Found problems: 85335
On a blackboard, there are $10$ numbers written: $1, 2, \dots, 10$. Nahyun repeatedly performs the following operations.
[b](Operation)[/b] Nahyun chooses two numbers from the 10 numbers on the blackboard that are not in a divisor-multiple relationship, erases them, and writes their GCD and LCM on the blackboard.
If every two numbers on the blackboard form a divisor-multiple relationship, Nahyun stops the process. What is the maximum number of operations Nahyun can perform?
(Note: $a, b$ are in a divisor-multiple relationship iff $a \mid b$ or $b \mid a$.)
Let $f(x)=x^2-3x+1,$ and let $\alpha_1, \alpha_2, \alpha_3, \alpha_4$ be the $4$ roots of $f(f(x))=x.$ Evaluate $\lfloor 10\alpha_1\rfloor+ $ $\lfloor 10\alpha_2\rfloor$ $+$ $\lfloor 10\alpha_3\rfloor+\lfloor 10\alpha_4\rfloor.$
Find all $\mathbb Q$-linear maps $\Phi:\mathbb Q[x]\to\mathbb Q[x]$ such that for any irreducible polynomial $p\in\mathbb Q[x]$ the polynomial $\Phi(p)$ is also irreducible.
For a natural number $n$, $f(n)$ is defined as the number of positive integers less than $n$ which are neither coprime to $n$ nor a divisor of it. Prove that for each positive integer $k$ there exist only finitely many $n$ satisfying $f(n) = k$.
For a finite set $C$ of integer numbers, we define $S(C)$ as the sum of the elements of $C$. Find two non-empty sets $A$ and $B$ whose intersection is empty, whose union is the set $\{1,2,\ldots, 2021\}$ and such that the product $S(A)S(B)$ is a perfect square.
Given $n$ students in the plane such that the $\frac{n(n-1)}{2}$ distances are pairwise distinct. Each student gives a candy each to the $k$ students closest to him. Given that each student receives the same amount of candies, determine all possible values of $n$ in terms of $k$.
[i]Proposed by Wong Jer Ren[/i]
Let $a,b$ and $c$ be positive real numbers. Prove that \[ \frac{a^2b(b-c)}{a+b}+\frac{b^2c(c-a)}{b+c}+\frac{c^2a(a-b)}{c+a} \ge 0. \]
The sign shown below consists of two uniform legs attached by a frictionless hinge. The coefficient of friction between the ground and the legs is $\mu$. Which of the following gives the maximum value of $\theta$ such that the sign will not collapse?
$\textbf{(A) } \sin \theta = 2 \mu \\
\textbf{(B) } \sin \theta /2 = \mu / 2\\
\textbf{(C) } \tan \theta / 2 = \mu\\
\textbf{(D) } \tan \theta = 2 \mu \\
\textbf{(E) } \tan \theta / 2 = 2 \mu$
A large golden square land lot of dimension $100 \times 100$ m was subdivided into $100$ square lots, each measured $10\times10$ m. A king of landfill had his men dump wastes onto some of the lots. There was a practice that if a particular lot was not dumped and twoof its adjacents had waste materials, then the lot would be filled with wastes the next day by the people. One day if all the lotswere filled with wastes, the king would claim his ownership ofthe whole land lot. At least how many lots should have the kind had his men dump wastes onto?
Vu Ha Van, Mathematics Faculty, Yale University, USA.
Find the value of $S_n= \arctan \frac 12 + \arctan \frac 18+ \arctan \frac {1}{18} + \cdots + \arctan \frac {1}{2n^2}.$ Also find $\lim_{n \to \infty} S_n.$
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 $7$ ?
In the cyclic quadrilateral $ABCD$, the diagonal $BD$ is divided in half by the diagonal $AC$. The points $E, F, G$ and $H{}$ are the midpoints of the sides $AB, BC, CD{}$ and $DA$ respectively. Let $P = AD \cap BC$ and $Q = AB \cap CD{}$. The bisectors of the angles $APC$ and $AQC$ intersect the segments $EG$ and $FH$ at the points $X{}$ and $Y{}$ respectively. Prove that $XY \parallel BD$.
