Found problems: 85335
Find the largest natural number $n$ such that for all real numbers $a, b, c, d$ the following holds:
$$(n + 2)\sqrt{a^2 + b^2} + (n + 1)\sqrt{a^2 + c^2} + (n + 1)\sqrt{a^2 + d^2} \ge n(a + b + c + d)$$
Show that there is some rational number in the interval $(0,1)$ that can be expressed as a sum of $2021$ reciprocals of positive integers, but cannot be expressed as a sum of $2020$ reciprocals of positive integers.
Suppse that $n\geq 2$ and $0<x_1<x_2<...<x_n$ are integer numbers. We denote that :\[ S_k=\sum_{A\subset \{x_1,x_2,...,x_n\}} \frac{1}{\prod_{a\in A}a} , k=1,2,...,n. \] (where $A$ is a non-empty subset). Show that if $S_n ,S_{n-1}$ were positive integer numbers , then $\forall k : S_k$ is a positive integer.
Find all pairs $(x,y)$ of positive integers such that $x(x+y)=y^2+1.$
In the triangle $ABC$, let $A'$ be the intersection of the perpendicular bisector of $AB$ and the angle bisector of $\angle BAC$ and define $B', C'$ analogously. Prove that
a) The triangle $ABC$ is equilateral if and only if $A' =B'.$
b) If $A', B'$ and $C'$ are distinct, we have $\angle B' A' C' = 90^{\circ} - \frac{1}{2} \angle BAC.$
Let $p$ be an odd prime. The numbers $1, 2, \ldots, d$ are written on a blackboard, where $d \geq p-1$ is a positive integer. A valid operation is to delete two numbers $x$ and $y$ and write $x + y - c \cdot xy$ in their place, where $c$ is a positive integer. One moment there is only one number $A$ left on the board. Show that if there is an order of operations such that $p$ divides $A$, then $p | d$ or $p | d + 1$.
Calculate the number of ways of choosing $4$ numbers from the set ${1,2,\cdots ,11}$ such that at least $2$ of the numbers are consecutive.
Compute the number of ordered pairs of positive integers $(a, b)$ satisfying the equation
$$gcd(a, b) \cdot a + b^2 = 10000.$$
The square $ABCD$ has side length $1$. The point $E$ lies on the side $CD$. The line through $A$ and $E$ intersects the line through $B$ and $C$ at the point $F$. Prove that $$\frac{1}{|AE|^2}+\frac{1}{|AF|^2}= 1.$$
[img]https://cdn.artofproblemsolving.com/attachments/5/8/4e803eb7748f7a72783065717044cfc06f565f.png[/img]
Show that there do not exist positive integers $m$ and $n$ such that \[ \dfrac{m}{n} + \dfrac{n+1}{m} = 4 . \]
Let $CX, CY$ be the tangents from vertex $C$ of triangle $ABC$ to the circle passing through the midpoints of its sides. Prove that lines $XY , AB$ and the tangent to the circumcircle of $ABC$ at point $C$ concur.
Let $n$ be a natural number. We define sequences $\langle a_i\rangle$ and $\langle b_i\rangle$ of integers as follows. We let $a_0=1$ and $b_0=n$. For $i>0$, we let $$\left( a_i,b_i\right)=\begin{cases} \left(2a_{i-1}+1,b_{i-1}-a_{i-1}-1\right) & \text{if } a_{i-1}<b_{i-1},\\
\left( a_{i-1}-b_{i-1}-1,2b_{i-1}+1\right) & \text{if } a_{i-1}>b_{i-1},\\
\left(a_{i-1},b_{i-1}\right) & \text{if } a_{i-1}=b_{i-1}.\end{cases}$$
Given that $a_k=b_k$ for some natural number $k$, prove that $n+3$ is a power of two.
Let $ABC$ be an acute triangle such that $AC<BC$ and $\omega$ its circumcircle. $M$ is the midpoint of $BC$. Points $F$ and $E$ are chosen in $AB$ and $BC$, respectively, such that $AC=CF$ and $EB=EF$. The line $AM$ intersects $\omega$ in $D\neq A$. The line $DE$ intersects the line $FM$ in $G$. Prove that $G$ lies on $\omega$.
Let $k$ be a positive integer. Determine the least positive integer $n$, with $n\geq k+1$, for which the game below can be played indefinitely:
Consider $n$ boxes, labelled $b_1,b_2,...,b_n$. For each index $i$, box $b_i$ contains exactly $i$ coins. At each step, the following three substeps are performed in order:
[b](1)[/b] Choose $k+1$ boxes;
[b](2)[/b] Of these $k+1$ boxes, choose $k$ and remove at least half of the coins from each, and add to the remaining box, if labelled $b_i$, a number of $i$ coins.
