Found problems: 85335
Prove that no number of the form $ 2^n $, where $ n $ is a natural number, is the sum of two or more consecutive natural numbers.
For $n\in\mathbb{N}$, $n\geq 2$, $a_{i}, b_{i}\in\mathbb{R}$, $1\leq i\leq n$, such that \[\sum_{i=1}^{n}a_{i}^{2}=\sum_{i=1}^{n}b_{i}^{2}=1, \sum_{i=1}^{n}a_{i}b_{i}=0. \] Prove that
\[\left(\sum_{i=1}^{n}a_{i}\right)^{2}+\left(\sum_{i=1}^{n}b_{i}\right)^{2}\leq n. \]
[i]Cezar Lupu & Tudorel Lupu[/i]
Let $X, Y$ , and $Z$ be concentric circles with radii $1$, $13$, and $22$, respectively. Draw points $A, B$, and $C$ on $X$, $Y$ , and $Z$, respectively, such that the area of triangle $ABC$ is as large as possible. If the area of the triangle is $\Delta$, find $\Delta^2$.
Circles $W_1,W_2$ meet at $D$and $P$. $A$ and $B$ are on $W_1,W_2$ respectively, such that $AB$ is tangent to $W_1$ and $W_2$. Suppose $D$ is closer than $P$ to the line $AB$. $AD$ meet circle $W_2$ for second time at $C$. Let $M$ be the midpoint of $BC$. Prove that $\angle{DPM}=\angle{BDC}$.
Does there exist a bijection $f:\mathbb{N}^{+} \rightarrow \mathbb{N}^{+}$, such that there exist a positive integer $k$, and it's possible to have each positive integer colored by one of $k$ chosen colors, such that for any $x \neq y$ , $f(x)+y$ and $f(y)+x$ are not the same color?
Nora drove $82$ miles in $90$ minutes. She averaged $50$ miles per hour for the fi rst half-hour and averaged $55$ miles per hour for the last half-hour. What was her average speed in miles per hour over the middle half-hour (during the $30$ minutes beginning after the fi rst half-hour)?
The positive integers are partitioned into two infinite sets so that the sum of any $2023$ distinct integers in one set is also in that set. Prove that one set contains all the odd positive integers, and one set contains all the even positive integers.
[i]Proposed by Evan Chang (squareman), USA[/i]
For all positive numbers $x,y,z$ the product $(x+y+z)^{-1}(x^{-1}+y^{-1}+z^{-1})(xy+yz+xz)^{-1}[(xy)^{-1}+(yz)^{-1}+(xz)^{-1}]$ equals
$\text{(A)}\ x^{-2}y^{-2}z^{-2} \qquad \text{(B)}\ x^{-2}+y^{-2}+z^{-2} \qquad \text{(C)}\ (x+y+z)^{-1}$
$\text{(D)}\ \frac{1}{xyz} \qquad \text{(E)}\ \frac{1}{xy+yz+xz}$
$(a)$ Let $a_1,a_2,a_3$ be three real numbers. Prove that
$(a_1-a_2)(a_1-a_3)+(a_2-a_1)(a_2-a_3)+(a_3-a_1)(a_2-a_2)\geq 0$.
$(b)$ Prove that the inequality above doesn't hold if we use four number instead of three.
$ABC$ is a triangle, $G$ its centroid and $A',B',C'$ the midpoints of its sides $BC,CA,AB$, respectively. Prove that if the quadrilateral $AC'GB'$ is cyclic then $AB \cdot CC' = AC \cdot BB'$:
Let $f:[1,\infty)\to(0,\infty)$ be a continuous function. Assume that for every $a>0$, the equation $f(x)=ax$ has at least one solution in the interval $[1,\infty)$.
(a) Prove that for every $a>0$, the equation $f(x)=ax$ has infinitely many solutions.
(b) Give an example of a strictly increasing continuous function $f$ with these properties.
From vertex $A$ in square $ABCD$ (of side length $1$ ) two lines are drawn , one intersecting side $BC$ and the other intersecting side $CD$. The angle between these lines is $\theta$. From vertices $B$ and $D$ we construct perpendiculars to each of these lines . Find the area of the quadrilateral whose vertices are the four feet of these perpendiculars.
A cryptographer devises the following method for encoding positive integers. First, the integer is expressed in base $5$. Second, a 1-to-1 correspondence is established between the digits that appear in the expressions in base $5$ and the elements of the set $\{V, W, X, Y, Z\}$. Using this correspondence, the cryptographer finds that three consecutive integers in increasing order are coded as $VYZ$, $VYX$, $VVW$, respectively. What is the base-10 expression for the integer coded as $XYZ$?
