Found problems: 85335
Find $x\in\mathbb{R}$ for which
\[\log_2(x^2 + 4) - \log_2x + x^2 - 4x + 2 = 0.\]
Let $q$ be a positive integer. Prove there exists a constant $C_q$ such that the following inequality holds for any finite set $A$ of integers:
\[|A+qA|\ge (q+1)|A|-C_q.\]
[i]Proposed by Antal Balog.[/i]
Let $ABCDEF$ be a convex hexagon such that all of its vertices are on a circle. Prove that $AD$, $BE$ and $CF$ are concurrent if and only if $\frac {AB}{BC}\cdot\frac {CD}{DE}\cdot\frac {EF}{FA}= 1$.
Find all functions $f$ from the set of real numbers into the set of real numbers which satisfy for all $x$, $y$ the identity \[ f\left(xf(x+y)\right) = f\left(yf(x)\right) +x^2\]
[i]Proposed by Japan[/i]
Two players play a game on a pile of $n$ beans. On each player's turn, they may take exactly $1$, $4$, or $7$ beans from the pile. One player goes first, and then the players alternate until somebody wins. A player wins when they take the last bean from the pile. For how many $n$ between $2014$ and $2050$ (inclusive) does the second player win?
For every $x$, $y$, $z>{3\over 2}$ prove the inequality
$$
x^{24} + \root 5\of {y^{60}+z^{40}} \geq
\left(x^4 y^3 + {1\over 3} y^2 z^2 + {1\over 9} x^3 z^3 \right)^2.
$$
The $28$ students of a class are seated in a circle. They then all claim that 'the two students next to me are of different genders'. It is known that all boys are lying while exactly $3$ girls are lying. How many girls are there in the class?
Given six three-element subsets of a finite set $X$, show that it is possible to color the elements of $X$ in two colors so that none of the given subsets is in one color
Show that we cannot find $171$ binary sequences (sequences of $0$’s and $1$’s), each of length $12$ such that any two of them differ in at least four positions.
Triangle $ABC$ has an incenter $I$. Let points $X$, $Y$ be located on the line segments $AB$, $AC$ respectively, so that $BX \cdot AB = IB^2$ and $CY \cdot AC = IC^2$. Given that the points $X, I, Y$ lie on a straight line, find the possible values of the measure of angle $A$.
Consider a table with $n$ rows and $2n$ columns. we put some blocks in some of the cells. After putting blocks in the table we put a robot on a cell and it starts moving in one of the directions right, left, down or up. It can change the direction only when it reaches a block or border. Find the smallest number $m$ such that we can put $m$ blocks on the table and choose a starting point for the robot so it can visit all of the unblocked cells. (the robot can't enter the blocked cells.)
Proposed by Seyed Mohammad Seyedjavadi and Alireza Tavakoli
A quadrilateral $ABCD$ is inscribed, as shown, in a square of area one unit. Prove that $$2\le |AB|^2+|BC|^2+|CD|^2+|DA|^2\le 4$$
[asy]
size(6cm);
draw((0,0)--(10,0));
draw((10,0)--(10,10));
draw((0,10)--(10,10));
draw((0,0)--(0,10));
dot((0,8.5)); dot((3.5,10)); dot((10,3.5)); dot((3.5,0));
label("$D$",(0,8.5),W);
label("$A$",(3.5,10),NE);
label("$B$",(10,3.5),E);
label("$C$",(3.5,0),S);
draw((0,8.5)--(3.5,10));
draw((3.5,10)--(10,3.5));
draw((10,3.5)--(3.5,0));
draw((3.5,0)--(0,8.5));
[/asy]
A group of persons is called [i]good[/i] if its members can be distributed to several rooms so that nobody is acquainted with any person in the same room
but it is possible to choose a person from each room so that all the chosen persons are acquainted with each other.
A group is called [i]perfect[/i] if it is good and every set of its members is also good.
A perfect group planned a party. However one of its members, Alice, brought here acquaintance Bob, who was not originally expected, and introduced him to all her other acquaintances. Prove that the new group is also perfect.
