Found problems: 85335
Let $ p$ be the product of two consecutive integers greater than 2. Show that there are no integers $ x_1, x_2, \ldots, x_p$ satisfying the equation
\[ \sum^p_{i \equal{} 1} x^2_i \minus{} \frac {4}{4 \cdot p \plus{} 1} \left( \sum^p_{i \equal{} 1} x_i \right)^2 \equal{} 1
\]
[b]OR[/b]
Show that there are only two values of $ p$ for which there are integers $ x_1, x_2, \ldots, x_p$ satisfying
\[ \sum^p_{i \equal{} 1} x^2_i \minus{} \frac {4}{4 \cdot p \plus{} 1} \left( \sum^p_{i \equal{} 1} x_i \right)^2 \equal{} 1
\]
Determine the set of integers $n$ for which $n^2+19n+92$ is a square.
In acute triangle $ ABC,$ points $ P,Q$ lie on its sidelines $ AB,AC,$ respectively. The circumcircle of triangle $ ABC$ intersects of triangle $ APQ$ at $ X$ (different from $ A$). Let $ Y$ be the reflection of $ X$ in line $ PQ.$ Given $ PX>PB.$ Prove that $ S_{\bigtriangleup XPQ}>S_{\bigtriangleup YBC}.$ Where $ S_{\bigtriangleup XYZ}$ denotes the area of triangle $ XYZ.$
Let $Q=\{0,1\}^n$, and let $A$ be a subset of $Q$ with $2^{n-1}$ elements. Prove that there are at least $2^{n-1}$ pairs $(a,b)\in A\times (Q\setminus A)$ for which sequences $a$ and $b$ differ in only one term.
Show that if the point $ M $ is situated in the interior of a square $ ABCD, $ then, among the segments $ MA,MB,MC,MD, $
[b]a)[/b] at most one of them is greater with a factor of $ \sqrt 5/2 $ than the side of the square.
[b]b)[/b] at most two of them are greater than the side of the square.
[b]c)[/b] at most three of them are greater with a factor of $ \sqrt 2/2 $ than the side of the square.
Let $O$ be the circumcenter of the acute triangle $ABC$ ($AB < AC$). Let $A_1$ and $P$ be the feet of the perpendicular lines drawn from $A$ and $O$ to $BC$, respectively. The lines $BO$ and $CO$ intersect $AA_1$ in $D$ and $E$, respectively. Let $F$ be the second intersection point of $\odot ABD$ and $\odot ACE$. Prove that the angle bisector od $\angle FAP$ passes through the incenter of $\triangle ABC$.
For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2015$ good partitions.
A safe is protected with a number of locks. Eleven members of the committee have keys for some of the locks. What is the smallest number of locks necessary so that every six members of the committee can open the safe, but no five members can do it? How should the keys be distributed among the committee members if the number of locks is the smallest?
Given $n$ points in the three dimensional space such, that the arbitrary triangle with the vertices in three of those points contains an angle greater than $120$ degrees. Prove that you can rearrange them to make a polyline (unclosed) with all the angles between the sequent links greater than $120$ degrees.
Solve the system of equations: \[ x+y+z=a \] \[ x^2+y^2+z^2=b^2 \] \[ xy=z^2 \] where $a$ and $b$ are constants. Give the conditions that $a$ and $b$ must satisfy so that $x,y,z$ are distinct positive numbers.
Let $a$ be a real number. Let $(f_n(x))_{n\ge 0}$ be a sequence of polynomials such that $f_0(x)=1$ and $f_{n+1}(x)=xf_n(x)+f_n(ax)$ for all non-negative integers $n$.
a) Prove that $f_n(x)=x^nf_n\left(x^{-1}\right)$ for all non-negative integers $n$.
b) Find an explicit expression for $f_n(x)$.
A function $ f$ has the property that $ f(3x \minus{} 1) \equal{} x^{2} \plus{} x \plus{} 1$ for all real numbers $ x$. What is $ f(5)$?