Find all positive integer solutions $x,y,z$ such that $1/x +2/y - 3/z=1$
find all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that
$f(-f(x)-f(y)) = 1-x-y$ $\quad \forall x,y \in \mathbb{Z}$
Consider $2009$ cards, each having one gold side and one black side, lying on parallel on a long table. Initially all cards show their gold sides. Two player, standing by the same long side of the table, play a game with alternating moves. Each move consists of choosing a block of $50$ consecutive cards, the leftmost of which is showing gold, and turning them all over, so those which showed gold now show black and vice versa. The last player who can make a legal move wins.
(a) Does the game necessarily end?
(b) Does there exist a winning strategy for the starting player?
[i]Proposed by Michael Albert, Richard Guy, New Zealand[/i]
A square and equilateral triangle have the same perimeter. If the triangle has area $16\sqrt3$, what is the area of the square?
[i]Proposed by Evan Chen[/i]
Prove that for any natural number $n > 1$, \[ \frac{1}{2} < \frac{1}{n^2+1} + \frac{2}{n^2 +2} + \ldots + \frac{n}{n^2 + n} < \frac{1}{2} + \frac{1}{2n}. \]
Does there exist a pair $(g,h)$ of functions $g,h:\mathbb{R}\rightarrow\mathbb{R}$ such that the only function $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfying $f(g(x))=g(f(x))$ and $f(h(x))=h(f(x))$ for all $x\in\mathbb{R}$ is identity function $f(x)\equiv x$?
In unit cube $ABCDEFGH$ (with faces $ABCD$, $EFGH$ and connecting vertices labeled so that $\overline{AE}$, $\overline{BF}$, $\overline{CG}$, $\overline{DH}$ are edges of the cube), $L$ is the midpoint of $GH$. The area of $\vartriangle CAL$ can be written in the form $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.
Let be given a real number $\alpha\ne0$. Show that there is a unique point $P$ in the coordinate plane, such that for every line through $P$ which intersects the parabola $y=\alpha x^2$ in two distinct points $A$ and $B$, segments $OA$ and $OB$ are perpendicular (where $O$ is the origin).
A $\pm 1$-[i]sequence[/i] is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$
In the expansion of $(ax+b)^{2000},$ where $a$ and $b$ are relatively prime positive integers, the coefficients of $x^{2}$ and $x^{3}$ are equal. Find $a+b.$
The pentagon $ABCDE$ is inscirbed in a circle $w$. Suppose that $w_a,w_b,w_c,w_d,w_e$ are reflections of $w$ with respect to sides $AB,BC,CD,DE,EA$ respectively. Let $A'$ be the second intersection point of $w_a,w_e$ and define $B',C',D',E'$ similarly. Prove that
\[2\le \frac{S_{A'B'C'D'E'}}{S_{ABCDE}}\le 3,\]
where $S_X$ denotes the surface of figure $X$.
[i]Proposed by Morteza Saghafian, Ali khezeli[/i]
Alice and Bob stand atop two different towers in the Arctic. Both towers are a positive integer number of meters tall and are a positive (not necessarily integer) distance away from each other. One night, the sea between them has frozen completely into reflective ice. Alice shines her flashlight directly at the top of Bob's tower, and Bob shines his flashlight at the top of Alice's tower by first reflecting it off the ice. The light from Alice's tower travels $16$ meters to get to Bob's tower, while the light from Bob's tower travels $26$ meters to get to Alice's tower. Assuming that the lights are both shown from exactly the top of their respective towers, what are the possibilities for the height of Alice's tower?
The sum of two numbers is $ S$. Suppose 3 is added to each number and then each of the resulting numbers is doubled. What is the sum of the final two numbers?
$ \textbf{(A)} \ 2S \plus{} 3 \qquad \textbf{(B)} \ 3S \plus{} 2 \qquad \textbf{(C)} \ 3S \plus{} 6 \qquad \textbf{(D)} \ 2S \plus{} 6 \qquad \textbf{(E)} \ 2S \plus{} 12$