Found problems: 85335
$2.$ Let $ABC$ be a triangle and $AD$ the angle bisector ($D\in BC$). The perpendicular from $B$ to $AD$ cuts the circumcircle of triangle $ABD$ at $E$. If $O$ is the center of the circle around $ABC$ , prove $A,O,E$ are collinear.
[hide]https://artofproblemsolving.com/community/c6h605458p3596629
https://artofproblemsolving.com/community/c6h1294020p6857833[/hide]
Does there exist a subset $E$ of the set $N$ of all positive integers such that none of the elements in $E$ can be presented as a sum of at least two other (not necessarily distinct) elements from $E$ ?
(E. Barabanov)
$O$ -circumcenter of $ABCD$. $AC$ and $BD$ intersect in $E$, $AD$ and $BC$ in $F$. $X,Y$ - midpoints of $AD$ and $BC$. $O_1$ -circumcenter of $EXY$. Prove that $OF \parallel O_1E$
Construct a point $Q$ in triangle $ABC$ such that at least two of the segments $CQ, BQ, AQ$, divide the inscribed circle in half. For which triangles is this possible?
Determine all functions $ f : \mathbb{R} \to \mathbb{R} $ continuous at zero and such that for every real number $ x $ the equality holds $$ 2f(2x) = f(x) + x.$$
Let $P \in \mathbb{R}[x]$. Suppose that the multiset of real roots (where roots are counted with multiplicity) of $P(x)-x$ and $P^3(x)-x$ are distinct. Prove that for all $n\in \mathbb{N}$, $P^n(x)-x$ has at least $\sigma(n)-2$ distinct real roots.
(Here $P^n(x):=P(P^{n-1}(x))$ with $P^1(x) = P(x)$, and $\sigma(n)$ is the sum of all positive divisors of $n$).
[i]Proposed by Malay Mahajan[/i]
How many ordered triples $(a,b,c)$ of integers satisfy the inequality \[a^2+b^2+c^2 \leq a+b+c+2?\]
Let $T = TNYWR$. David rolls a standard $T$-sided die repeatedly until he first rolls $T$, writing his rolls in order on a chalkboard. What is the probability that he is able to erase some of the numbers he's written such that all that's left on the board are the numbers $1, 2, \dots, T$ in order?
Is it true that if a function $f: R \to R$ is continuous and takes rational values at rational points,
then at least at one point it is differentiable?
On a board there are $n$ nails, each two connected by a rope. Each rope is colored in one of $n$ given distinct colors. For each three distinct colors, there exist three nails connected with ropes of these three colors.
a) Can $n$ be $6$ ?
b) Can $n$ be $7$ ?
Prove that a quadratic trinomial $x^2 + ax + b (a, b \in R)$ cannot attain at ten consecutive integral points values equal to powers of $2$ with non-negative integral exponent.
[i](F. Petrov )[/i]
Let $\{a_n\}_{n\geq 0}$ be an arithmetic sequence with difference $d$ and $1\leq a_0\leq d$. Denote the sequence as $S_0$, and define $S_n$ recursively by two operations below:
Step $1$: Denote the first number of $S_n$ as $b_n$, and remove $b_n$.
Step $2$: Add $1$ to the first $b_n$ numbers to get $S_{n+1}$.
Prove that there exists a constant $c$ such that $b_n=[ca_n]$ for all $n\geq 0$, where $[]$ is the floor function.
The decimal representation of an integer uses only two different digits. The number is at least $10$ digits long, and any two neighbouring digits are distinct. What is the greatest power of two that can divide this number?
$ a$ is a real number. $ x_1$ and $ x_2$ are the distinct roots of $ x^2 \plus{} ax \plus{} 2 \equal{} x$. $ x_3$ and $ x_4$ are the distinct roots of $ (x \minus{} a)^2 \plus{} a(x \minus{} a) \plus{} 2 \equal{} x$. If $ x_3 \minus{} x_1 \equal{} 3(x_4 \minus{} x_2)$, then $ x_4 \minus{} x_2$ will be ?
