Found problems: 85335
In acute angled triangle $ABC$, $BC=a$,$CA=b$,$AB=c$, and $a>b>c$. $I,O,H$ are the incentre, circumcentre and orthocentre of $\triangle{ABC}$ respectively. Point $D \in BC$, $E \in CA$ and $AE=BD$, $CD+CE=AB$. Let the intersectionf of $BE$ and $AD$ be $K$. Prove that $KH \parallel IO$ and $KH = 2IO$.
For each positive integer $n$, put
$$p_n =\left(1+\frac{1}{n}\right)^{n},\; P_n =\left(1+\frac{1}{n}\right)^{n+1}, \; h_n = \frac{2 p_n P_{n}}{ p_n + P_n }.$$
Prove that $h_1 < h_2 < h_3 <\ldots$
Ten distinct numbers are chosen at random from the set $\{1, 2, 3, ... , 37\}$. Show that one can select four distinct numbers out of those ten so that the sum of two of them is equal to the sum of the other two.
Given the segments $AB$ and $CD$ not necessarily on the same plane. Point $X$ is the midpoint of the segment $AB$, and the point $Y$ is the midpoint of $CD$. Given that point $X$ is not on line $CD$, and that point $Y$ is not on line $AB$, prove that $2 | XY | \le | AD | + | BC |$. When is equality achieved?
Let $ p\in\mathbb{R}_\plus{}$ and $ k\in\mathbb{R}_\plus{}$. The polynomial $ F(x)\equal{}x^4\plus{}a_3x^3\plus{}a_2x^2\plus{}a_1x\plus{}k^4$ with real coefficients has $ 4$ negative roots. Prove that $ F(p)\geq(p\plus{}k)^4$
Let $\triangle ABC$ be an acute angled triangle. Let $G$ be the centroid and let $D$ be the foot of the altitude from $A$ onto $BC$. Let ray $GD$ intersect $(ABC)$ at $X$ and let $AG$ intersect nine point circle at $Y$ not on $BC$. Let $Z$ be the intersection of the $\text{A-tangent}$ to $(ABC)$ and $\text{A-midline}$. Prove that perpendicular from $Z$ to the Euler line, $AX$ and $DY$ concur.
The line joining the midpoints of $AB$ and $AC$ is called the $\text{A-midline}$.
$(ABC)$ denotes the circumcircle of $\triangle ABC$
A right cone in $xyz$-space has its apex at $(0,0,0)$, and the endpoints of a diameter on its base are $(12,13,-9)$ and $(12,-5,15)$. The volume of the cone can be expressed as $a\pi$. What is $a$?
Which positive integers satisfy the following three conditions?
a) The number consists of at least two digits.
b) The last digit is not zero.
c) Inserting a zero between the last two digits yields a number divisible by the original number.
Prove that if the product $1\cdot 2\cdot ...\cdot n$ ($n> 3$) is not divisible by $n + 1$, then $n + 1$ is prime.
[b]p1.[/b] In the group of five people any subgroup of three persons contains at least two friends. Is it possible to divide these five people into two subgroups such that all members of any subgroup are friends?
[b]p2.[/b] Coefficients $a,b,c$ in expression $ax^2+bx+c$ are such that $b-c>a$ and $a \ne 0$. Is it true that equation $ax^2+bx+c=0$ always has two distinct real roots?
[b]p3.[/b] Point $D$ is a midpoint of the median $AF$ of triangle $ABC$. Line $CD$ intersects $AB$ at point $E$. Distances $|BD|=|BF|$. Show that $|AE|=|DE|$.
[b]p4.[/b] Real numbers $a,b$ satisfy inequality $a+b^5>ab^5+1$. Show that $a+b^7>ba^7+1$.
[b]p5.[/b] A positive number was rounded up to the integer and got the number that is bigger than the original one by $28\%$. Find the original number (find all solutions).
[b]p6.[/b] Divide a $5\times 5$ square along the sides of the cells into $8$ parts in such a way that all parts are different.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
An acute triangle is a triangle that has all angles less than $90^{\circ}$ ($90^{\circ}$ is a Right Angle). Let $ABC$ be an acute triangle with altitudes $AD$, $BE$, and $CF$ meeting at $H$. The circle passing through points $D$, $E$, and $F$ meets $AD$, $BE$, and $CF$ again at $X$, $Y$, and $Z$ respectively. Prove the following inequality: $$\frac{AH}{DX}+\frac{BH}{EY}+\frac{CH}{FZ} \geq 3.$$
Sebastian has a certain number of rectangles with areas that sum up to 3 and with side lengths all less than or equal to $1$. Demonstrate that with each of these rectangles it is possible to cover a square with side $1$ in such a way that the sides of the rectangles are parallel to the sides of the square.
