Found problems: 85335
Find all pairs $(m, n)$ of integers for which $$\sqrt{m^2-6}<2\sqrt{n}-m<\sqrt{m^2-2}.$$
It is given that the number $4^{11}+1$ is divisible by some prime greater than $1000$. Determine this prime.
[i] Proposed by David Altizio [/i]
Consider in a plane $ P$ the points $ O,A_1,A_2,A_3,A_4$ such that \[ \sigma(OA_iA_j) \geq 1 \quad \forall i, j \equal{} 1, 2, 3, 4, i \neq j.\] where $ \sigma(OA_iA_j)$ is the area of triangle $ OA_iA_j.$ Prove that there exists at least one pair $ i_0, j_0 \in \{1, 2, 3, 4\}$ such that \[ \sigma(OA_iA_j) \geq \sqrt{2}.\]
The right triangles $ABC$ and $AB_1C_1$ are similar and have opposite orientation. The right angles are at $C$ and $C_1$, and we also have $ \angle CAB = \angle C_1AB_1$. Let $M$ be the point of intersection of the lines $BC_1$ and $B_1C$. Prove that if the lines $AM$ and $CC_1$ exist, they are perpendicular.
Given $P$ a point m inside a triangle acute-angled $ABC$ and $DEF$ intersections of lines with that $AP$, $BP$, $CP$ with$\left[ BC \right],\left[ CA \right],$respective $\left[ AB \right]$
a) Show that the area of the triangle $DEF$ is at most a quarter of the area of the triangle $ABC$
b) Show that the radius of the circle inscribed in the triangle $DEF$ is at most a quarter of the radius of the circle circumscribed on triangle $4ABC.$
Consider the acute-angled triangle $ ABC$, altitude $ AD$ and point $ E$ - intersection of $ BC$ with diameter from $ A$ of circumcircle. Let $ M,N$ be symmetric points of $ D$ with respect to the lines $ AC$ and $ AB$ respectively. Prove that $ \angle{EMC} \equal{} \angle{BNE}$.
Find all composite numbers $n$ having the property that each proper divisor $d$ of $n$ has $n-20 \le d \le n-12$.
Six points on a circle are given. Four of the chords joining pairs of the six points are selected at random. What is the probability that the four chords are the sides of a convex quadrilateral?
$ \textbf{(A)}\ \frac{1}{15}\qquad
\textbf{(B)}\ \frac{1}{91}\qquad
\textbf{(C)}\ \frac{1}{273}\qquad
\textbf{(D)}\ \frac{1}{455}\qquad
\textbf{(E)}\ \frac{1}{1365}$
Given a polynomial $p(x)$ with real coefficients, we denote by $S(p)$ the sum of the squares of its coefficients. For example $S(20x+ 21)=20^2+21^2=841$.
Prove that if $f(x)$, $g(x)$, and $h(x)$ are polynomials with real coefficients satisfying the indentity $f(x) \cdot g(x)=h(x)^ 2$, then $$S(f) \cdot S(g) \ge S(h)^2$$
[i]Proposed by Bhavya Tiwari[/i]
One hundred students at Century High School participated in the AHSME last year, and their mean score was $100$. The number of non-seniors taking the AHSME was $50\%$ more than the number of seniors, and the mean score of the seniors was $50\%$ higher than that of the non-seniors. What was the mean score of the seniors?
$ \textbf{(A)}\ 100\qquad\textbf{(B)}\ 112.5\qquad\textbf{(C)}\ 120\qquad\textbf{(D)}\ 125\qquad\textbf{(E)}\ 150 $
Let $ABC$ be an arbitrary triangle. Let $E,E$ be two points on $AB,AC$ respectively such that their distance to the midpoint of $BC$ is equal. Let $P$ be the second intersection of the triangles $ABC,AEF$ circumcircles . The tangents from $E,F$ to the circumcircle of $AEF$ intersect each other at $K$. Prove that : $\angle KPA = 90$
A subset of $\left\{1,2,3,\ldots,2017,2018\right\}$ has the property that none of its members are $5$ times another. What is the maximum number of elements that such a subset could have?
[i]2018 CCA Math Bonanza Lightning Round #4.2[/i]
Show that there exists a tournament $ (T,\rightarrow)$ of cardinality $ \aleph_1$ containing no transitive subtournament of size $ \aleph_1$. ( A structure $ (T,\rightarrow)$ is a $ \textit{tournament}$ if $ \rightarrow$ is a binary, irreflexive, asymmetric and trichotomic relation. The tournament $ (T,\rightarrow)$ is transitive if $ \rightarrow$ is transitive, that is, if it orders $ T$.)
