Found problems: 85335
For every positive integer $m$ prove the inquality
$|\{\sqrt{m}\} - \frac{1}{2}| \geq \frac{1}{8(\sqrt m+1)} $
(The integer part $[x]$ of the number $x$ is the largest integer not exceeding $x$. The fractional part of the number $x$ is a number $\{x\}$ such that $[x]+\{x\}=x$.)
A. Golovanov
Let $p \neq 13$ be a prime number of the form $8k+5$ such that $39$ is a quadratic non-residue modulo $p$. Prove that the equation $$x_1^4+x_2^4+x_3^4+x_4^4 \equiv 0 \pmod p$$ has a solution in integers such that $p\nmid x_1x_2x_3x_4$.
he center of the circle passing through the midpoints of all sides of triangle $ABC$ lies on the bisector of its angle $C$. Find the side $AB$ if $BC = a$, $AC = b$ ($a$ is not equal to $b$).
Let $n$ be a positive integer, and let $F$ be a family of subsets of $\{1, 2, ... , 2^n\}$ such that for any non-empty $ A\in F$ there exists $B \in F$ so that $|A| = |B| + 1$ and $B \subset A$. Suppose that F contains all $(2^n - 1)$-element subsets of $\{1, 2, ... , 2^n\}$ Determine the minimal possible value of $|F|$.
Let $r_1$, $r_2$, $r_3$ be the distinct real roots of $x^3-2019x^2-2020x+2021=0$. Prove that $r_1^3+r_2^3+r_3^3$ is an integer multiple of $3$.
What is the smallest possible value of the least common multiple of $a, b, c, d$ if we know that these four numbers are distinct and $a + b + c + d = 1000$?
Let $d\geq 1$ and $n\geq 0$ be integers. Find the number of ways to write down a nonnegative integer in each square of a $d\times d$ grid such that the numbers in any set of $d$ squares, no two in the same row or column, sum to $n$.
Find all natural numbers $n$ such that $n$, $n^2+10$, $n^2-2$, $n^3+6$, and $n^5+36$ are all prime numbers.
Let $\mathbb{Z}$ be the set of integers. Determine all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that, for all integers $a$ and $b$, $$f(2a)+2f(b)=f(f(a+b)).$$
[i]Proposed by Liam Baker, South Africa[/i]
Give an example of a continuous real-valued function $f$ form $[0,1]$ to $[0,1]$ which takes on every value in $[0,1]$ an infinite number of times.
Let $a,b,c$ be positive real numbers . Prove that$$ \frac{1}{ab(b+1)(c+1)}+\frac{1}{bc(c+1)(a+1)}+\frac{1}{ca(a+1)(b+1)}\geq\frac{3}{(1+abc)^2}.$$
We may say concerning the solution of
\[ |x|^2 \plus{} |x| \minus{} 6 \equal{} 0
\]
that:
$ \textbf{(A)}\ \text{there is only one root}\qquad
\textbf{(B)}\ \text{the sum of the roots is }{\plus{}1}\qquad
\textbf{(C)}\ \text{the sum of the roots is }{0}\qquad \\
\textbf{(D)}\ \text{the product of the roots is }{\plus{}4}\qquad
\textbf{(E)}\ \text{the product of the roots is }{\minus{}6}$
Let $ \alpha$ be a projective plane and $ c$ a closed polygon on $ \alpha$. Prove that $ \alpha$ will be decomposed into two regions by $ c$ if and only if there exists a straight line $ g$ in $ \alpha$ which has an even number of points in common with $ c$.
For an odd prime $ p, $ show that $ \sum_{k=1}^{p-1} \frac{k^p-k}{p}\equiv \frac{1+p}{2}\pmod p . $
There is an isosceles triangle which follows the following:
$ \bar{AB}=\bar{AC}=5, \bar{BC}=6 $
D,E are points on $ \bar{AC} $ which follows $ \bar{AD}=1, \bar{EC}=2 $
If the extent of $ \triangle $ BDE = S, Find 15S.
Find all non negative integer pairs $(a,b)$ such that $3^a5^b-2024$ is a square of a positive integer.
