Found problems: 85335
Let $\{a_{n,1},\ldots, a_{n,n} \}_{n=1}^{\infty}$ integers such that $a_{n,i}\neq a_{n,j}$ for $1\le i<j\le n\, , n=2,3,\ldots$ and let $\left\langle y\right\rangle\in [0,1)$ denote the fractional part of the real number $y$. Show that there exists a real sequence $\{ x_n\}_{n=1}^{\infty}$ such that the numbers $\langle a_{n,1}x_n \rangle, \ldots, \langle a_{n,n}x_n \rangle$ are asymptotically uniformly distributed on the interval $[0,1]$.
(translated by L. Erdős)
A polynomial $p(x) = a_0x^k + a_1x^{k-1} + \cdots + a_k$ with integer coefficients is said to be divisible by an integer $m$ if $p(x)$ is divisible by m for all integers $x$. Prove that if $p(x)$ is divisible by $m$, then $k!a_0$ is also divisible by $m$. Also prove that if $a_0, k,m$ are non-negative integers for which $k!a_0$ is divisible by $m$, there exists a polynomial $p(x) = a_0x^k+\cdots+ a_k$ divisible by $m.$
All the numbers $2, 3, 4, 5, 6, 7$ are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. What is the greatest possible value of the sum of these eight products?
$\textbf{(A)}\ 312 \qquad
\textbf{(B)}\ 343 \qquad
\textbf{(C)}\ 625 \qquad
\textbf{(D)}\ 729 \qquad
\textbf{(E)}\ 1680$
Points $A, B$, and $C$ lie on a circle with centre $M$. The reflection of point $M$ in the line $AB$ lies inside triangle $ABC$ and is the intersection of the angle bisectors of angles $A$ and $B$. Line $AM$ intersects the circle again in point $D$.
Show that $|CA| \cdot |CD| = |AB| \cdot |AM|$.
A student wrote down the following sequence of numbers : the first number is 1, the second number is 2, and after that, each number is obtained by adding together all the previous numbers. Determine the 12th number in the sequence.
A circle of radius 2 has center at (2,0). A circle of radius 1 has center at (5,0). A line is tangent to the two circles at points in the first quadrant. Which of the following is closest to the $y$-intercept of the line?
$\text{(A)} \ \sqrt{2}/4 \qquad \text{(B)} \ 8/3 \qquad \text{(C)} \ 1 + \sqrt 3 \qquad \text{(D)} \ 2 \sqrt 2 \qquad \text{(E)} \ 3$
Let $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n$ be positive reals such that
\[a_1 + b_1 = a_2 + b_2 = \ldots + a_n + b_n\]
and
\[\sqrt[n]{\dfrac{a_1a_2\ldots a_n}{b_1b_2\ldots b_n}} \ge n.\]
Prove that
\[\sqrt[n]{\dfrac{a_1a_2\ldots a_n}{b_1b_2\ldots b_n}} \ge \dfrac{a_1+a_2+\ldots+a_n}{b_1+b_2+\ldots+b_n}.\]
All the roots of polynomial $z^6 - 10z^5 + Az^4 + Bz^3 + Cz^2 + Dz + 16$ are positive integers. What is the value of $B$?
$\textbf{(A)}\ -88 \qquad\textbf{(B)}\ -80 \qquad\textbf{(C)}\ -64\qquad\textbf{(D)}\ -41 \qquad\textbf{(E)}\ -40$
Let $ABCD$ be a cyclic quadrilateral, $E = AC \cap BD$, $F = AD \cap BC$. The bisectors of angles $AFB$ and $AEB$ meet $CD$ at points $X, Y$ . Prove that $A, B, X, Y$ are concyclic.
Given that $\frac{x}{\sqrt{x} + \sqrt{y}} = 18$ and $\frac{y}{\sqrt{x} + \sqrt{y}} = 2$, find $\sqrt{x} - \sqrt{y}$.
An integer $N \ge 2$ is given. A collection of $N(N + 1)$ soccer players, no two of whom are of the same height, stand in a row. Sir Alex wants to remove $N(N - 1)$ players from this row leaving a new row of $2N$ players in which the following $N$ conditions hold:
($1$) no one stands between the two tallest players,
($2$) no one stands between the third and fourth tallest players,
$\;\;\vdots$
($N$) no one stands between the two shortest players.
Show that this is always possible.
[i]Proposed by Grigory Chelnokov, Russia[/i]
For each $n\geq 1$ let
$$a_{n}=\sum_{k=0}^{\infty}\frac{k^{n}}{k!}, \;\; b_{n}=\sum_{k=0}^{\infty}(-1)^{k}\frac{k^{n}}{k!}.$$
Show that $a_{n}\cdot b_{n}$ is an integer.
