Found problems: 85335
Given an acute triangle $ ABC$ with $ \angle B > \angle C$. Point $ I$ is the incenter, and $ R$ the circumradius. Point $ D$ is the foot of the altitude from vertex $ A$. Point $ K$ lies on line $ AD$ such that $ AK \equal{} 2R$, and $ D$ separates $ A$ and $ K$. Lines $ DI$ and $ KI$ meet sides $ AC$ and $ BC$ at $ E,F$ respectively. Let $ IE \equal{} IF$.
Prove that $ \angle B\leq 3\angle C$.
[i]Author: Davoud Vakili, Iran[/i]
Prove that there are no $1999$ primes in an arithmetic progression that are all less than $12345$.
Let $n$ be a positive integer that is not a perfect cube. Define real numbers $a$, $b$, $c$ by \[a=\sqrt[3]{n}, \; b=\frac{1}{a-\lfloor a\rfloor}, \; c=\frac{1}{b-\lfloor b\rfloor}.\] Prove that there are infinitely many such integers $n$ with the property that there exist integers $r$, $s$, $t$, not all zero, such that $ra+sb+tc=0$.
$ ABC$ is a triangle with $ \angle BAC \equal{} 10{}^\circ$, $ \angle ABC \equal{} 150{}^\circ$. Let $ X$ be a point on $ \left[AC\right]$ such that $ \left|AX\right| \equal{} \left|BC\right|$. Find $ \angle BXC$.
$\textbf{(A)}\ 15^\circ \qquad\textbf{(B)}\ 20^\circ \qquad\textbf{(C)}\ 25^\circ \qquad\textbf{(D)}\ 30^\circ \qquad\textbf{(E)}\ 35^\circ$
In the following figure, the bigger wheel has circumference $12$m and the inscribed wheel has circumference $8 $m.
$P_{1}$ denotes a point on the bigger wheel and $P_{2}$ denotes a point on the smaller wheel. Initially $P_{1}$ and $P_{2}$ coincide as in the figure. Now we roll the wheels on a smooth surface and the smaller wheel also rolls in the bigger wheel smoothly. What distance does the bigger wheel have to roll so that the points will be together again?
Prove that among the seven natural numbers forming an arithmetic progression with difference $ 30 $ , one and only one is divisible by $ 7 $ .
$n$ being a given integer, find all functions $f\colon \mathbb{Z} \to \mathbb{Z}$, such that for all integers $x,y$ we have $f\left( {x + y + f(y)} \right) = f(x) + ny$.
$ABC$ is an acute angle triangle such that $AB>AC$ and $\hat{BAC}=60^{\circ}$. Let's denote by $O$ the center of the circumscribed circle of the triangle and $H$ the intersection of altitudes of this triangle. Line $OH$ intersects $AB$ in point $P$ and $AC$ in point $Q$. Find the value of the ration $\frac{PO}{HQ}$.
Let $E$ be a point lies inside the parallelogram $ABCD$ such that $\angle BCE = \angle BAE$.
Prove that the circumcenters of triangles $ABE,BCE,CDE,DAE$ are concyclic.
How many rectangles are in the following figure?
[asy]
size(80);
draw((0,0)--(3,0)--(3,4)--(0,4)--cycle);
draw((0,2)--(3,2));
draw((0.75,2)--(0.75,0));
draw((2.25,2)--(2.25,0));
[/asy]
$\textbf{(A) }5\qquad\textbf{(B) }6\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9$
Let $n{}$ be a positive integer. What is the smallest sum of digits that $5^n + 6^n + 2022^n$ can take?
Prove that for any finite set $A$ of positive integers, there exists a subset $B$ of $A$ satisfying the following conditions:
[list][*]if $b_1,b_2\in B$ are distinct, then neither $b_1$ and $b_2$ nor $b_1+1$ and $b_2+1$ are multiples of each other, and
[*] for any $a\in A$, we can find a $b\in B$ such that $a$ divides $b$ or $b+1$ divides $a+1$.[/list]
An investigator works out that he needs to ask at most $91$ questions on the basis that all the answers will be yes or no and all will be true. The questions may depend upon the earlier answers. Show that he can make do with $105$ questions if at most one answer could be a lie.
Given an ellipse that is not a circle.
(1) Prove that the rhombus tangent to the ellipse at all four of its sides with minimum area is unique.
(2) Construct this rhombus using a compass and a straight edge.
For some real number $r$, the polynomial $8x^3-4x^2-42x+45$ is divisible by $(x-r)^2$. Which of the following numbers is closest to $r$?
$\text{(A)} \ 1.22 \qquad \text{(B)} \ 1.32 \qquad \text{(C)} \ 1.42 \qquad \text{(D)} \ 1.52 \qquad \text{(E)} \ 1.62$
Let $F_1 = 0$, $F_2 = 1$, and $F_n = F_{n-1} + F_{n-2}$. Compute $$\sum_{n=1}^\infty \frac{\sum_{n=1}^\infty F_i}{3^n}.$$
The cells of an $8\times 8$ chessboard are all coloured in white. A move consists in inverting the colours of a rectangle $1 \times 3$ horizontal or vertical (the white cells become black and conversely). Is it possible to colour all the cells of the chessboard in black in a finite number of moves ?
A finite set $M$ of real numbers has the following properties: $M$ has at least $4$ elements, and there exists a bijective function $f:M\to M$, different from the identity, such that $ab\leq f(a)f(b)$ for all $a\neq b\in M.$ Prove that the sum of the elements of $M$ is $0.$
Let $(a_n)_{n\geq0}$ and $(b_n)_{n \geq 0}$ be two sequences of natural numbers. Determine whether there exists a pair $(p, q)$ of natural numbers that satisfy
\[p < q \quad \text{ and } \quad a_p \leq a_q, b_p \leq b_q.\]
Let $O$ be the circumcenter and $H$ the orthocenter of an acute-angled triangle $ABC$ such that $BC>CA$. Let $F$ be the foot of the altitude $CH$ of triangle $ABC$. The perpendicular to the line $OF$ at the point $F$ intersects the line $AC$ at $P$. Prove that $\measuredangle FHP=\measuredangle BAC$.
There exists a block of 1000 consecutive positive integers containing no prime numbers, namely, $1001!+2,1001!+3,...,1001!+1001$. Does there exist a block of 1000 consecutive positive intgers containing exactly five prime numbers?
How many ways can the vertices of a cube be colored red or blue so that the color of each vertex is the color of the majority of the three vertices adjacent to it?
[i]Proposed by Milan Haiman.[/i]
Let $ABC$ be a triangle with $\angle B = 90^o$. Given that there exists a point $D$ on $AC$ such that $AD = DC$ and $BD = BC$, compute the value of the ratio $\frac{AB}{BC}$ .
Compute the number of ordered pairs $(a,b)$ of positive integers satisfying $a^b=2^{100}$.
[i]Proposed by Nathan Xiong[/i]
Let $S$ be the set of all ordered pairs $(a,b)$ of integers with $0<2a<2b<2017$ such that $a^2+b^2$ is a multiple of $2017$. Prove that \[\sum_{(a,b)\in S}a=\frac{1}{2}\sum_{(a,b)\in S}b.\]
Proposed by Uwe Leck, Germany