Found problems: 85335
Find the pairs of integers $(a, b)$ such that $a^2 + 2b^2 + 2a +1$ is a divisor of $2ab$.
Let $c$ be a nonnegative integer, and define $a_n = n^2 + c$ (for $n \geq 1)$. Define $d_n$ as the greatest common divisor of $a_n$ and $a_{n + 1}$.
(a) Suppose that $c = 0$. Show that $d_n = 1,\ \forall n \geq 1$.
(b) Suppose that $c = 1$. Show that $d_n \in \{1,5\},\ \forall n \geq 1$.
(c) Show that $d_n \leq 4c + 1,\ \forall n \geq 1$.
Let $ABCD$ be a rectangle. Let $K,L$ be the midpoints of $BC, AD$ respectively. From point $B$ the perpendicular line on $AK$, intersects $AK$ at point $E$ and $CL$ at point $Z$.
a) Prove that the quadrilateral $AKZL$ is an isosceles trapezoid
b) Prove that $2S_{ABKZ}=S_{ABCD}$
c) If quadrilateral $ABCD$ is a square of side $a$, calculate the area of the isosceles trapezoid $AKZL$ in terms of side $BC=a$
In the land of Microbablia the alphabet has only two letters, ‘A’ and ‘B’. Not surprisingly, the inhabitants are obsessed with the band ABBA. Words in the local dialect with a high ABBA-factor are considered particularly lucky. To compute the ABBA-factor of a word you just count the number of occurrences of ABBA within the word (not necessarily consecutively). So for instance AABA has ABBA-factor $0$, ABBA has ABBA-factor $1$, AABBBA has ABBA-factor $6$, and ABBABBA has ABBA factor $8$. What is the greatest possible ABBA-factor for a $100$ letter word?
[Help me] Determine the smallest value of the sum M =xy-yz-zx where x; y; z are real numbers satisfying the following condition $x^2+2y^2+5z^2 = 22$.
A circle is circumscribed around triangle $ABC$ and a circle is inscribed in it, which touches the sides of the triangle $BC,CA,AB$ at points $A_1,B_1,C_1$, respectively. The line $B_1C_1$ intersects the line $BC$ at the point $P$, and $M$ is the midpoint of the segment $PA_1$. Prove that the segments of the tangents drawn from the point $M$ to the inscribed and circumscribed circle are equal.
Prove that among any $n$ positive integers one can always find some (at least one) whose sum is divisible by $n$.
The line $12x+5y=60$ forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle?
$\textbf{(A) } 20
\qquad\textbf{(B) } \dfrac{360}{17}
\qquad\textbf{(C) } \dfrac{107}{5}
\qquad\textbf{(D) } \dfrac{43}{2}
\qquad\textbf{(E) } \dfrac{281}{13}
$
Given a scalene triangle $\vartriangle ABC$ has orthocenter $H$ and circumcircle $\Omega$. The tangent lines passing through $A,B,C$ are $\ell_a,\ell_b,\ell_c$. Suppose that the intersection of $\ell_b$ and $\ell_c$ is $D$. The foots of $H$ on $\ell_a,AD$ are $P,Q$ respectively. Prove that $PQ$ bisects segment $BC$.
[img]https://4.bp.blogspot.com/-iiQoxMG8bEs/XnYNK7R8S3I/AAAAAAAALeY/FYvSuF6vQQsofASnXJUgKZ1T9oNnd-02ACK4BGAYYCw/s400/imoc2019g3.png[/img]
Prove that
\[ab+bc+ca\ge 2(a+b+c)\]
where $a,b,c$ are positive reals such that $a+b+c+2=abc$.
Find the value of
$(1+2)(1+2^2)(1+2^4)(1+2^8)...(1+2^{2048})$.
A $n+1$-tuple $\left(h_1,h_2, \cdots, h_{n+1}\right)$ where $h_i\left(x_1,x_2, \cdots , x_n\right)$ are $n$ variable polynomials with real coefficients is called [i]good[/i] if the following condition holds:
For any $n$ functions $f_1,f_2, \cdots ,f_n : \mathbb R \to \mathbb R$ if for all $1 \le i \le n+1$, $P_i(x)=h_i \left(f_1(x),f_2(x), \cdots, f_n(x) \right)$ is a polynomial with variable $x$, then $f_1(x),f_2(x), \cdots, f_n(x)$ are polynomials.
$a)$ Prove that for all positive integers $n$, there exists a [i]good[/i] $n+1$-tuple $\left(h_1,h_2, \cdots, h_{n+1}\right)$ such that the degree of all $h_i$ is more than $1$.
