Found problems: 85335
Given that $a>2b$ and $b>2c$ and $a$, $b$, and $c$ are nonzero, which of the following statements must be true?
$\textbf{(A) } a+b>c\qquad\textbf{(B) } a-c>0\qquad\textbf{(C) } abc>0\qquad\textbf{(D) } \frac{a}{b}>2\qquad\textbf{(E) } \text{none of these}$
Amy places positive integers in each of these cells so each row and column contains each of $1,2,3,4,5$ exactly once. Find the sum of the numbers in the gray cells.
[asy]
import graph; size(4cm);
fill((4,0)--(10,0)--(10,6)--(4,6)--cycle, gray(.7));
draw((0,0)--(10,0));
draw((0,2)--(10,2));
draw((0,4)--(10,4));
draw((0,6)--(10,6));
draw((0,8)--(10,8));
draw((0,10)--(10,10));
draw((0,0)--(0,10));
draw((2,0)--(2,10));
draw((4,0)--(4,10));
draw((6,0)--(6,10));
draw((8,0)--(8,10));
draw((10,0)--(10,10));
label("1",(1,9));
label("2",(3,9));
label("3",(1,7));
label("4",(3,7));
[/asy]
Determine all positive integers $n\ge 3$ such that the inequality
\[a_1a_2+a_2a_3+\ldots a_{n-1}a_n\le 0\]
holds for all real numbers $a_1,a_2,\ldots , a_n$ which satisfy $a_1+a_2+\ldots +a_n=0$.
For any nonnegative reals $x, y$ show the inequality $$x^2y^2 + x^2y + xy^2 \le x^4y + x + y^4$$.
Suppose $JHMT$ is a convex quadrilateral with perimeter $68$ and satisfies $\angle HJT = 120^\circ,$ $HM = 20,$ and $JH + JT = JM > HM.$ Furthermore, $\overrightarrow{JM}$ bisects $\angle HJT.$ Compute $JM.$
Let $n$ be an integer greater than $1$, such that $3n + 1$ is a perfect square. Prove that $n + 1$ can be expressed as a sum of three perfect squares.
Let $n \ge 3$ be an integer. Prove that there exist positive integers $\ge 2$, $a_1,a_2,..,a_n$, such that $a_1 a_2 ... \widehat{a_i}... a_n \equiv 1$ (mod $a_i$), for $i = 1,..., n$. Here $\widehat{a_i}$ means the term $a_i$ is omitted.
The price of a diamond is proportional to the square of its weight. Show that, breaking it into two parts, there is a depreciation of its value. When is it the maximum depreciation?
What is the surface area of a cube with volume $64$?
Consider the recursive sequence defined by $a_{n+1}=a_n^n+1,$ with $a_1=0.$ What is the last digit of $a_{2024}$?
Let $a_1,a_2,a_3, \ldots$ be a sequence of positive integers and let $b_1,b_2,b_3,\ldots$ be the sequence of real numbers given by
$$b_n = \dfrac{a_1a_2\cdots a_n}{a_1+a_2+\cdots + a_n},\ \mbox{for}\ n\geq 1$$
Show that, if there exists at least one term among every million consecutive terms of the sequence $b_1,b_2,b_3,\ldots$ that is an integer, then there exists some $k$ such that $b_k > 2021^{2021}$.
$20$ players enter a chess tournament in which each player will play every other player exactly once. Some competitors are cheaters and will cheat in every game they play, but the rest of the competitors are not cheaters. A game is cheating if both players cheat, and a game is half-cheating if one player cheats and one player does not. If there were $68$ more half-cheating games than cheating games, how many of the players are cheaters?
Show that there exist integers $m$ and $n$ such that \[\dfrac{m}{n}=\sqrt[3]{\sqrt{50}+7}-\sqrt[3]{\sqrt{50}-7}.\]
Let $n$ be a positive integer and $(x_1, x_2, ..., x_{2n})$, $x_i = 0$ or $1, i = 1, 2, ... , 2n$ be a sequence of $2n$ integers. Let $S_n$ be the sum $S_n = x_1x_2 + x_3x_4 + ... + x_{2n-1}x_{2n}$.
If $O_n$ is the number of sequences such that $S_n$ is odd and $E_n$ is the number of sequences such that $S_n$ is even, prove that $$\frac{O_n}{E_n}=\frac{2^n - 1}{2^n + 1}$$
On a line $n+1$ segments are marked such that one of the points of the line is contained in all of them. Prove that one can find $2$ distinct segments $I, J$ which intersect at a segment of length at least $\frac{n-1}{n}d$, where $d$ is the length of the segment $I$.
Using each of the eight digits $1,3,4,5,6,7,8$ and $9$ exactly once, a three-digit number $A$, two two-digit numbers $B$ and $C$, $B<C$, and a one digit number $D$ are formed. The numbers are such that $A+D=B+C=143$. In how many ways can this be done?
Determine the largest $3$ digit prime number that is a factor of ${2000 \choose 1000}$.
Determine all integers $m \geq 2$ such that every $n$ with $\frac{m}{3} \leq n \leq \frac{m}{2}$ divides the binomial coefficient $\binom{n}{m-2n}$.
The number of solution-pairs in the positive integers of the equation $3x+5y=501$ is:
$\textbf{(A)}\ 33\qquad
\textbf{(B)}\ 34\qquad
\textbf{(C)}\ 35\qquad
\textbf{(D)}\ 100\qquad
\textbf{(E)}\ \text{none of these}$
If you were to randomly select an answer to this question, what is the probability it would be correct?
$\textbf{(A) }0\%\qquad\textbf{(B) }20\%\qquad\textbf{(C) }40\%\qquad\textbf{(D) }80\%\qquad\textbf{(E) }100\%$
Prove that there are infinitely many positive integers $ n$ such that $ n(n\plus{}1)$ can be represented as a sum of two positive squares in at least two different ways. (Here $ a^{2}\plus{}b^{2}$ and $ b^{2}\plus{}a^{2}$ are considered as the same representation.)
$(HUN 4)$IMO2 If $a_1, a_2, . . . , a_n$ are real constants, and if $y = \cos(a_1 + x) +2\cos(a_2+x)+ \cdots+ n \cos(a_n + x)$ has two zeros $x_1$ and $x_2$ whose difference is not a multiple of $\pi$, prove that $y = 0.$
For a positive integer $x$, denote its $n^{\mathrm{th}}$ decimal digit by $d_n(x)$, i.e. $d_n(x)\in \{ 0,1, \dots, 9\}$ and $x=\sum_{n=1}^{\infty} d_n(x)10^{n-1}$. Suppose that for some sequence $(a_n)_{n=1}^{\infty}$, there are only finitely many zeros in the sequence $(d_n(a_n))_{n=1}^{\infty}$. Prove that there are infinitely many positive integers that do not occur in the sequence $(a_n)_{n=1}^{\infty}$.
(Proposed by Alexander Bolbot, State University, Novosibirsk)
The tangent lines from a point $P$ meet a circle $k$ at $A$ and $B$. Let $X$ be an arbitrary point on the shorter arc $AB$, and $C$ and $D$ be the orthogonal projections of $P$ onto the lines $AX$ and $BX$, respectively. Prove that the line $CD$ passes through a fixed point $Y$ as $X$ moves along the arc $AB$.
Let $n$ be an integer greater than $1$, and let $a_1$, $a_2$, ..., $a_n$ be not all identical positive integers. Prove that there are infinitely many primes $p$ such that $p$ divides $a_1^k+a_2^k+...+a_n^k$ for some positive integer $k$.