$ \textbf{(A)}\ 7 \qquad \textbf{(B)}\ 13 \qquad \textbf{(C)}\ 31 \qquad \textbf{(D)}\ 111 \qquad \textbf{(E)}\ 211$
Let $n$ be a positive integer relatively prime to $6$. We paint the vertices of a regular $n$-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours.
The diagram below shows a large square divided into nine congruent smaller squares. There are circles inscribed in five of the smaller squares. The total area covered by all the five circles is $20\pi$. Find the area of the large square.
[asy]
size(80);
defaultpen(linewidth(0.6));
pair cent[] = {(0,0),(0,2),(1,1),(2,0),(2,2)};
for(int i=0;i<=3;++i)
{
draw((0,i)--(3,i));
}
for(int j=0;j<=3;++j)
{
draw((j,0)--(j,3));
}
for(int k=0;k<=4;++k)
{
draw(circle((cent[k].x+.5,cent[k].y+.5),.5));
}
[/asy]
Prove that for every convex polygon, we can choose three of its consecutive vertices, such that the circle, defined by them, covers the the entire polygon.
(proposed by J. Tabov)
How many integers $0 \leq x < 125$ are there such that $x^3 - 2x + 6 \equiv 0 \pmod {125}$?
$
\textbf{a)}\ 0
\qquad\textbf{b)}\ 1
\qquad\textbf{c)}\ 2
\qquad\textbf{d)}\ 3
\qquad\textbf{e)}\ \text{None of above}
$
Find all natural numbers $n$ such that $\frac{\sqrt{n}}{2}+\frac{10}{\sqrt{n}}$ is a natural number.
In triangle $ABC$, $\angle ACB=50^{\circ}$ and $\angle CBA=70^{\circ}$. Let $D$ be a foot of perpendicular from point $A$ to side $BC$, $O$ circumcenter of $ABC$ and $E$ antipode of $A$ in circumcircle $ABC$. Find $\angle DAE$
In a sequence of numbers, a term is called [i]golden [/i] if it is divisible by the term immediately before it. What is the maximum possible number of golden terms in a permutation of $1, 2, 3, . . . , 2021$?
The negative real numbers $x, y, z, t$ satisfy simultaneously equalities, $$x + y + z = t, \,\,\,\,\frac{1}{x}+ \frac{1}{y}+\frac{1}{z}= \frac{1}{t}, \\,\,\,\, x^3 + y^3 + z^3 = 1000^3$$ Compute $x + y + z + t$.
Let $P(x)$ be a polynomial of degree $5$ and suppose that a and b are real numbers different from zero. Suppose the remainder when $P(x)$ is divided by $x^3 + ax + b$ equals the remainder when $P(x)$ is divided by $x^3 + ax^2 + b$. Then determine $a + b$.
Let $E$ be the point of intersection of the diagonals of the cyclic quadrilateral $ABCD$, and let $K, L, M$ and $N$ be the midpoints of the sides $AB, BC, CD$ and $DA$, respectively. Prove that the radii of the circles circumscribed around the triangles $KLE$ and $MNE$ are equal.
Prove that exist different natural numbers $x$, $y$, $z$, $t$ for which $$256\times 2019^{180n+1}=2x^9-2y^6+z^5-t^4$$for all $n\in \mathbb{N}^*$
What is the maximum number of diagonals of a regular $12$-gon which can be selected such that no two of the chosen diagonals are perpendicular?
Note: sides are not diagonals and diagonals which intersect outside the $12$-gon at right angles are still considered perpendicular.
[i]2018 CCA Math Bonanza Tiebreaker Round #1[/i]
Let $ABC$ be a triangle with circumcenter $O$, angle bisector $AD$ with $D \in BC$ and altitude $AE$ with $E \in BC$. The lines $AO$ and $BC$ meet at $I$. The circumcircle of $\triangle ADE$ meets $AB, AC$ at $F, G$ and $FG$ meets $BC$ at $H$. The circumcircles of triangles $AHI$ and $ABC$ meet at $J$. Show that $AJ$ is a symmedian in $\triangle ABC$