[b](3)[/b] If one of the boxes is left empty, the game ends; otherwise, go to the next step.
[i]Proposed by Demetres Christofides, Cyprus[/i]
What is the largest positive integer $n$ for which there is a unique integer $k$ such that $\frac{8}{15} < \frac{n}{n + k} < \frac{7}{13}$?
Let $BB',CC'$ be altitudes of $\triangle ABC$ and assume $AB$ ≠ $AC$.Let $M$ be the midpoint of $BC$ and $H$ be orhocenter of $\triangle ABC$ and $D$ be the intersection of $BC$ and $B'C'$.Show that $DH$ is perpendicular to $AM$.
Kathy has $5$ red cards and $5$ green cards. She shuffles the $10$ cards and lays out $5$ of the cards in a row in a random order. She will be happy if and only if all the red cards laid out are adjacent and all the green cards laid out are adjacent. For example, card orders RRGGG, GGGGR, or RRRRR will make Kathy happy, but RRRGR will not. The probability that Kathy will be happy is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Solve the following equation in integers:
$4n^4+7n^2+3n+6=m^3$.
Let $B = (-1, 0)$ and $C = (1, 0)$ be fixed points on the coordinate plane. A nonempty, bounded subset $S$ of the plane is said to be [i]nice[/i] if
$\text{(i)}$ there is a point $T$ in $S$ such that for every point $Q$ in $S$, the segment $TQ$ lies entirely in $S$; and
$\text{(ii)}$ for any triangle $P_1P_2P_3$, there exists a unique point $A$ in $S$ and a permutation $\sigma$ of the indices $\{1, 2, 3\}$ for which triangles $ABC$ and $P_{\sigma(1)}P_{\sigma(2)}P_{\sigma(3)}$ are similar.
Prove that there exist two distinct nice subsets $S$ and $S'$ of the set $\{(x, y) : x \geq 0, y \geq 0\}$ such that if $A \in S$ and $A' \in S'$ are the unique choices of points in $\text{(ii)}$, then the product $BA \cdot BA'$ is a constant independent of the triangle $P_1P_2P_3$.
Find the minimum value of $m + n$, where $m$ and $n$ are positive integers satisfying:
$2023 \vert m + 2025n$
$2025 \vert m + 2023n$
[i]Proposed by Prudencio Guerrero Fernández [/i]
For constant $k>1$, 2 points $X,\ Y$ move on the part of the first quadrant of the line, which passes through $A(1,\ 0)$ and is perpendicular to the $x$ axis, satisfying $AY=kAX$. Let a circle with radius 1 centered on the origin $O(0,\ 0)$ intersect with line segments $OX,\ OY$ at $P,\ Q$ respectively. Express the maximum area of $\triangle{OPQ}$ in terms of $k$.
[i]2011 Tokyo Institute of Technology entrance exam, Problem 3[/i]
The squares in a $7\times7$ grid are colored one of two colors: green and purple. The coloring has the property that no green square is directly above or to the right of a purple square. Find the total number of ways this can be done.
Nine cards can be numbered using positive half-integers $(1/2, 1, 3/2, 2, 5/2, \dots )$ so that the sum of the numbers on a randomly chosen pair of cards gives an integer from $2$ to $12$ with the same frequency of occurrence as rolling that sum on two standard dice. What are the numbers on the nine cards and how often does each number appear on the cards?
Let
\[f = X^{n}+a_{n-1}X^{n-1}+\ldots+a_{1}X+a_{0}\]
be an integer polynomial of degree $n \geq 3$ such that $a_{k}+a_{n-k}$ is even for all $k \in \overline{1,n-1}$ and $a_{0}$ is even.
Suppose that $f = gh$, where $g,h$ are integer polynomials and $\deg g \leq \deg h$ and all the coefficients of $h$ are odd.
Prove that $f$ has an integer root.
On the domain $0\leq \theta \leq 2\pi:$
(a) Prove that $\sin^{2}\theta \cdot \sin 2\theta$ takes its maximum at $\frac{\pi}{3}$ and $\frac{4 \pi}{3}$ (and hence its minimum at $\frac{2 \pi}{3}$ and $\frac{ 5 \pi}{3}$).
(b) Show that
$$| \sin^{2} \theta \cdot \sin^{3} 2\theta \cdot \sin^{3} 4 \theta \cdots \sin^{3} 2^{n-1} \theta \cdot \sin 2^{n} \theta |$$
takes its maximum at $\frac{4 \pi}{3}$ (the maximum may also be attained at other points).
(c) Derive the inequality:
$$ \sin^{2} \theta \cdot \sin^{2} 2\theta \cdots \sin^{2} 2^{n} \theta \leq \left( \frac{3}{4} \right)^{n}.$$