Found problems: 85335
Find the positive integers $n$ that are not divisible by $3$ if the number $2^{n^2-10}+2133$ is a perfect cube.
[hide="Note"]
[color=#BF0000]The wording of this problem is perhaps not the best English. As far as I am aware, just solve the diophantine equation $x^3=2^{n^2-10}+2133$ where $x,n \in \mathbb{N}$ and $3\nmid n$.[/color][/hide]
Determine all polynomials $P(x)$ with real coefficients such that
\[(x+1)P(x-1)-(x-1)P(x)\]
is a constant polynomial.
$A,\ B,\ C$ and $D$ are points on a given straight line, in that order. Show how to construct a square $PQRS$, with all of $P,\ Q,\ R$ and $S$ on the same side of $AD$, such that $A,\ B,\ C$ and $D$ lie on $PQ,\ SR,\ QR$ and $PS$ produced respectively.
3. Show that there are infinitely many triples (x,y,z) of integers such that $x^3 + y^4 = z^{31}$.
Does there exist a convex pentagon, all of whose vertices are lattice points in the plane, with no lattice point in the interior?
Let $a, b$ be two positive integers such that $b + 1|a^2 + 1$,$ a + 1|b^2 + 1$. Prove that $a, b$ are odd numbers.
Find all three digit natural numbers of the form $(abc)_{10}$ such that $(abc)_{10}$, $(bca)_{10}$ and $(cab)_{10}$ are in geometric progression. (Here $(abc)_{10}$ is representation in base $10$.)
Let $C$ be the circle of radius $12$ centered at $(0, 0)$. What is the length of the shortest path in the plane between $(8\sqrt3, 0)$ and $(0, 12 \sqrt2)$ that does not pass through the interior of $C$?
A convex pentagon $ABCDE$ satisfies that the sidelengths and $AC,AD\leq \sqrt 3.$ Let us choose $2011$ distinct points inside this pentagon. Prove that there exists an unit circle with centre on one edge of the pentagon, and which contains at least $403$ points out of the $2011$ given points.
{Edited}
{I posted it correctly before but because of a little confusion deleted the sidelength part, sorry.}
Let $D$ be the set of positive reals different from $1$ and let $n$ be a positive integer.
If for $f: D\rightarrow \mathbb{R}$ we have $x^n f(x)=f(x^2)$,
and if $f(x)=x^n$ for $0<x<\frac{1}{1989}$ and for $x>1989$,
then prove that $f(x)=x^n$ for all $x \in D$.
In triangle $ABC$, let $P$ and $R$ be the feet of the perpendiculars from $A$ onto the external and internal bisectors of $\angle ABC$, respectively; and let $Q$ and $S$ be the feet of the perpendiculars from $A$ onto the internal and external bisectors of $\angle ACB$, respectively. If $PQ = 7, QR = 6$ and $RS = 8$, what is the area of triangle $ABC$?
Line $\ell$ is perpendicular to one of the medians of the triangle. The perpendicular bisectors of the sides of this triangle intersect line $\ell$ at three points. Prove that one of them is the midpoint of the segment formed by the remaining two.
For which integers $N\ge 3$ can we find $N$ points on the plane such that no three are collinear, and for any triangle formed by three vertices of the points’ convex hull, there is exactly one point within that triangle?
Let $f(x)$ be a function satisfying $f(x+1)-f(x)=2x+1 (x \in \mathbb{R})$.In addition, $|f(x)|\le 1$ holds for $x\in [0,1]$. Prove that $|f(x)|\le 2+x^2$ holds for $x \in \mathbb{R}$.
24) A ball of mass m moving at speed $v$ collides with a massless spring of spring constant $k$ mounted on a stationary box of mass $M$ in free space. No mechanical energy is lost in the collision. If the system does not rotate, what is the maximum compression $x$ of the spring?
