Found problems: 85335
Prove that there is no positive integer $n$ such that, for $k = 1,2,\ldots,9$, the leftmost digit (in decimal notation) of $(n+k)!$ equals $k$.
What is the smallest positive integer $k$ such that $k(3^3 + 4^3 + 5^3) = a^n$ for some positive integers $a$ and $n$, with $n > 1$?
A set S consisting of $2019$ (different) positive integers has the following property:
[i]the product of every 100 elements of $S$ is a divisor of the product of the remaining $1919$ elements[/i].
What is the maximum number of prime numbers that $S$ can contain?
A $10$ digit number is called interesting if its digits are distinct and is divisible by $11111$. Then find the number of interesting numbers.
Let $ABCDEF$ be a convex hexagon satisfying $AC = DF, CE = FB$ and $EA = BD$. Prove that the lines connecting the midpoints of opposite sides of the hexagon $ABCDEF$ intersect in one point.
Let be two distinct $ 2\times 2 $ real matrices having the property that there exists a natural number such that these matrices raised to this number are equal, and these matrices raised to the successor of this number are also equal.
Prove that these matrices raised to any power greater than $ 2 $ are equal.
A prime is called [i]perfect[/i] if there is a permutation $a_1, a_2, \cdots, a_{\frac{p-1}{2}}, b_1, b_2, \cdots, b_{\frac{p-1}{2}}$ of $1, 2, \cdots, p-1$ satisfies $$b_i \equiv a_i + \frac{1}{a_i} \pmod p$$ for all $1 \le i \le \frac{p-1}{2}$. Show that there are infinitely many primes that are not perfect.
[i]Proposed By - CSJL[/i]
Let S be the point of intersection of the two lines $ l_1 : 7x \minus{} 5y \plus{} 8 \equal{} 0$ and $ l_2 : 3x \plus{} 4y \minus{} 13 \equal{} 0.$ Let $ P \equal{} (3, 7), Q \equal{} (11, 13),$ and let $ A$ and $ B$ be points on the line $ PQ$ such that $ P$ is between $ A$ and $ Q,$ and $ B$ is between $ P$ and $ Q,$ and such that \[ \frac{PA}{AQ} \equal{} \frac{PB}{BQ} \equal{} \frac{2}{3}.\] Without finding the coordinates of $ B$ find the equations of the lines $ SA$ and $ SB.$
For a positive integer $n$ let $t_n$ be the number of unordered triples of non-empty and pairwise disjoint subsets of a given set with $n$ elements. For example, $t_3 = 1$. Find a closed form formula for $t_n$ and determine the last digit of $t_{2022}$.
(I also give here that $t_4 = 10$, for a reader to check his/her understanding of the problem statement.)
Determine all positive integers $x$, $y$ and $z$ such that $x^5 + 4^y = 2013^z$.
([i]Serbia[/i])
Consider a parallelogram $[ABCD]$ such that $\angle DAB$ is an acute angle. Let $G$ be a point in line $AB$ different from $B$ such that $\overline{BC}=\overline{GC}$, and let $H$ be a point in line $BC$ different from $B$ such that $\overline{AB}=\overline{AH}$. Prove that triangle $[GDH]$ is isosceles.
Let $f:[0,1]\to\mathbb R$ be a function satisfying
$$|f(x)-f(y)|\le|x-y|$$for every $x,y\in[0,1]$. Show that for every $\varepsilon>0$ there exists a countable family of rectangles $(R_i)$ of dimensions $a_i\times b_i$, $a_i\le b_i$ in the plane such that
$$\{(x,f(x)):x\in[0,1]\}\subset\bigcup_iR_i\text{ and }\sum_ia_i<\varepsilon.$$(The edges of the rectangles are not necessarily parallel to the coordinate axes.)
A uniform disk is being pulled by a force F through a string attached to its center of mass. Assume
that the disk is rolling smoothly without slipping. At a certain instant of time, in which region of the
disk (if any) is there a point moving with zero total acceleration?
