Found problems: 85335
In a given right triangle $ABC$, the hypotenuse $BC$, of length $a$, is divided into $n$ equal parts ($n$ and odd integer). Let $\alpha$ be the acute angel subtending, from $A$, that segment which contains the mdipoint of the hypotenuse. Let $h$ be the length of the altitude to the hypotenuse fo the triangle. Prove that: \[ \tan{\alpha}=\dfrac{4nh}{(n^2-1)a}. \]
Eight spheres of radius 1, one per octant, are each tangent to the coordinate planes. What is the radius of the smallest sphere, centered at the origin, that contains these eight spheres?
$ \textbf{(A)}\ \sqrt 2\qquad
\textbf{(B)}\ \sqrt 3\qquad
\textbf{(C)}\ 1 \plus{} \sqrt 2\qquad
\textbf{(D)}\ 1 \plus{} \sqrt 3\qquad
\textbf{(E)}\ 3$
The length of each side and each non-principal diagonal of a convex hexagon does not exceed $1$. Prove that this hexagon contains a principal diagonal whose length does not exceed $\frac{2}{\sqrt3}$.
Let $A$ be a set of $2n$ points in the plane, no three of which are collinear. Suppose that $n$ of them are colored red and the remaining $n$ blue. Prove or disprove: there are $n$ closed straight line segments, no two with a point in common, such that the endpoints of each segment are points of $A$ having different colors.
Let $n>2,n \in \mathbb{N}$ and $a>0,a \in \mathbb{R}$ such that $2^a + \log_2 a = n^2$. Prove that: \[ 2 \cdot \log_2 n>a>2 \cdot \log_2 n -\frac{1}{n} . \]
[i]Radu Gologan[/i]
Prove that if $ 0 \leq \alpha < \frac{\pi}{2} $ and $ 0 \leq \beta < \frac{\pi}{2} $, then
$$ tg \frac{\alpha + \beta}{2} \leq \frac{tg \alpha + tg \beta}{2}.$$
Find an integer pair of solutions $(x, y)$ to the following system of equations.
$$\log_2 (y^x) = 16$$
$$\log_2 (x^y) = 8$$
I have a set $A$ containing $n$ distinct integers. This set has the property that if $a,b \in A$, then $12 \nmid |a+b|$ and $12 \nmid |a-b|$. What is the largest possible value of $n$?
In a regular hexagon $ABCDEF$ of side length $8$ and center $K$, points $W$ and $U$ are chosen on $\overline{AB}$ and $\overline{CD}$ respectively such that $\overline{KW} = 7$ and $\angle WKU = 120^{\circ}$. Find the area of pentagon $WBCUK$.
[i]Proposed by Bradley Guo[/i]
Prove that $p(x)=1+x+x^2 +\ldots+x^n$ is reducible over $\mathbb{F}_{2}$ in case $n+1$ is composite. If $n+1$ is prime, is $p(x)$ irreducible over $\mathbb{F}_{2}$ ?
For $2n$ numbers in a row, Bob could perform the following operation:
$$S_i=(a_1,a_2,\ldots,a_{2n})\mapsto S_{i+1}=(a_1,a_3,\ldots,a_{2n-1},a_2,a_4,\ldots,a_{2n}).$$
Let $T$ be the order of this operation. In other words, $T$ is the smallest positive integer such that $S_i=S_{i+T}$. Prove that $T<2n$.
The rectangular table has four rows. The first one contains arbitrary natural numbers (some of them may be equal). The consecutive lines are filled according to the rule: we look through the previous row from left to the certain number $n$ and write the number $k$ if $n$ was met $k$ times. Prove that the second row coincides with the fourth one.
Let $ A,B,C,Q$ be fixed points on plane. $ M,N,P$ are intersection points of $ AQ,BQ,CQ$ with $ BC,CA,AB$. $ D',E',F'$ are tangency points of incircle of $ ABC$ with $ BC,CA,AB$. Tangents drawn from $ M,N,P$ (not triangle sides) to incircle of $ ABC$ make triangle $ DEF$. Prove that $ DD',EE',FF'$ intersect at $ Q$.
Let $ f$ and $ g$ be continuous positive functions defined on the interval $ [0, +\infty)$, and let $ E \subset[0,+\infty)$ be a set of positive measure. Prove that the range of the function defined on $ E \times E$ by the relation \[ F(x,y)= %Error. "dispalymath" is a bad command.
\int_0^xf(t)dt+ %Error. "dispalymath" is a bad command.
\int_0^y g(t)dt\] has a nonvoid interior.