$b)$ Prove that there doesn't exist any integer $n>1$ that for which there is a [i]good[/i] $n+1$-tuple $\left(h_1,h_2, \cdots, h_{n+1}\right)$ such that all $h_i$ are symmetric polynomials.
[i]Proposed by Alireza Shavali[/i]
A sequence \( a_1, a_2, \ldots \) of real numbers satisfies the following condition: for every positive integer \( k \geq 2 \), there exists a positive integer \( i < k \) such that \( a_i + a_k = k \). It is known that for some \( j \), the fractional parts of the numbers \( a_j \) and \( a_{j+1} \) are equal. Prove that for some positive integers \( x \neq y \), the equality
\[
a_x - a_y = x - y
\]
holds.
[i]The fractional part of a real number \( a \) is defined as the number \( \{a\} \in [0, 1) \), which satisfies the condition \( a = n + \{a\} \), where \( n \) is an integer. For example, \( \{-3\} = 0 \), \( \{3.14\} = 0.14 \), and \( \{-3.14\} = 0.86 \).[/i]
[i]Proposed by Mykhailo Shtandenko[/i]
Let $ABC$ be a triangle whose inscribed circle is $\omega$. Let $r_1$ be the line parallel to $BC$ and tangent to $\omega$, with $r_1 \ne BC$ and let $r_2$ be the line parallel to $AB$ and tangent to $\omega$ with $r_2 \ne AB$. Suppose that the intersection point of $r_1$ and $r_2$ lies on the circumscribed circle of triangle $ABC$. Prove that the sidelengths of triangle $ABC$ form an arithmetic progression.
Show that for any positive integers $a$ and $b$, $(36a+b)(a+36b)$ cannot be a power of $2$.
Let $k$ be a positive real number. Suppose that the set of real numbers $x$ such that $x^2+k|x| \leq 2019$ is an interval of length $6$. Compute $k$.
[i]Proposed by Luke Robitaille[/i]
In convex quadrilateral $ ABCD$, $ CB,DA$ are external angle bisectors of $ \angle DCA,\angle CDB$, respectively. Points $ E,F$ lie on the rays $ AC,BD$ respectively such that $ CEFD$ is cyclic quadrilateral. Point $ P$ lie in the plane of quadrilateral $ ABCD$ such that $ DA,CB$ are external angle bisectors of $ \angle PDE,\angle PCF$ respectively. $ AD$ intersects $ BC$ at $ Q.$ Prove that $ P$ lies on $ AB$ if and only if $ Q$ lies on segment $ EF$.
Let $a$ and $b$ be real numbers such that $a^5b^8=12$ and $a^8b^{13}=18$. Find $ab$.
Suppose $\tan \alpha = \dfrac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. Prove that the number $\tan \beta$ for which $\tan {2 \beta} = \tan {3 \alpha}$ is rational only when $p^2 + q^2$ is the square of an integer.
In this problem, we consider integers consisting of $5$ digits, of which the rst and last one are nonzero. We say that such an integer is a palindromic product if it satis es the following two conditions:
- the integer is a palindrome, (i.e. it doesn't matter if you read it from left to right, or the other way around);
- the integer is a product of two positive integers, of which the fi rst, when read from left to right, is equal to the second, when read from right to left, like $4831$ and $1384$.
For example, $20502$ is a palindromic product, since $102 \cdot 201 = 20502$, and $20502$ itself is a palindrome.
Determine all palindromic products of $5$ digits.
Two mathematicians discuss two positive integers. One of them states that the square of the ratio between their product and their sum is exactly one more than this ratio. What is the smaller of these two numbers?
Let $ABC$ be a triangle such that $AB = 26, AC = 30,$ and $BC = 28$. Let $C'$ and $B'$ be the reflections of the circumcenter $O$ over $AB$ and $AC$, respectively. The length of the portion of line segment $B'C'$ inside triangle $ABC$ can be written as $\frac{p}{q}$, where $p,q$ are relatively prime positive integers. Compute $p+q$.
Show that in a right-angled triangle the bisector of the right angle divides into equal parts the angle between the altitude and the median, drawn from the same vertex.
A sequence of $28$ letters consists of $14$ of each of the letters $A$ and $B$ arranged in random order. The expected number of times that $ABBA$ appears as four consecutive letters in that sequence is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Let $n$ be a positive integer of the form $4k+1$, $k\in \mathbb N$ and $A = \{ a^2 + nb^2 \mid a,b \in \mathbb Z\}$. Prove that there exist integers $x,y$ such that $x^n+y^n \in A$ and $x+y \notin A$.