$ \textbf{(A)}\ 48 \qquad\textbf{(B)}\ 71 \qquad\textbf{(C)}\ 82 \qquad\textbf{(D)}\ 108 \qquad\textbf{(E)}\ 113$
There are $2012$ backgammon checkers (stones, pieces) with one side is black and the other side is white.
These checkers are arranged into a line such that no two consequtive checkers are in same color. At each move, we are chosing two checkers. And we are turning upside down of the two checkers and all of the checkers between the two. At least how many moves are required to make all checkers same color?
$ \textbf{(A)}\ 1006 \qquad \textbf{(B)}\ 1204 \qquad \textbf{(C)}\ 1340 \qquad \textbf{(D)}\ 2011 \qquad \textbf{(E)}\ \text{None}$
The vertex $A$, center $O$ and Euler line $\ell$ of a triangle $ABC$ is given. It is known that $\ell$ intersects $AB,AC$ at two points equidistant from $A$. Restore the triangle.
Let $n,k$ be positive integers such that $n^k>(k+1)!$ and consider the set
\[ M=\{(x_1,x_2,\ldots,x_n)\dvd x_i\in\{1,2,\ldots,n\},\ i=\overline{1,k}\}. \]
Prove that if $A\subset M$ has $(k+1)!+1$ elements, then there are two elements $\{\alpha,\beta\}\subset A$, $\alpha=(\alpha_1,\alpha_2,\ldots,\alpha_n)$, $\beta=(\beta_1,\beta_2,\ldots,\beta_n)$ such that
\[ (k+1)! \left| (\beta_1-\alpha_1)(\beta_2-\alpha_2)\cdots (\beta_k-\alpha_k) \right. .\]
Suppose we climb a mountain that is a cone with radius $100$ and height $4$. We start at the bottom of the mountain (on the perimeter of the base of the cone), and our destination is the opposite side of the mountain, halfway up (height $z = 2$). Our climbing speed starts at $v_0=2$ but gets slower at a rate inversely proportional to the distance to the mountain top (so at height $z$ the speed $v$ is $(h-z)v_0/h$). Find the minimum time needed to get to the destination.
In an acute triangle $ABC$ the points $D, E$ are the feet of the altitudes through $B, C$ respectively. Let $L$ be the point on segment $BD$ such that $AD=DL$. Similarly, let $K$ be the point on segment $CE$ such that $AE=EK$. Let $M$ be the midpoint of $KL$. The circumcircle of $ABC$ intersect the lines $AL$ and $AK$ for a second time at $T, S$ respectively. Prove that the lines $BS, CT, AM$ are concurrent.
Determine all $6$-tuples $(x,y,z,u,v,w)$ of integers satisfying the equation
\[x^3+7y^3+49z^3=2u^3+14v^3+98w^3.\]
Let real $a$, $b$, and $c$ satisfy $$abc+a+b+c=ab+bc+ca+5.$$ Find the least possible value of $a^2+b^2+c^2$.
Triangle $ABC$ is not isosceles. The incircle of $\triangle ABC$ touches the sides $BC$, $CA$, $AB$ in the points $K$, $L$, $M$. The parallel with $LM$ through $B$ meets $KL$ at $D$, the parallel with $LM$ through $C$ meets $KM$ at $E$.
Prove that $DE$ passes through the midpoint of $\overline{LM}$.
ِDo there exist non-zero reals numbers $a_1, a_2, ....., a_{10}$ for which \[(a_1+\frac{1}{a_1})(a_2+\frac{1}{a_2}) \cdots(a_{10}+\frac{1}{a_{10}})= (a_1-\frac{1}{a_1})(a_2-\frac{1}{a_2})\cdots(a_{10}-\frac{1}{a_{10}}) \ ? \]
Let $g: R \to R$ be a linear function such that $g (1) = 0$. If $f: R \to R$ is a quadratic function such what $g (x^2) = f (x)$ and $f (x + 1) - f (x - 1) = x$ for all $x \in R$. Determine the value of $f (2019)$.
If a right triangle is drawn in a semicircle of radius $1/2$ with one leg (not the hypotenuse) along the diameter, what is the triangle's maximum possible area?
The arithmetic mean (ordinary average) of the fifty-two successive positive integers beginning with $2$ is:
$\textbf{(A) }27\qquad
\textbf{(B) }27\tfrac14\qquad
\textbf{(C) }27\tfrac12\qquad
\textbf{(D) }28\qquad
\textbf{(E) }28\tfrac12$