Found problems: 85335
Let $n$ be a positive integer. Prove that $n\sqrt{19}\{n\sqrt{19}\} > 1$, where $\{x\}$ denotes the fractional part of $x$.
Let $f(x)$ be a twice continuously differentiable function on closed interval $[a,b]$
Prove that
$\lim_{n \to \infty} n^2[\int_{a}^{b}f(x)dx-\frac{b-a}{n}\sum_{k=1}^{n}f(a+\frac{2k-1}{2n}(b-a))]=\frac{(b-a)^2}{24}[f'(b)-f'(a)]$
In a country there are infinitely many towns and for every pair of towns there is one road connecting them. Initially there are $n$ coin in each city. Every day traveller Hong starts from one town and moves on to another, but if Hong goes from town $A$ to $B$ on the $k$-th day, he has to send $k$ coins from $B$ to $A$, and he can no longer use the road connecting $A$ and $B$. Prove that Hong can't travel for more than $n+2n^\frac{2}{3}$ days.
Show that the number $1989$ as well as each of its powers $1989^n$ ($n \in N$), can be expressed as a sum of two positive squares in at least two ways.
Do there exist infinitely many triplets of positive integers $(a, b, c)$ such that
the following two conditions hold:
1. $\gcd(a, b, c) = 1$;
2. $a+b+c, a^2+b^2+c^2$ and $abc$ are all perfect squares?
[i](Proposed by Ivan Chan Guan Yu)[/i]
The sides of a $10$ by $10$ square $ABCD$ are reflective on the inside. A beam of light enters the square via the vertex $A$ and heads to the point $P$ on $CD$ with $CP = 3$ and $PD = 7$. In $P$ it naturally reflects on the $CD$ side. The light beam can only leave the square via one of the angular points $A, B, C$ or $D$.
What is the distance that the light beam travels within the square before it leaves the square again?
By which vertex does that happen?
Consider a triangle $ABC$ and a point $D$ on its side $\overline{AB}$. Let $I$ be a point inside $\triangle ABC$ on the angle bisector of $ACB$. The second intersections of lines $AI$ and $CI$ with circle $ACD$ are $P$ and $Q$, respectively. Similarly, the second intersection of lines $BI$ and $CI$ with circle $BCD$ are $R$ and $S$, respectively. Show that if $P\neq Q$ and $R\neq S$, then lines $AB$, $PQ$ and $RS$ pass through a point or are parallel.
Let $n > 1$ be an integer.
A set $S \subseteq \{ 0, 1, 2, \cdots , 4n - 1 \}$ is called ’sparse’ if for any $k \in \{ 0, 1, 2, \cdots , n - 1 \}$ the following two conditions are satisfied:
$(a)$ The set $S \cap \{4k - 2, 4k - 1, 4k, 4k + 1, 4k + 2 \}$ has at most two elements;
$(b)$ The set $S \cap \{ 4k +1, 4k +2, 4k +3 \}$ has at most one element.
Prove that there are exactly $8 \cdot 7^{n-1}$ sparse subsets.
A trapezoid $ABCD$ with bases $AB$ and $CD$ is such that the circumcircle of the triangle $BCD$ intersects the line $AD$ in a point $E$, distinct from $A$ and $D$. Prove that the circumcircle oF the triangle $ABE$ is tangent to the line $BC$.
Let $a_1$, $a_2$, ..., $a_{10}$ be positive integers such that $a_1<a_2<...<a_{10}$. For every $k$, denote by $b_k$ the greatest divisor of $a_k$ such that $b_k<a_k$. Assume that $b_1>b_2>...>b_{10}$. Show that $a_{10}>500$.
If $b$ and $c$ are constants and \[(x + 2)(x + b) = x^2 + cx + 6,\] then $c$ is
$ \textbf{(A)}\ -5\qquad\textbf{(B)}\ -3\qquad\textbf{(C)}\ -1\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 5 $
What is the smallest positive integer $n$ such that $2^n - 1$ is a multiple of $2015$?
a) Prove that when you divide any prime number by $30$, the remainder is either $1$ or is a prime number!
b) Does this also apply when dividing a prime number by $60$? Justify your answer!
Prove or disprove that there exists an integer which is doubled when the initial digit is transferred to the end.
Prove that every nonzero coefficient of the Taylor series of $(1-x+x^2)e^x$ about $x=0$ is a rational number whose numerator (in lowest terms) is either $1$ or a prime number.
Given a triangle $ABC$, $A',B',C'$ are the midpoints of $\overline{BC},\overline{AC},\overline{AB}$, respectively. $B^*,C^*$ lie in $\overline{AC},\overline{AB}$, respectively, such that $\overline{BB^*},\overline{CC^*}$ are the altitudes of the triangle $ABC$. Let $B^{\#},C^{\#}$ be the midpoints of $\overline{BB^*},\overline{CC^*}$, respectively. $\overline{B'B^{\#}}$ and $\overline{C'C^{\#}}$ meet at $K$, and $\overline{AK}$ and $\overline{BC}$ meet at $L$. Prove that $\angle{BAL}=\angle{CAA'}$
Compute $$\frac{20+\frac{1}{25-\frac{1}{20}}}{25+\frac{1}{20-\frac{1}{25}}}.$$
Let $a, \overline{bcd}, \overline{aef}, \overline{cfg}, \overline{hci}, \overline{dea}, \overline{ifd}, \overline{jgf}, \overline{bfeg},\ldots$ be an increasing arithmetic progression. Find the $16$th term of this sequence.
Let $ f: \mathbb{R}^2\to\mathbb{R}$ be a function such that $ f(x,y)\plus{}f(y,z)\plus{}f(z,x)\equal{}0$ for real numbers $ x,y,$ and $ z.$ Prove that there exists a function $ g: \mathbb{R}\to\mathbb{R}$ such that $ f(x,y)\equal{}g(x)\minus{}g(y)$ for all real numbers $ x$ and $ y.$
Find all the pime numbers $(p,q)$ such that :
$p^{3}+p=q^{2}+q$
Three balloon vendors each offer two types of balloons - one offers red & blue, one offers blue & yellow, and one offers yellow & red. I like each vendor the same, so I must buy $7$ balloons from each. How many different possible triples $(x,y,z)$ are there such that I could buy $x$ blue, $y$ yellow, and $z$ red balloons?
A cyclic quadrilateral $ABCD$ has circumcenter $O$, its diagonals $AC$ and $BD$ intersect at $G$. Let $P, Q, R, S$ be the circumcenters of $\vartriangle AGB, \vartriangle BGC, \vartriangle CGD, \vartriangle DGA$ respectively. Lines $P R$ and $QS$ intersect at $M$. Show that $M$ is the midpoint of $OG$.
$f$ is a differentiable function such that $f(f(x))=x$ where $x \in [0,1]$.Also $f(0)=1$.Find the value of
$$\int_0^1(x-f(x))^{2016}dx$$
Alexa wrote the first $16$ numbers of a sequence:
\[1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 10, 11, …\]
Then she continued following the same pattern, until she had $2015$ numbers in total.
What was the last number she wrote?
Assign to each side $b$ of a convex polygon $P$ the maximum area of a triangle that has $b$ as a side and is contained in $P$. Show that the sum of the areas assigned to the sides of $P$ is at least twice the area of $P$.