Let $ X$ be a point on the side $ AB$ of a triangle $ ABC$, different from $ A$ and $ B$. Let $ P$ and $ Q$ be the incenters of the triangles $ ACX$ and $ BCX$ respectively, and let $ M$ be the midpoint of $ PQ$. Prove that: $ MC>MX$.
For arbitrary pairwise distinct positive real numbers $a, b, c$, prove the inequality
$$\frac{(a^2- b^2)^3 + (b^2-c^2)^3+(c^2-a^2)^3}{(a- b)^3 + (b-c)^3+(c-a)^3}> 8abc$$
Given a circle $\omega$ with radius $1$. Let $T$ be a set of triangles good, if the following conditions apply:
(a) the circumcircle of each triangle in the set $T$ is $\omega$;
(b) The interior of any two triangles in the set $T$ has no common point.
Find all positive real numbers $t$, for which for each positive integer $n$ there is a good set of $n$ triangles, where the perimeter of each triangle is greater than $t$.
Show that if $t_1 , t_2, t_3, t_4, t_5$ are real numbers, then
$$\sum_{j=1}^{5} (1-t_j )\exp \left( \sum_{k=1}^{j} t_k \right) \leq e^{e^{e^{e}}}.$$
A square with horizontal and vertical sides is drawn on the plane. It held several segments parallel to the sides, and there are no two segments which lie on one line or intersect at an interior point for both segments. It turned out that the segments cuts square into rectangles, and any vertical line intersecting the square and not containing segments of the partition intersects exactly $ k $ rectangles of the partition, and any horizontal line intersecting the square and not containing segments of the partition intersects exactly $\ell$ rectangles. How much the number of rectangles can be?
[i]I. Bogdanov, D. Fon-Der-Flaass[/i]
In $\Delta ABC$, $\angle ABC=120^\circ$. The internal bisector of $\angle B$ meets $AC$ at $D$. If $BD=1$, find the smallest possible value of $4BC+AB$.
On the table, there are $1024$ marbles and two students, $A$ and $B$, alternatively take a positive number of marble(s). The student $A$ goes first, $B$ goes after that and so on. On the first move, $A$ takes $k$ marbles with $1 < k < 1024$. On the moves after that, $A$ and $B$ are not allowed to take more than $k$ marbles or $0$ marbles. The student that takes the last marble(s) from the table wins. Find all values of $k$ the student $A$ should choose to make sure that there is a strategy for him to win the game.
In the figure, $\triangle ABC$ has $\angle A =45^{\circ}$ and $\angle B =30^{\circ}$. A line $DE$, with $D$ on $AB$ and $\angle ADE =60^{\circ}$, divides $\triangle ABC$ into two pieces of equal area. (Note: the figure may not be accurate; perhaps $E$ is on $CB$ instead of $AC$.) The ratio $\frac{AD}{AB}$ is
[asy]
size((220));
draw((0,0)--(20,0)--(7,6)--cycle);
draw((6,6)--(10,-1));
label("A", (0,0), W);
label("B", (20,0), E);
label("C", (7,6), NE);
label("D", (9.5,-1), W);
label("E", (5.9, 6.1), SW);
label("$45^{\circ}$", (2.5,.5));
label("$60^{\circ}$", (7.8,.5));
label("$30^{\circ}$", (16.5,.5));
[/asy]
$ \textbf{(A)}\ \frac{1}{\sqrt{2}} \qquad\textbf{(B)}\ \frac{2}{2+\sqrt{2}} \qquad\textbf{(C)}\ \frac{1}{\sqrt{3}} \qquad\textbf{(D)}\ \frac{1}{\sqrt[3]{6}} \qquad\textbf{(E)}\ \frac{1}{\sqrt[4]{12}} $
Let $p \in \mathbb{N} \setminus \{0, 1\}$ be a fixed positive integer. Prove that for every $K > 0$, there exist infinitely many $n$ and $N$ such that there are atleast $\dfrac{KN}{\log(N)}$ primes among the following $N$ numbers given by
\[n + 1, n + 2^p, n + 3^p, \cdots, n + N^p.\]
[i]Proposed by Bimit Mandal[/i]