Found problems: 85335
A bean packing plant has a machine that puts a certain amount of beans into bags and then puts a certain amount of bags into boxes, which are then shipped to customers. One day, the machine broke down and the first n bags came out empty, the next $n$ bags came out with $1$ bean, the next $n$ bags came out with $2$ beans,..., and the last $n$ bags came out with $2006$ beans. To provide each customer with the agreed quantity of bags of beans, the person responsible for the unit intends to distribute the bags among the $2007$ boxes that day so that all boxes contain the same number of bags and all boxes contain the same number. number of beans. For what values of $n$ is this possible?
66 players take part in the chess tournament, each player plays one game against each other, and the games take place in four cities. Prove that three players play all their games in the same city.
Let $ ABCD$ be a regular tetrahedron, and let $ O$ be the centroid of triangle $ BCD$. Consider the point $ P$ on $ AO$ such that $ P$ minimizes $ PA \plus{} 2(PB \plus{} PC \plus{} PD)$. Find $ \sin \angle PBO$.
Higher Secondary P5
Let $x>1$ be an integer such that for any two positive integers $a$ and $b$, if $x$ divides $ab$ then $x$ either divides $a$ or divides $b$. Find with proof the number of positive integers that divide $x$.
The trapezoid $ABCD$, of bases $AB$ and $CD$, is inscribed in a circumference $\Gamma$. Let $X$ a variable point of the arc $AB$ of $\Gamma$ that does not contain $C$ or $D$. We denote $Y$ to the point of intersection of $AB$ and $DX$, and let Z be the point of the segment $CX$ such that $\frac{XZ}{XC}=\frac{AY}{AB}$ . Prove that the measure of $\angle AZX$ does not depend on the choice of $X.$
Given a plane with two circles, one with points $A$ and $B$, and the other with points $C$ and $D$ are shown in the figure. The line $AB$ passes through the center of the first circle and touches the second circle while the line $CD$ passes through the center of the second circle and touches the first circle. Prove that the lines $AD$ and $BC$ are parallel.
[img]https://cdn.artofproblemsolving.com/attachments/e/e/92f7b57751e7828a6487a052d4869e27e658b2.png[/img]
Let $RANDOM$ be a regular hexagon with side length $1.$ Points $I$ and $T$ lie on segments $\overline{RA}$ and $\overline{DO},$ respectively, such that $MI=MT$ and $\angle{TMI}=90^\circ.$ Compute the area of triangle $MIT.$
Given points $O$ and $A$ in the plane. Every point in the plane is colored with one of a finite number of colors. Given a point $X$ in the plane, the circle $C(X)$ has center $O$ and radius $OX+{\angle AOX\over OX}$, where $\angle AOX$ is measured in radians in the range $[0,2\pi)$. Prove that we can find a point $X$, not on $OA$, such that its color appears on the circumference of the circle $C(X)$.
Consider the following graph algorithm (where $V$ is the set of vertices and $E$ the set of edges in $G$).
$\textbf{procedure }\textsc{s}(G)$
$\qquad \textbf{if } |V| = 0\textbf{ then return true}$
$\qquad \textbf{for }(u,v)\textbf{ in }E\textbf{ do}$
$\qquad\qquad H\gets G-u-v$
$\qquad\qquad\textbf{if } \textsc{s}(H)\textbf{ then return true}$
$\qquad\textbf{return false}$
Here $G - u - v$ means the subgraph of $G$ which does not contain vertices $u,v$ and all edges using them. How many graphs $G$ with vertex set $\{1,2,3,4,5,6\}$ and [i]exactly[/i] $6$ edges satisfy $s(G)$ being true?
What is the units digit of $19^{19} + 99^{99}$?
$\text{(A)}\ 0 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 2 \qquad \text{(D)}\ 8 \qquad \text{(E)}\ 9$
Let $a, b, c,$ be nonnegative reals with $ a+b+c=3 $, find the largest positive real $ k $ so that for all $a,b,c,$ we have $$ a^2+b^2+c^2+k(abc-1)\ge 3 $$
Given an integer $a>1$. Prove that there exists a sequence of positive integers
\[ n_1, n_2, n_3, \ldots \]
Such that
\[ \gcd(a^{n_i+1} + a^{n_i} - 1, \ a^{n_j + 1} + a^{n_j} - 1) =1 \] For every $i \neq j$.
