Found problems: 85335
Consider points $C_1, C_2$ on the side $AB$ of a triangle $ABC$, points $A_1, A_2$ on the side $BC$ and points $B_1 , B_2$ on the side $CA$ such that these points divide the corresponding sides to three equal parts. It is known that all the points $A_1, A_2, B_1, B_2 , C_1$ and $C_2$ are concyclic. Prove that triangle $ABC$ is equilateral.
$ABCDA'B'CD'$ is a rectangular parallelepiped with $AA'= 2AB = 8a$ , $E$ is the midpoint of $(AB)$ and $M$ is the point of $(DD')$ for which $DM = a \left( 1 + \frac{AD}{AC}\right)$.
a) Find the position of the point. $F$ on the segment $(AA')$ for which the sum $CF + FM$ has the minimum possible value.
b) Taking $F$ as above, compute the measure of the angle of the planes $(D, E, F)$ and $(D, B', C')$.
c) Knowing that the straight lines $AC'$ and $FD$ are perpendicular, compute the volume of the parallelepiped $ABCDA'B'C'D'$.
$n$ is natural number and $p$ is prime number. If $1+np$ is square of natural number then prove that $n+1$ is equal to some sum of $p$ square of natural numbers.
Let $AH_a$ and $BH_b$ be altitudes, $AL_a$ and $BL_b$ be angle bisectors of a triangle $ABC$. It is known that $H_aH_b // L_aL_b$. Is it necessarily true that $AC = BC$?
(B. Frenkin)
Find all polynomials of degree 3, such that for each $x,y\geq 0$: \[p(x+y)\geq p(x)+p(y)\]
Joey and Chloe and their daughter Zoe all have the same birthday. Joey is 1 year older than Chloe, and Zoe is exactly 1 year old today. Today is the first of the 9 birthdays on which Chloe's age will be an integral multiple of Zoe's age. What will be the sum of the two digits of Joey's age the next time his age is a multiple of Zoe's age?
$\textbf{(A) } 7 \qquad \textbf{(B) } 8 \qquad \textbf{(C) } 9 \qquad \textbf{(D) } 10 \qquad \textbf{(E) } 11 $
Find the sum of all non-zero digits that can repeat at the end of a perfect square. (For example, if $811$ were a perfect square, $1$ would be one of these non-zero digits.)
Let $n \geq 5$ be a given integer. Determine the greatest integer $k$ for which there exists a polygon with $n$ vertices (convex or not, with non-selfintersecting boundary) having $k$ internal right angles.
[i]Proposed by Juozas Juvencijus Macys, Lithuania[/i]
nic$\kappa$y is drawing kappas in the cells of a square grid. However, he does not want to draw kappas in three consecutive cells (horizontally, vertically, or diagonally). Find all real numbers $d>0$ such that for every positive integer $n,$ nic$\kappa$y can label at least $dn^2$ cells of an $n\times n$ square.
[i]Proposed by Mihir Singhal and Michael Kural[/i]
A convex quadrillateral $ABCD$ is given and the intersection point of the diagonals is denoted by $O$. Given that the perimeters of the triangles $ABO, BCO, CDO,ADO$ are equal, prove that $ABCD$ is a rhombus.
Is it possible to find $100$ positive integers not exceeding $25,000$, such that all pairwise sums of them are different?
In three years, Xingyou’s age in years will be twice his current height in feet. If Xingyou’s current age in years is also his current height in feet, what is Xingyou’s age in years right now?
Let be two real numbers $ a<b, $ a natural number $ n\ge 2, $ and a continuous function $ f:[a,b]\longrightarrow (0,\infty ) $ whose image contains $ 1 $ and that admits a primitive $ F:[a,b]\longrightarrow [a,b] . $ Prove that there is a real number $ c\in (a,b) $ such that
$$ (\underbrace{F\circ\cdots\circ F}_{\text{n times}} )(b) -(\underbrace{F\circ\cdots\circ F}_{\text{n times}} )(a) =(f(c))^{n+1} (b-a) $$
[i]Vlad Mihaly[/i]
Find the value of $ k\ (0<k<5)$ such that $ \int_0^{\infty} \frac{x^k}{2\plus{}4x\plus{}3x^2\plus{}5x^3\plus{}3x^4\plus{}4x^5\plus{}2x^6}\ dx$ is minimal.
