Found problems: 85335
Let $f$ be a differentiable function such that $f'(0)=4$ and $f(0)=3$. Compute
$$\lim_{x\rightarrow\infty}\left(\frac{f\left(\frac{1}{x}\right)}{f(0)}\right)^x.$$
Find all 4-digit numbers $n$, such that $n=pqr$, where $p<q<r$ are distinct primes, such that $p+q=r-q$ and $p+q+r=s^2$, where $s$ is a prime number.
Let $ABC$ be a triangle in which $BC<AC$. Let $M$ be the mid-point of $AB$, $AP$ be the altitude from $A$ on $BC$, and $BQ$ be the altitude from $B$ on to $AC$. Suppose that $QP$ produced meets $AB$ (extended) at $T$. If $H$ is the orthocenter of $ABC$, prove that $TH$ is perpendicular to $CM$.
Given $5n$ real numbers $r_i, s_i, t_i, u_i, v_i \geq 1 (1 \leq i \leq n)$, let $R = \frac {1}{n} \sum_{i=1}^{n} r_i$, $S = \frac {1}{n} \sum_{i=1}^{n} s_i$, $T = \frac {1}{n} \sum_{i=1}^{n} t_i$, $U = \frac {1}{n} \sum_{i=1}^{n} u_i$, $V = \frac {1}{n} \sum_{i=1}^{n} v_i$. Prove that $\prod_{i=1}^{n}\frac {r_i s_i
t_i u_i v_i + 1}{r_i s_i t_i u_i v_i - 1} \geq \left(\frac {RSTUV +1}{RSTUV - 1}\right)^n$.
Trapezoid $ABCD,$ with $AB \parallel CD,$ has side lengths $AB=11, BC=8, CD=19,$ and $DA=4.$ Compute the area of the convex quadrilateral whose vertices are the circumcenters of $\triangle{ABC}, \triangle{BCD}, \triangle{CDA},$ and $\triangle{DAB}.$
Let $\phi(n)$ denote the number of positive integers less than or equal to $n$ that are relatively prime to $n$, and let $d(n)$ denote the number of positive integer divisors of $n$. For example, $\phi(6) = 2$ and $d(6) = 4$. Find the sum of all odd integers $n \le 5000$ such that $n \mid \phi(n) d(n)$.
[i]Alex Zhu[/i]
The following operation is performed over points $O_1, O_2, O_3$ and $A$ in space. The point $A$ is reflected with respect to $O_1$, the resultant point $A_1$ is reflected through $O_2$, and the resultant point $A_2$ through $O_3$. We get some point $A_3$ that we will also consecutively reflect through $O_1, O_2, O_3$. Prove that the point obtained last coincides with $A$..
We will say that a set of real numbers $A = (a_1,... , a_{17})$ is stronger than the set of real numbers $B = (b_1, . . . , b_{17})$, and write $A >B$ if among all inequalities $a_i > b_j$ the number of true inequalities is at least $3$ times greater than the number of false. Prove that there is no chain of sets $A_1, A_2, . . . , A_N$ such that $A_1>A_2> \cdots A_N>A_1$.
Remark: For 11.4, the constant $3$ is changed to $2$ and $N=3$ and $17$ is changed to $m$ and $n$ in the definition (the number of elements don't have to be equal).
Evaluate the sum $$\frac{1}{3^1} + \frac{2}{3^2} + \frac{3}{3^3} + \cdots + \frac{k}{3^k} + \cdots$$
$\text{(A) }\frac{5}{9}\qquad\text{(B) }\frac{5}{8}\qquad\text{(C) }\frac{2}{3}\qquad\text{(D) }\frac{3}{4}\qquad\text{(E) }\frac{7}{9}$
Let $c>0$ be a given positive real and $\mathbb{R}_{>0}$ be the set of all positive reals. Find all functions $f \colon \mathbb{R}_{>0} \to \mathbb{R}_{>0}$ such that \[f((c+1)x+f(y))=f(x+2y)+2cx \quad \textrm{for all } x,y \in \mathbb{R}_{>0}.\]
Let $a_1,a_2,a_3, \cdots $ be distinct positive integers, and $0<c<\frac{3}{2}$ . Prove that : There exist infinitely many positive integers $k$, such that $[a_k,a_{k+1}]>ck $.
In a triangle $ABC$, $\angle ABC= 60^o$, point $I$ is the incenter. Let the points $P$ and $T$ on the sides $AB$ and $BC$ respectively such that $PI \parallel BC$ and $TI \parallel AB$ , and points $P_1$ and $T_1$ on the sides $AB$ and $BC$ respectively such that $AP_1 = BP$ and $CT_1 = BT$. Prove that point $I$ lies on segment $P_1T_1$.