[b]Note:[/b] The rectangles can overlap and they can protrude over the sides of the square.
Let $N$ be the set of positive integers. If $A,B,C \ne \emptyset$, $A \cap B = B \cap C = C \cap A = \emptyset$ and $A \cup B \cup C = N$, we say that $A,B,C$ are partitions of $N$. Prove that there are no partitions of $N, A,B,C$, that satis fy the following:
(i) $\forall a \in A, b \in B$, we have $a + b + 1 \in C$
(ii) $\forall b \in B, c \in C$, we have $b + c + 1 \in A$
(iii) $\forall c \in C, a \in A$, we have $c + a + 1 \in B$
The graph of $ x^2 \plus{} y \equal{} 10$ and the graph of $ x \plus{} y \equal{} 10$ meet in two points. The distance between these two points is:
$ \textbf{(A)}\ \text{less than 1} \qquad
\textbf{(B)}\ 1\qquad
\textbf{(C)}\ \sqrt{2}\qquad
\textbf{(D)}\ 2\qquad
\textbf{(E)}\ \text{more than 2}$
Evaluate $\int_0^1 \{x(1-x)\}^{\frac 32}dx.$
[i]2010 Hirosaki University School of Medicine entrance exam[/i]
Assume $p$ is a prime number. If there is a positive integer $a$ such that $p!|(a^p + 1)$, prove that :
(1) $(a+1, \frac{a^p+1}{a+1}) = p$
(2) $\frac{a^p+1}{a+1}$ has no prime factors less than $p$.
(3) $p!|(a +1) $.
Find all the pairs of positive integers $ (a,b)$ such that $ a^2 \plus{} b \minus{} 1$ is a power of prime number $ ; a^2 \plus{} b \plus{} 1$ can divide $ b^2 \minus{} a^3 \minus{} 1,$ but it can't divide $ (a \plus{} b \minus{} 1)^2.$
Determine all integers $n\geqslant 2$ with the following property: every $n$ pairwise distinct integers whose sum is not divisible by $n$ can be arranged in some order $a_1,a_2,\ldots, a_n$ so that $n$ divides $1\cdot a_1+2\cdot a_2+\cdots+n\cdot a_n.$
[i]Arsenii Nikolaiev, Anton Trygub, Oleksii Masalitin, and Fedir Yudin[/i]
A hyperbola in the coordinate plane passing through the points $(2,5)$, $(7,3)$, $(1,1)$, and $(10,10)$ has an asymptote of slope $\frac{20}{17}$. The slope of its other asymptote can be expressed in the form $-\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$.
[i]Proposed by Michael Ren[/i]
Prove that for any integer $n>3$ there exist infinitely many non-constant arithmetic progressions of length $n-1$ whose terms are positive integers whose product is a perfect $n$-th power.
Let $x_1, x_2,... , x_{84}$ be the roots of the equation $x^{84} + 7x - 6 = 0$. Compute $\sum_{k=1}^{84} \frac{x_k}{x_k-1}$.
Show that a triangle whose side lengths are prime numbers cannot have integer area.
For distinct complex numbers $z_1,z_2,\dots,z_{673}$, the polynomial
\[ (x-z_1)^3(x-z_2)^3 \cdots (x-z_{673})^3 \]
can be expressed as $x^{2019} + 20x^{2018} + 19x^{2017}+g(x)$, where $g(x)$ is a polynomial with complex coefficients and with degree at most $2016$. The value of
\[ \left| \sum_{1 \le j <k \le 673} z_jz_k \right| \]
can be expressed in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
If a function $f$ defined on all real numbers has at least two centers of symmetry, show that this function can be written as sum of a linear function and a periodic function.
[For every real number $x$, if there is a real number $a$ such that $f(a-x) + f(a+x) =2f(a)$, the point $(a,f(a))$ is called a center of symmetry of the function $f$.]
Let $P_1 , P_2 , ...$ be arbitrary points and A be a connected compact set in the plane with a diameter greater than 4. Show that for some point P in A , $\overline {PP_1} \cdot \overline {PP_2} \cdots \overline {PP_n}>1$. Furthermore, prove that this is no longer necessarily true for compact sets of diameter 4.