Let $a_1,a_2,\ldots a_n,k$, and $M$ be positive integers such that
$$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=k\quad\text{and}\quad a_1a_2\cdots a_n=M.$$
If $M>1$, prove that the polynomial
$$P(x)=M(x+1)^k-(x+a_1)(x+a_2)\cdots (x+a_n)$$
has no positive roots.
Let $N$ be the smallest positive integer divisble by $10^{2023} - 1$ that only has the digits $4$ and $8$ in decimal form (these digits may be repeated). Compute the sum of the digits of $\frac{N}{10^{2023}-1}$ .
On the base $AB$ of the isosceles triangle $ABC$, lies the point $P$ such that $AP : PB = 1 : 2$. Determine the minimum of $\angle ACP$.
Let $ I$ be the incenter of triangle $ ABC$, and let $ A_1$, $ B_1$, $ C_1$ be arbitrary points on the segments $ (AI)$, $ (BI)$, $ (CI)$, respectively. The perpendicular bisectors of $ AA_1$, $ BB_1$, $ CC_1$ intersect each other at $ A_2$, $ B_2$, and $ C_2$. Prove that the circumcenter of the triangle $ A_2B_2C_2$ coincides with the circumcenter of the triangle $ ABC$ if and only if $ I$ is the orthocenter of triangle $ A_1B_1C_1$.
Let $ q \equal{} 2p\plus{}1$, $ p, q > 0$ primes. Prove that there exists a multiple of $ q$ whose digits sum in decimal base is positive and at most $ 3$.
For every natural number $n$ let:
$a_n=ln(1+2e+4e^4+\dots+2ne^{n^2})$. Find:
\[ \displaystyle{\lim_{n \to \infty}\frac{a_n}{n^2}} \].
A pair of positive integers $(m,n)$ is called [i]compatible[/i] if $m \ge \tfrac{1}{2} n + 7$ and $n \ge \tfrac{1}{2} m + 7$. A positive integer $k \ge 1$ is called [i]lonely[/i] if $(k,\ell)$ is not compatible for any integer $\ell \ge 1$. Find the sum of all lonely integers.
[i]Proposed by Evan Chen[/i]
Let $Q(x)$ be a quadratic trinomial. Given that the function $P(x)=x^{2}Q(x)$ is increasing in the interval $(0,\infty )$, prove that:
\[P(x) + P(y) + P(z) > 0\]
for all real numbers $x,y,z$ such that $x+y+z>0$ and $xyz>0$.
Find the sum of all real numbers $x$ such that $x^4-2x^3+3x^2-2x-2014=0$.
Triangle $ABC$ is an isosceles right triangle with $AB=AC=3$. Let $M$ be the midpoint of hypotenuse $\overline{BC}$. Points $I$ and $E$ lie on sides $\overline{AC}$ and $\overline{AB}$, respectively, so that $AI>AE$ and $AIME$ is a cyclic quadrilateral. Given that triangle $EMI$ has area $2$, the length $CI$ can be written as $\frac{a-\sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers and $b$ is not divisible by the square of any prime. What is the value of $a+b+c$?
$
\textbf{(A) }9 \qquad
\textbf{(B) }10 \qquad
\textbf{(C) }11 \qquad
\textbf{(D) }12 \qquad
\textbf{(E) }13 \qquad
$
Call a positive integer an uphill integer if every digit is strictly greater than the previous digit. For example, $1357, 89,\text{and } 5$ are all uphill integers, but $32, 1240, \text{and } 466$ are not. How many uphill integers are divisible by $15$?
$\textbf{(A)}\ 4 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ 8$
There is a set of $2n$ chips of $n$ different colors, two chips of each color. The chips are randomly placed in a row. Prove that the probability that there are two adjacent chips of the same color in a row is greater than $1/2$.
[i]From the folklore[/i]
Let $n \geq 2$ be an integer and $x_1, x_2, \ldots, x_n$ be positive real numbers such that $\sum_{i=1}^nx_i=1$. Show that $$\bigg(\sum_{i=1}^n\frac{1}{1-x_i}\bigg)\bigg(\sum_{1 \leq i < j \leq n}x_ix_j\bigg) \leq \frac{n}{2}.$$