc $ABC$ of $ \Omega $ then prove that $RP=RQ$.
M. Kungodjin
Let the medians of the triangle $ABC$ intersect at point $M$. A line $t$ through $M$ intersects the circumcircle of $ABC$ at $X$ and $Y$ so that $A$ and $C$ lie on the same side of $t$. Prove that $BX\cdot BY=AX\cdot AY+CX\cdot CY$.
What is the area of the region bounded by the graphs of $y = |x + 2| - |x - 2|$ and $y = |x + 1| - |x - 3|$?
Does there exist a $4$-digit integer which cannot be changed into a multiple of $1992$ by changing $3$ of its digits?
Let $a$ and $b$ be positive integers with $a>b$. Suppose that
$$\sqrt{\sqrt{a}+\sqrt{b}}+\sqrt{\sqrt{a}-\sqrt{b}}$$
is an integer.
(a) Must $\sqrt{a}$ be an integer?
(b) Must $\sqrt{b}$ be an integer?
Find all right triangles with integer side lengths in which two congruent circles with prime radius can be inscribed such that they are externally tangent, both touch the hypotenuse, and each is tangent to another leg of the right triangle.
In the parallelogram $ABCD$, the length of side $AB$ is half the length of side $BC$. The bisector of angle $\angle ABC$ intersects side $AD$ at point $K$ and diagonal $AC$ at point $L$. The bisector of angle $\angle ADC$ intersects the extension of side $AB$ at point $M$, with $B$ between $A$ and $M$. The line $ML$ intersects side $AD$ at point $F$. Calculate the ratio $\frac{AF}{AD}$.
Find all functions $f:\mathbb{Z}\rightarrow\mathbb{C}$ such that $f(x(2y+1))=f(x(y+1))+f(x)f(y)$ holds for any two integers $x,y$.
Let $x,y,z$ be positive real numbers such that $xyz=1$. Prove that
$\frac{1}{(x+1)^2+y^2+1}$ $+$ $\frac{1}{(y+1)^2+z^2+1}$ $+$ $\frac{1}{(z+1)^2+x^2+1}$ $\leq$ ${\frac{1}{2}}$.
Suppose that $m$ and $k$ are positive integers. Determine the number of sequences $x_1, x_2, x_3, \dots , x_{m-1}, x_m$ with
[list]
[*]$x_i$ an integer for $i = 1, 2, 3, \dots , m$,
[*]$1\le x_i \le k$ for $i = 1, 2, 3, \dots , m$,
[*]$x_1\neq x_m$, and
[*]no two consecutive terms equal.[/list]
Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that
$$x^2f(x)+yf(y^2)=f(x+y)f(x^2-xy+y^2)$$
for all $x,y\in\mathbb{R}$.
How many nonnegative integers can be written in the form $$a_7\cdot3^7+a_6\cdot3^6+a_5\cdot3^5+a_4\cdot3^4+a_3\cdot3^3+a_2\cdot3^2+a_1\cdot3^1+a_0\cdot3^0,$$
where $a_i\in \{-1,0,1\}$ for $0\le i \le 7$?
$\textbf{(A) } 512 \qquad
\textbf{(B) } 729 \qquad
\textbf{(C) } 1094 \qquad
\textbf{(D) } 3281 \qquad
\textbf{(E) } 59,048 $
For how many integer $ n$, $ P \equal{} n^4 \plus{} 4n^3 \plus{} 3n^2 \minus{} 2n \plus{} 7$ is prime?
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{Infinitely many}$
How many of the numbers
\[
a_1\cdot 5^1+a_2\cdot 5^2+a_3\cdot 5^3+a_4\cdot 5^4+a_5\cdot 5^5+a_6\cdot 5^6
\]
are negative if $a_1,a_2,a_3,a_4,a_5,a_6 \in \{-1,0,1 \}$?
$
\textbf{(A)}\ 121
\qquad\textbf{(B)}\ 224
\qquad\textbf{(C)}\ 275
\qquad\textbf{(D)}\ 364
\qquad\textbf{(E)}\ 375
$
The sequence $a_1,a_2…$ , is defined by the equations $a_1=1$ and $a_n=n.a_{[n/2]}$ for $n>1$. Prove that $a_n>n^2$ for $n>11$.
Express $\frac{2^3-1}{2^3+1}\times\frac{3^3-1}{3^3+1}\times\frac{4^3-1}{4^3+1}\times\dots\times\frac{16^3-1}{16^3+1}$ as a fraction in lowest terms.
There exists a unique triple of integers $(B,M, T)$ such that $B > T > M$ and $$3B^2(3T -M) + 8M^2(B - T) + 3T^2(5M - B) - (2B^3 + 3M^3 + 4T^3) + 15BMT = 2023.$$ Compute $B +M + T$.
If $s$ and $d$ are positive integers such that $\frac{1}{s} + \frac{1}{2s} + \frac{1}{3s} = \frac{1}{d^2 - 2d},$ then the smallest possible value of $s + d$ is
$ \mathrm{(A) \ } 6 \qquad \mathrm{(B) \ } 8 \qquad \mathrm {(C) \ } 10 \qquad \mathrm{(D) \ } 50 \qquad \mathrm{(E) \ } 96$
Let $a,b,c$ be positive real numbers. Prove that
\[(a+b)(b+c)(c+a)\ge 8(a+b-c)(b+c-a)(c+a-b)\]
and determine when equality occurs.
A wobbly number is a positive integer whose digits are alternately zero and non-zero with the last digit non-zero (for example, 201). Find all positive integers which do not divide any wobbly number.
The sequence $1, 2, \dots, 2023, 2024$ is written on a whiteboard. Every second, Megavan chooses two integers $a$ and $b$, and four consecutive numbers on the whiteboard. Then counting from the left, he adds $a$ to the 1st and 3rd of those numbers, and adds $b$ to the 2nd and 4th of those numbers. Can he achieve the sequence $2024, 2023, \dots, 2, 1$ in a finite number of moves?
[i](Proposed by Avan Lim Zenn Ee)[/i]