If $f(x, y) = 3x^2 + 3xy + 1$ and $f(a, b) + 1 = f(b, a) = 42$, then determine $|a + b|$.
Let $n \geq 2$ be an integer. Positive reals $x_1, x_2, \cdots, x_n$ satisfy $x_1x_2 \cdots x_n = 1$.
Show: $$\{x_1\} + \{x_2\} + \cdots + \{x_n\} < \frac{2n-1}{2}$$
Where $\{x\}$ denotes the fractional part of $x$.
Let $n \ge 3$ and $\sigma \in S_n$ a permutation of the first $n$ positive integers. Prove that the numbers $\sigma (1), 2\sigma (2), 3\sigma(3), ... , n\sigma (n)$ cannot form an arithmetic, nor a geometric progression.
Let $f (x) = x^2 + ax + b$ be a quadratic function with real coefficients $a, b$. It is given that the equation $f (f (x)) = 0$ has $4$ distinct real roots and the sum of $2$ roots among these roots is equal to $-1$. Prove that $b \le -\frac14$
Does there exist an integer such that its cube is equal to $3n^2 +3n+7$, where $n$ is integer?
Prove that there exists a constant $K$ such that the following inequality holds for any sequence of positive numbers $a_1 , a_2 , a_3 , \ldots:$
$$\sum_{n=1}^{\infty} \frac{n}{a_1 + a_2 +\ldots + a_n } \leq K \sum_{n=1}^{\infty} \frac{1}{a_{n}}.$$
Does there exist a natural number $n \geq 2$ such that:
a) $\frac{2^{n-1}+1}{n}$ is a natural number?
b) $\frac{2^{2n-1}-1}{n}$ is a prime number?
Let be a natural number $ n\ge 2 $ and $ n $ positive real numbers $ a_1,a_n,\ldots ,a_n $ that satisfy the inequalities
$$ \sum_{j=1}^i a_j\le a_{i+1} ,\quad \forall i\in\{ 1,2,\ldots ,n-1 \} . $$
Prove that
$$ \sum_{k=1}^{n-1} \frac{a_k}{a_{k+1}}\le n/2 . $$
Find the limit $\lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^n k\ln \left(\frac{n^2+(k-1)^2}{n^2+k^2}\right).$
[i]2010 Yokohama National University entrance exam/Engineering, 2nd exam[/i]
Let $\omega_1$ and $\omega_2$ be two non-intersecting circles. Suppose the following three conditions hold:
[list]
[*]The smallest of a common internal tangent of $\omega_1$ and $\omega_2$ is equal to $19.$
[*]The length of a common external tangent of $\omega_1$ and $\omega_2$ is equal to $37.$
[*]If two points $X$ and $Y$ are selected on $\omega_1$ and $\omega_2,$ respectively, uniformly at random, then the expected value of $XY^2$ is $2023.$
[/list]
Compute the distance between the centers of $\omega_1$ and $\omega_2.$
Let $O$ and $H$ be the circumcenter and the orthocenter, respectively, of an acute triangle $ABC$. Points $D$ and $E$ are chosen from sides $AB$ and $AC$, respectively, such that $A$, $D$, $O$, $E$ are concyclic. Let $P$ be a point on the circumcircle of triangle $ABC$. The line passing $P$ and parallel to $OD$ intersects $AB$ at point $X$, while the line passing $P$ and parallel to $OE$ intersects $AC$ at $Y$. Suppose that the perpendicular bisector of $\overline{HP}$ does not coincide with $XY$, but intersect $XY$ at $Q$, and that points $A$, $Q$ lies on the different sides of $DE$. Prove that $\angle EQD = \angle BAC$.
[i]Proposed by Shuang-Yen Lee[/i]