$\textbf{(A)}\ \frac {a}{2} \qquad\textbf{(B)}\ \frac {a}{3} \qquad\textbf{(C)}\ \frac {2a}{3} \qquad\textbf{(D)}\ \frac {3a}{2} \qquad\textbf{(E)}\ \text{None}$
In a triangle with sides of lengths $a,b,$ and $c,$ $(a+b+c)(a+b-c) = 3ab.$ The measure of the angle opposite the side length $c$ is
$\displaystyle \text{(A)} \ 15^\circ \qquad \text{(B)} \ 30^\circ \qquad \text{(C)} \ 45^\circ \qquad \text{(D)} \ 60^\circ \qquad \text{(E)} \ 150^\circ$
Let $ABC$ be a right-angled triangle with right-angle at $A$. Let $X$ be the foot of the perpendicular from $A$ to $BC$, and $Y$ the mid-point of $XC$. Let $AB$ be extended to $D$ so that $|AB|=|BD|$. Prove that $DX$ is perpendicular to $AY$.
Seongcheol has $3$ red shirts and $2$ green shirts, such that he cannot tell the difference between his three red shirts and he similarly cannot tell the difference between his two green shirts. In how many ways can he hang them in a row in his closet, given that he does not want the two green shirts next to each other?
Consider two coprime integers $p{}$ and $q{}$ which are greater than $1{}$ and differ from each other by more than $1{}$. Prove that there exists a positive integer $n{}$ such that \[\text{lcm}(p+n, q+n)<\text{lcm}(p,q).\]
JHMT Pizzeria messed up ordering boxes to put their $10$ inch pizza in. ($10$ inch pizza means the diameter of the pizza is $10$ inches) They accidentally ordered an $8\, in \times 8\, in$. box and they immediately need to deliver $10$ inch pizzas to customers, and they decide to cut the pizza minimally so that the most part of the pizza ts in to the $8\, in \times 8\, in$ box. The area they need to cut out from the original $10$ inch pizza can be written in form of $\alpha \arccos( \beta ) - \gamma$ . Find the value of $\alpha \beta \gamma$, where $\alpha$ and $\gamma$ are integers and $\beta$ is a rational number strictly between $\frac12$ and $1$.
How many distinct triangles can be drawn using three of the dots below as vertices?
[asy]dot(origin^^(1,0)^^(2,0)^^(0,1)^^(1,1)^^(2,1));[/asy]
$ \textbf{(A)}\ 9\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 20\qquad\textbf{(E)}\ 24 $
Given positive integers $a,b,c$ which are pairwise relatively prime, show that \[2abc-ab-bc-ac \] is the biggest number that can't be expressed in the form $xbc+yca+zab$ with $x,y,z$ being natural numbers.
Prove that for every integer $n\ge 3$ there are such positives integers $x$ and $y$ such that $2^n = 7x^2 + y^2$
Let $n$ be an integer greater than $1$. Define
\[x_1 = n, y_1 = 1, x_{i+1} =\left[ \frac{x_i+y_i}{2}\right] , y_{i+1} = \left[ \frac{n}{x_{i+1}}\right], \qquad \text{for }i = 1, 2, \ldots\ ,\]
where $[z]$ denotes the largest integer less than or equal to $z$. Prove that
\[ \min \{x_1, x_2, \ldots, x_n \} =[ \sqrt n ]\]
Given are positive integers $ n>1$ and $ a$ so that $ a>n^2$, and among the integers $ a\plus{}1, a\plus{}2, \ldots, a\plus{}n$ one can find a multiple of each of the numbers $ n^2\plus{}1, n^2\plus{}2, \ldots, n^2\plus{}n$. Prove that $ a>n^4\minus{}n^3$.
Let $a + bi$ and $c + di$ be two roots of the equation $x^n = 1990$, where $n \geq 3$ is an integer and $a,b,c,d \in \mathbb R$. Under the linear transformation $f =\left(\begin{array}{cc}a&c\\b &d\end{array}\right)$, we have $(2, 1) \to (1, 2)$. Denote $r$ to be the distance from the image of $(2, 2)$ to the origin. Find the range of $r.$
Show that the only real solution to
\[\left\{ \begin{array}{l}
x(x+y)^2 = 9 \\
x(y^3 - x^3) = 7 \\
\end{array} \right.
\]
is $x = 1$, $y = 2$.