A Region I
B Region II
C Region III
D Region IV
E All points on the disk have a nonzero acceleration
Let $ABC$ be an acute triangle. The arcs $AB$ and $AC$ of the circumcircle of the triangle are reflected over the lines AB and $AC$, respectively. Prove that the two arcs obtained intersect in another point besides $A$.
The vertex $C$ of equilateral triangles $ABC$ and $CDE$ lies on the segment $AE$, and the vertices $B$ and $D$ lie on the same side with respect to this segment. The circumcircles of these triangles centered at $O_1$ and $O_2$ meet for the second time at point $F$. The lines $O_1O_2$ and $AD$ meet at point $K$. Prove that $AK = BF$.
Given a fixed rational number $q$. Let's call a number $x$ [i]charismatic [/i] if we can find a natural number $n$ and integers $a_1, a_2,.., a_n$ such that
$$x = (q + 1)^{a_1} \cdot (q + 2)^{a_2} \cdot ... \cdot(q + n)^{a_n} .$$
i) Prove that one can find a $q$ such that all positive rational numbers are charismatic.
ii) Is it true that for all $q$, if the number $x$ is charismatic, then $x + 1$ is also charismatic?
Let $A$ be the product of all positive integers less than $1000$ whose ones or hundreds digit is $7$. Compute the remainder when $A/101$ is divided by $101$.
let $m,n\in \mathbb{N}$ and $p(x),q(x),h(x)$ are polynomials with real Coefficients such that $p(x)$ is Descending.
and for all $x\in \mathbb{R}$
$p(q(nx+m)+h(x))=n(q(p(x))+h(x))+m$ .
prove that dont exist function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all $x\in \mathbb{R}$
$f(q(p(x))+h(x))=f(x)^{2}+1$
Find all integers $a$ such that there is a prime number of $p\ge 5$ that divides
${p-1 \choose 2}$ $+ {p-1 \choose 3} a$ $+{p-1 \choose 4} a^2$+ ...+$ {p-1 \choose p-3} a^{p-5} .$
Do there exist $10$ real numbers, not all of which are equal, each of which is equal to the square of the sum of the remaining $9$ numbers?
[i]Proposed by Bogdan Rublov[/i]
There are 100 boxers, each of them having different strengths, who participate in a tournament. Any of them fights each other only once. Several boxers form a plot. In one of their matches, they hide in their glove a horse shoe. If in a fight, only one of the boxers has a horse shoe hidden, he wins the fight; otherwise, the stronger boxer wins. It is known that there are three boxers who obtained (strictly) more wins than the strongest three boxers. What is the minimum number of plotters ?
[i]Proposed by N. Kalinin[/i]
Find all functions $f : R \to R$ satisfying the following conditions
(a) $f(1) = 1$,
(b) $f(x + y) = f(x) + f(y)$, $\forall (x,y) \in R^2$
(c) $f\left(\frac{1}{x}\right) =\frac{ f(x)}{x^2 }$, $\forall x \in R -\{0\}$
Trần Nam Dũng
For every positive integer $n$, find the maximum power of $2$ that divides the number $$1 + 2019 + 2019^2 + 2019^3 +.. + 2019^{n-1}.$$
Find all nonzero subsets $A$ of the set $\left\{2,3,4,5,\cdots\right\}$ such that $\forall n\in A$, we have that $n^{2}+4$ and $\left\lfloor{\sqrt{n}\right\rfloor}+1$ are both in $A$.
Let $ABC$ be a triangle, and let $D, E$ and $F$ be the feet of the altitudes from $A, B$ and $C,$ respectively. A circle $\omega_A$ through $B$ and $C$ crosses the line $EF$ at $X$ and $X'$. Similarly, a circle $\omega_B$ through $C$ and $A$ crosses the line $FD$ at $Y$ and $Y',$ and a circle $\omega_C$ through $A$ and $B$ crosses the line $DE$ at $Z$ and $Z'$. Prove that $X, Y$ and $Z$ are collinear if and only if $X', Y'$ and $Z'$ are collinear.
[i]Vlad Robu[/i]