[i]A. Hajnal[/i]
For a given natural number $n > 3$, the real numbers $x_1, x_2, \ldots, x_n, x_{n + 1}, x_{n + 2}$ satisfy the conditions $0
< x_1 < x_2 < \cdots < x_n < x_{n + 1} < x_{n + 2}$. Find the minimum possible value of
\[\frac{(\sum _{i=1}^n \frac{x_{i + 1}}{x_i})(\sum _{j=1}^n \frac{x_{j + 2}}{x_{j +
1}})}{(\sum _{k=1}^n \frac{x_{k + 1} x_{k + 2}}{x_{k + 1}^2 + x_k
x_{k + 2}})(\sum _{l=1}^n \frac{x_{l + 1}^2 + x_l x_{l + 2}}{x_l
x_{l + 1}})}\] and find all $(n + 2)$-tuplets of real numbers $(x_1, x_2, \ldots, x_n, x_{n + 1}, x_{n + 2})$ which gives this value.
A $3\times2\times2$ right rectangular prism has one of its edges with length $3$ replaced with an edge of length $5$ parallel to the original edge. The other $11$ edges remain the same length, and the $6$ vertices that are not endpoints of the replaced edge remain in place. The resulting convex solid has $8$ faces, as shown below.
Find the volume of the solid.
[i]Proposed by Justin Hsieh[/i]
There exist integers $a$ and $b$ such that $(1 +\sqrt2)^{12}= a + b\sqrt2$. Compute the remainder when $ab$ is divided by $13$.
Let $a, b, c$ be positive real numbers such that $abc=8$. Prove that
\[
\frac{a^2}{\sqrt{(1+a^3)(1+b^3)}} +\frac{b^2}{\sqrt{(1+b^3)(1+c^3)}} +\frac{c^2}{\sqrt{(1+c^3)(1+a^3)}} \geq \frac{4}{3}
\]
Prove that there exist infinitely many positive integers $d$ such that we can find a polynomial $P\in\mathbb{Z}[x]$ of degree $d$ and $N\in\mathbb{N}$ such that for all integers $x>N$ and any prime $p$, we have $$\nu_p(P(x)^3+3P(x)^2-3)<\frac{d\cdot\log(x)}{2024^{2024}}.$$
[i]Proposed by Marius Cerlat[/i]
Point \( H \) is the orthocenter of the acute triangle \( ABC \), and \( AD \) is its altitude. Tangents are drawn from points \( B \) and \( C \) to the circle with center \( A \) and radius \( AD \), which do not coincide with the line \( BC \). These tangents intersect at point \( P \). Prove that the radius of the incircle of \( \triangle BCP \) is equal to \( HD \).
[i]Proposed by Danylo Khilko[/i]
Let $ABC$ be a triangle, and $C^{\prime}$ and $A^{\prime}$ the midpoints of its sides $AB$ and $BC$. Consider two lines $g$ and $g^{\prime}$ which both pass through the vertex $A$ and are symmetric to each other with respect to the angle bisector of the angle $CAB$. Further, let $Y$ and $Y^{\prime}$ be the orthogonal projections of the point $B$ on these lines $g$ and $g^{\prime}$.
Show that the points $Y$ and $Y^{\prime}$ are symmetric to each other with respect to the line $C^{\prime}A^{\prime}$.
On the cartesian plane we draw circunferences of radii 1/20 centred in each lattice point. Show that any circunference of radii 100 in the cartesian plane intersect at least one of the small circunferences.
Find all triplets of positive integers $(x,y,z)$ such that $x^{2}+y^{2}+z^{2}=2011$.
The vertices of a regular $2002$-sided polygon are numbered $1$ through $2002$, clockwise. Given an integer $ n$, $1 \le n \le 2002$, color vertex $n$ blue, then, going clockwise, count$ n$ vertices starting at the next of $n$, and color $n$ blue. And so on, starting from the vertex that follows the last vertex that was colored, n vertices are counted, colored or uncolored, and the number $n$ is colored blue. When the vertex to be colored is already blue, the process stops. We denote $P(n)$ to the set of blue vertices obtained with this procedure when starting with vertex $n$. For example, $P(364)$ is made up of vertices $364$, $728$, $1092$, $1456$, $1820$, $182$, $546$, $910$, $1274$, $1638$, and $2002$.
Determine all integers $n$, $1 \le n \le 2002$, such that $P(n)$ has exactly $14 $ vertices,
Let $n$ be a positive integer with exactly twelve positive divisors $1=d_1 < \cdots < d_{12}=n$. We say $n$ is [i]trite[/i] if \[
5 + d_6(d_6+d_4) = d_7d_4.
\] Compute the sum of the two smallest trite positive integers.
[i]Proposed by Brandon Wang[/i]
There are $110$ guinea pigs for each of the $110$ species, arranging as a $110\times 110$ array. Find the maximum integer $n$ such that, no matter how the guinea pigs align, we can always find a column or a row of $110$ guinea pigs containing at least $n$ different species.