A) $x = v\sqrt{\frac{mM}{(m + M)k}}$
B) $x = v\sqrt{\frac{m}{k}}$
C) $x = v\sqrt{\frac{M}{k}}$
D) $x = v\sqrt{\frac{m + M}{k}}$
E) $x = v\sqrt{\frac{(m + M)^3}{mMk}}$
The graph of the degree $2021$ polynomial $P(x)$, which has real coefficients and leading coefficient 1, meets the x-axis at the points $(1, 0)$, $(2, 0)$, $(3, 0)$ , $...$ , $(2020, 0)$ and nowhere else. The mean of all possible values of $P(2021)$ can be written in the form $a!/b$, where $a$ and $b$ are positive integers and $a$ is as small as possible. Compute $a + b$.
How many ordered integer pairs $ \left(x,y\right)$ are there satisfying following equation:
\[ y \equal{} \sqrt{x\plus{}1998\plus{}\sqrt{x\plus{}1998\plus{}\sqrt{x\plus{}1997\plus{}\sqrt{x\plus{}1997\plus{}\ldots \plus{}\sqrt{x\plus{}1\plus{}\sqrt{x\plus{}1\plus{}\sqrt{x\plus{}\sqrt{x} } } } } } } }.\]
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 1998 \qquad\textbf{(D)}\ 3996 \qquad\textbf{(E)}\ \text{Infinitely many}$
Find all $m,n\in\mathbb N$ and primes $p\geq 5$ satisfying
\[m(4m^2+m+12)=3(p^n-1).\]
Let $a,b>1$ be odd positive integers. A board with $a$ rows and $b$ columns without fields $(2,1),(a-2,b)$ and $(a,b)$ is tiled with $2\times 2$ squares and $2\times 1$ dominoes (that can be rotated). Prove that the number of dominoes is at least $$\frac{3}{2}(a+b)-6.$$
Prove that there exists a positive integer $N$ such that for any positive integer $n \ge N$, there are at least $2015$ non-empty subsets $S$ of $\{ n^2 + 1, n^2 + 2, \dots, n^2 + 3n \}$ with the property that the product of the elements of $S$ is a perfect square.
Determine the number of triangles, of any size and shape, in the following figure:
[asy]
size(4cm);
draw(2*dir(0)--dir(120)--dir(240)--cycle);
draw(dir(60)--2*dir(180)--dir(300)--cycle);
[/asy]
[i]Proposed by William Yue[/i]
In the magic square shown, the sums of the numbers in each row, column, and diagonal are the same. Five of these numbers are represented by $ v$, $ w$, $ x$, $ y$, and $ z$. Find $ y \plus{} z$.
$ \textbf{(A)}\ 43 \qquad
\textbf{(B)}\ 44 \qquad
\textbf{(C)}\ 45 \qquad
\textbf{(D)}\ 46 \qquad
\textbf{(E)}\ 47$
[asy]unitsize(10mm);
defaultpen(linewidth(1pt));
for(int i=0; i<=3; ++i)
{
draw((0,i)--(3,i));
draw((i,0)--(i,3));
}
label("$25$",(0.5,0.5));
label("$z$",(1.5,0.5));
label("$21$",(2.5,0.5));
label("$18$",(0.5,1.5));
label("$x$",(1.5,1.5));
label("$y$",(2.5,1.5));
label("$v$",(0.5,2.5));
label("$24$",(1.5,2.5));
label("$w$",(2.5,2.5));[/asy]
Keith decides that a sequence of digits is [i]slick[/i] if every pair of adjacent digits in the sequence is divisible by either $23$ or $17.$ What is the greatest possible number of $2$s in a $2021$-digit long slick sequence?
Let $\mathcal P$ be the parabola $y^2=2px$, and let $T_0$ be a point on it. Point $T_0'$ is such that the midpoint of the segment $T_0T_0'$ lies on the axis of the parabola. For a variable point $T$ on $\mathcal P$, the perpendicular from $T_0'$ to the line $T_0T$ intersects the line through $T$ parallel to the axis of $\mathcal P$ at a point $T'$. Find the locus of $T'$.
For a positive integer $n$, let $f(n)$ be the number obtained by
writing $n$ in binary and replacing every 0 with 1 and vice
versa. For example, $n=23$ is 10111 in binary, so $f(n)$ is 1000 in
binary, therefore $f(23) =8$. Prove that
\[\sum_{k=1}^n f(k) \leq \frac{n^2}{4}.\]
When does equality hold?
(Proposed by Stephan Wagner, Stellenbosch University)