[i]L. Losonczi[/i]
Assume $x_{1},x_{2},\dots,x_{n}\in\mathbb R^{+}$, $\sum_{i=1}^{n}x_{i}^{2}=n$, $\sum_{i=1}^{n}x_{i}\geq s>0$ and $0\leq\lambda\leq1$. Prove that at least $\left\lceil\frac{s^{2}(1-\lambda)^{2}}n\right\rceil$ of these numbers are larger than $\frac{\lambda s}{n}$.
Several children were playing in the ugly tree when suddenly they all fell.
$\bullet$ Roger hit branches $A$, $B$, and $C$ in that order on the way down.
$\bullet$ Sue hit branches $D$, $E$, and $F$ in that order on the way down.
$\bullet$ Gillian hit branches $G$, $A$, and $C$ in that order on the way down.
$\bullet$ Marcellus hit branches $B$, $D$, and $H$ in that order on the way down.
$\bullet$ Juan-Phillipe hit branches $I$, $C$, and $E$ in that order on the way down.
Poor Mikey hit every branch A through $I$ on the way down. Given only this information, in how many different orders could he have hit these 9 branches on the way down?
The fuction $f(0,\infty)\rightarrow\mathbb{R}$ verifies $f(x)+f(y)=2f(\sqrt{xy}), \forall x,y>0$. Show that for every positive integer $n>2$ the following relation takes place $$f(x_1)+f(x_2)+\ldots+f(x_n)=nf(\sqrt[n]{x_1x_2\ldots x_n}),$$ for every positive integers $x_1,x_2,\ldots,x_n$.
From the symmetry center of two congruent intersecting circles, two rays are drawn that intersect the circles at four non-collinear points. Prove that these points lie on one circle.
Let $ABC$ be a triangle and its bisector at $A$ cuts its circumcircle at $D.$ Let $I$ be the incenter of triangle $ABC,$ $M$ be the midpoint of $BC,$ $P$ is the symmetric to $I$ with respect to $M$ (Assuming $P$ is in the circumcircle). Extend $DP$ until it cuts the circumcircle again at $N.$ Prove that among segments $AN, BN, CN$, there is a segment that is the sum of the other two.
We write $\{a,b,c\}$ for the set of three different positive integers $a, b$, and $c$. By choosing some or all of the numbers a, b and c, we can form seven nonempty subsets of $\{a,b,c\}$. We can then calculate the sum of the elements of each subset. For example, for the set $\{4,7,42\}$ we will find sums of $4, 7, 42,11, 46, 49$, and $53$ for its seven subsets. Since $7, 11$, and $53$ are prime, the set $\{4,7,42\}$ has exactly three subsets whose sums are prime. (Recall that prime numbers are numbers with exactly two different factors, $1$ and themselves. In particular, the number $1$ is not prime.)
What is the largest possible number of subsets with prime sums that a set of three different positive integers can have? Give an example of a set $\{a,b,c\}$ that has that number of subsets with prime sums, and explain why no other three-element set could have more.
When Ann meets new people, she tries to find out who is acquainted with who. In order to memorize it she draws a circle in which each person is depicted by a chord; moreover, chords corresponding to acquainted persons intersect (possibly at the ends), while the chords corresponding to non-acquainted persons do not. Ann believes that such set of chords exists for any company. Is her judgement correct? (5)
a) Prove that, for any real $x>0$, it is true that $x^3-3x\ge -2$ .
b) Prove that, for any real $x,y,z>0$, it is true that
$$\frac{x^2y}{z}+\frac{y^2z}{x}+\frac{z^2x}{y}+2\left(\frac{y}{xz}+\frac{z}{xy}+\frac{x}{yz} \right)\ge 9$$ . When we have equality ?
Addison and Emerson are playing a card game with three rounds. Addison has the cards $1, 3$, and $5$, and Emerson
has the cards $2, 4$, and $6$. In advance of the game, both designate each one of their cards to be played for either round one, two, or three. Cards cannot be played for multiple rounds. In each round, both show each other their designated card for that round, and the person with the higher-numbered card wins the round. The person who wins the most rounds wins the game. Let $m/n$ be the probability that Emerson wins, where $m$ and $n$ are relatively prime positive integers. Find $m +n$.
[i]Proposed by Ada Tsui[/i]
$2016$ numbers written on the board: $\frac{1}{2016}, \frac{2}{2016}, \frac{3}{2016}, ..., \frac{2016}{2016}$. In one move, it is allowed to choose any two numbers $a$ and $b$ written on the board, delete them, and write the number $3ab - 2a - 2b + 2$ instead. Determine what number will remain written on the board after $2015$ moves.
Let $n,k$ be positive integers such that n is not divisible by 3 and $k \geq n$. Prove that there exists a positive integer $m$ which is divisible by $n$ and the sum of its digits in decimal representation is $k$.