[i]Author: C. Berge[/i]
In triangle $ABC$, $P$ is a point on altitude $AD$. $Q,R$ are the feet of the perpendiculars from $P$ to $AB,AC$, and $QP,RP$ meet $BC$ at $S$ and $T$ respectively. the circumcircles of $BQS$ and $CRT$ meet $QR$ at $X,Y$.
a) Prove $SX,TY, AD$ are concurrent at a point $Z$.
b) Prove $Z$ is on $QR$ iff $Z=H$, where $H$ is the orthocenter of $ABC$.
[i]Ray Li.[/i]
Volume $A$ equals one fourth of the sum of the volumes $B$ and $C$, while volume $B$ equals one sixth of the sum of the volumes $A$ and $C$. There are relatively prime positive integers $m$ and $n$ so that the ratio of volume $C$ to the sum of the other two volumes is $\frac{m}{n}$. Find $m+n$.
Let $S$ denote the sum of all rational numbers of the form $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive divisors of $1300$. If $S$ can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, then find $m+n$.
[i]Proposed by Ephram Chun[/i]
Let $n$ be an integer greater than $2$. Consider a set of $n$ different points, with no three collinear, in the plane. Prove that we can label the points $P_1,~ P_2, \dots , P_n$ such that $P_1P_2 \dots P_n$ is not a self-intersecting polygon. ([i]A polygon is self-intersecting if one of its side intersects the interior of another side. The polygon is not necessarily convex[/i] )
A triangle with sides $a,b,c$ and perimeter $2p$ is given. Is possible, a new triangle with sides $p-a$, $p-b$, $p-c$ is formed. The process is then repeated with the new triangle. For which original triangles can this process be repeated indefinitely?
Find all functions $ f:\mathbb{Q}\longrightarrow\mathbb{R} $ satisfying the equation
$$ f(x+y)+f(x-y)=f(x)+f(y) +f(f(x+y)) , $$
for any rational numbers $ x,y. $
[i]Mihai Onucu Drîmbe[/i]
How many of the numbers $2,6,12,20,\ldots,14520$ are divisible by $120$?
$\text{(A) }2\qquad\text{(B) }8\qquad\text{(C) }12\qquad\text{(D) }24\qquad\text{(E) }32$
Call arrangement of $m$ number on the circle [b]$m$-negative[/b], if all numbers are equal to $-1$. On the first step Andrew chooses one number on circle and multiplies it by $-1$. All other steps are similar: instead of the next number(clockwise) he writes its product with the number, written on the previous step. Prove that if $n$-negative arrangement in $k$ steps becomes $n$-negative again, then $(2^n - 1)$-negative after $(2^k - 1)$ steps becomes $(2^n - 1)$-negative again.
At an elementary school, the students in third grade, fourth grade, and fifth grade run an average of 12, 15, and 10 minutes per day, respectively. There are twice as many third graders as fourth graders, and twice as many fourth graders as fifth graders. What is the average number of minutes run per day by these students?
$ \textbf{(A)}\ 12 \qquad
\textbf{(B)}\ \frac{37}{3} \qquad
\textbf{(C)}\ \frac{88}{7}\qquad
\textbf{(D)}\ 13 \qquad
\textbf{(E)}\ 14$
Let, $a\geq b\geq c >0$ be real numbers such that for all natural number $n$, there exist triangles of side lengths $a^{n} , b^{n} ,c^{n}$.
Prove that the triangles are isosceles.
Let $A$ be a point and let k be a circle through $A$. Let $B$ and $C$ be two more points on $k$. Let $X$ be the intersection of the bisector of $\angle ABC$ with $k$. Let $Y$ be the reflection of $A$ wrt point $X$, and $D$ the intersection of the straight line $YC$ with $k$. Prove that point $D$ is independent of the choice of $B$ and $C$ on the circle $k$.
Given a convex quadrilateral $ABCD$. The medians of the triangle $ABC$ intersect at point $M$, and the medians of the triangle $ACD$ at point$ N$. The circle, circumscibed around the triangle $ACM$, intersects the segment $BD$ at the point $K$ lying inside the triangle $AMB$ . It is known that $\angle MAN = \angle ANC = 90^o$. Prove that $\angle AKD = \angle MKC$.