About $5$ years ago, Joydip was researching on the number $2017$. He understood that $2017$ is a prime number. Then he took two integers $a,b$ such that $0<a,b <2017$ and $a+b\neq 2017.$ He created two sequences $A_1,A_2,\dots ,A_{2016}$ and $B_1,B_2,\dots, B_{2016}$ where $A_k$ is the remainder upon dividing $ak$ by $2017$, and $B_k$ is the remainder upon dividing $bk$ by $2017.$ Among the numbers $A_1+B_1,A_2+B_2,\dots A_{2016}+B_{2016}$ count of those that are greater than $2017$ is $N$. Prove that $N=1008.$
In isosceles triangle $ABC$, $AB=AC$. Extend segment $BC$ through $C$ to $P$. Points $X$ and $Y$ lie on lines $AB$ and $AC$, respectively, such that $PX \parallel AC$ and $PY \parallel AB$. Point $T$ lies on the circumcircle of triangle $ABC$ such that $PT \perp XY$. Prove that $\angle BAT = \angle CAT$.
The positive integers are arranged in a row in some order, each occuring exactly once. Does there always exist an adjacent block of at least two numbers somewhere in this row such that the sum of the numbers in the block is a prime number?
At most how many elements does a set have such that all elements are less than $102$ and it doesn't contain the sum of any two elements?
$
\textbf{(A)}\ 49
\qquad\textbf{(B)}\ 50
\qquad\textbf{(C)}\ 51
\qquad\textbf{(D)}\ 54
\qquad\textbf{(E)}\ 62
$
Let $a_1,a_2,...,a_9$ be nine real numbers, not necessarily distinct, with average $m$. Let $A$ denote the number of triples $1 \le i < j < k \le 9$ for which $a_i + a_j + a_k \ge 3m$. What is the minimum possible value of $A$?
[i]Proposed by Ray Li[/i]
Let $a, b, c \ge 0$ so that $ab + bc + ca = 3$. Prove that:
$$\frac{a}{a^2+7}+\frac{b}{b^2+7}+\frac{c}{c^2+7}\le \frac38$$
A rectangle of dimensions $2024 \times 2023$ is filled with pieces of the following types:
[asy]
size(200);
// Figure (A)
draw((0,0)--(4,0)--(4,1)--(0,1)--cycle);
draw((1,0)--(1,1));
draw((2,0)--(2,1));
draw((3,0)--(3,1));
// Figure (B)
draw((6,0)--(8,0)--(8,2)--(6,2)--cycle);
draw((7,0)--(7,2));
draw((6,1)--(8,1));
// Figure (C)
draw((10,0)--(12,0)--(12,1)--(11,1)--(11,2)--(9,2)--(9,1)--(10,1)--cycle);
draw((10,0)--(10,1));
draw((11,0)--(11,1));
draw((10,1)--(11,1));
draw((9,1)--(9,2));
draw((10,1)--(10,2));
draw((11,0)--(12,0));
draw((10,1)--(12,1));
// Labeling
label("(A)", (2, -0.5));
label("(B)", (7, -0.5));
label("(C)", (10.5, -0.5));
[/asy]
Each piece can be turned arround, and each square has side length $1$.
Is it possible to use exactly 2023 pieces of type $(A)$?
Kimia has a weird clock; the clock's second hand moves 34 or 47 seconds forward instead of each regular second, at random. As an example, if the clock displays the time as $\text{12:23:05}$, the following times could be displayed in this order:
$$\text{12:23:39, 12:24:13, 12:25:00, 12:25:34, 12:26:21,\dots}$$
Prove that the clock's second hand would eventually land on a perfect square.
Given triangle $ ABC$. Point $ O$ is the center of the excircle touching the side $ BC$. Point $ O_1$ is the reflection of $ O$ in $ BC$. Determine angle $ A$ if $ O_1$ lies on the circumcircle of $ ABC$.
Richard starts with the string $HHMMMMTT$. A move consists of replacing an instance of $HM$ with $MH$, replacing an instance of $MT$ with $TM$, or replacing an instance of $TH$ with $HT$. Compute the number of possible strings he can end up with after performing zero or more moves.
Let $a_1 = 1$, $a_2 = 1$, and for $n \ge 2$, let $$a_{n+1} =\frac{1}{n} a_n + a_{n-1}.$$ What is $a_{12}$?
A trapezoid $ABCD$ is given where $AB$ and $CD$ are parallel. Let $M$ be the midpoint of the segment $AB$. Point $N$ is located on the segment $CD$ such that $\angle ADN = \frac{1}{2} \angle MNC$ and $\angle BCN = \frac{1}{2} \angle MND$. Prove that $N$ is the midpoint of the segment $CD$.
[i]Proposed by Alireza Dadgarnia[/i]
[b]a)[/b] Show that the solution of the equation $ |z-i|=1 $ in $ \mathbb{C} $ is the set $ \{ 2e^{i\alpha} \sin\alpha |\alpha\in [0,\pi ) \} . $
[b]b)[/b] Let be $ n\ge 1 $ complex numbers $ z_1,z_2,\ldots ,z_n $ that verify the inequalities
$$ \left| z_k-i \right|\le 1,\quad\forall k\in\{ 1,2,\ldots ,n \} . $$
Prove that there exists a complex number $ w $ such that $ |w-i|\le 1 $ and $ w^n=z_1z_2\cdots z_n. $
[i]Dan-Ștefan Marinescu[/i]