(Anton Trygub)
Let $a_1,a_2,a_3,\dots$ be a sequence of real numbers satisfying the inequality \[ |a_{k+m}-a_k-a_m| \leq 1 \quad \text{for all} \ k,m \in \mathbb{Z}_{>0}. \] Show that the following inequality holds for all positive integers $k,m$ \[ \left| \frac{a_k}{k}-\frac{a_m}{m} \right| < \frac{1}{k}+\frac{1}{m}. \]
Consider a cyclic quadrilateral $ABCD$. The extensions of its sides $AB,DC$ meet at the point $P$ and the extensions of its sides $AD,BC$ meet at the point $Q$. Suppose $\quad QE,QF$ are tangents to the circumcircle of quadrilateral $ABCD$ at $E,F$ respectively. Show that $P,E,F$ are collinear.
Let $a,b,c,d$ be positive reals such taht $a+b+c+d=1$.
Prove that: \[6(a^{3}+b^{3}+c^{3}+d^{3})\geq a^{2}+b^{2}+c^{2}+d^{2}+\frac{1}{8}.\]
With six rods a piece like the one in the figure is constructed. The three outer rods are equal to each other. The three inner rods are equal to each other. You want to paint each rod a single color so that at each joining point, the three arriving rods have a different color. The rods can only be painted blue, white, red or green. In how many ways can the piece be painted?
[img]https://cdn.artofproblemsolving.com/attachments/1/1/91e6b388498613486477ab6b51735055e920cc.gif[/img]
Find all positive integers $n$ such that $$\lfloor \sqrt{n} \rfloor +
\left\lfloor \frac{n}{\lfloor \sqrt{n} \rfloor} \right \rfloor> 2\sqrt{n}.$$
If $k$ is a real number, then $\lfloor k \rfloor$ means the floor of $k$, this is the greatest integer less than or equal to $k$.
Determine the set $S$ with minimal number of points defining $7$ distinct lines
In the cube $ABCDA'B'C'D'$, with side $a$, the plane $(AB'D')$ intersects the planes $(A'BC),(A'CD),(A'DB)$ after the lines $d_1,d_2$ and $d_3$ respectively.
a) Show that the lines $d_1,d_2,d_3$ intersect pairwise.
b) Determine the area of the triangle formed by these three lines.
A $1 \times 5$ rectangle is split into five unit squares (cells) numbered 1 through 5 from left to right. A frog starts at cell 1. Every second it jumps from its current cell to one of the adjacent cells. The frog makes exactly 14 jumps. How many paths can the frog take to finish at cell 5?
Let $ABCD$ be a square. Let $E, F, G$ and $H$ be the centers, respectively, of equilateral triangles with bases $\overline{AB}, \overline{BC}, \overline{CD},$ and $\overline{DA},$ each exterior to the square. What is the ratio of the area of square $EFGH$ to the area of square $ABCD$?
$\textbf{(A)}\ 1\qquad\textbf{(B)}\ \frac{2+\sqrt{3}}{3} \qquad\textbf{(C)}\ \sqrt{2} \qquad\textbf{(D)}\ \frac{\sqrt{2}+\sqrt{3}}{2} \qquad\textbf{(E)}\ \sqrt{3}$
Establish if there exists a finite set $M$ of points in space, not all situated in the same plane, so that for any straight line $d$ which contains at least two points from M there exists another straight line $d'$, parallel with $d,$ but distinct from $d$, which also contains at least two points from $M$.
Let $a,b,c$ be positive reals such that $a^2+b^2+c^2=3abc$. Prove that
\[\frac{a}{b^2c^2}+\frac{b}{c^2a^2}+\frac{c}{a^2b^2} \geq \frac{9}{a+b+c}\]
Two polynomials with real coefficients have the leading coefficients equal to 1 . Each polynomial has an odd degree that is equal to the number of its distinct real roots. The product of the values of the first polynomial at the roots of the second polynomial is equal to 2024. Find the product of the values of the second polynomial at the roots of the first one.
Sergey Yanzhinov
Let $ABC$ be a triangle with altitudes $AD, BE, CF$ and orthocenter $H$. The perpendicular bisector of $HD$ meets $EF$ at $P$ and $N$ is the center of the nine-point circle. Let $L$ be a point on the circumcircle of $ABC$ such that $\angle PLN=90^{\circ}$ and $A, L$ are in distinct sides of the line $PN$. Show that $ANDL$ is cyclic.