Found problems: 85335
Let $ a$ and $b$ be positive integers. Prove that there exist positive integers $ x $ and $ y $ such that
\[ \binom{x+y}{2} = ax + by . \]
Let $ABC$ be an equilateral triangle of side length $15.$ Let $A_b$ and $B_a$ be points on side $AB,$ $A_c$ and $C_a$ be points on $AC,$ and $B_c$ and $C_b$ be points on $BC$ such that $\triangle{AA_bA_c}, \triangle{BB_cB_a},$ and $\triangle{CC_aC_b}$ are equilateral triangles with side lengths $3,4,$ and $5,$ respectively. Compute the radius of the circle tangent to segments $\overline{A_bA_c}, \overline{B_aB_c},$ and $\overline{C_aC_b}.$
Find the positive real numbers $x,y$ for which $\frac{x+y}{2},\sqrt{xy},\frac{2xy}{x+y},\sqrt{\frac{x^2 +y^2}{2}}$ are integers whose sum is $66$.
For two different prime numbers $p, q$, defi ne $S_{p,q} = \{p,q,pq\}$. If two elements in $S_{p,q}$ are numbers in the form of $x^2 + 2005y^2, (x, y \in Z)$, prove that all three elements in $S_{p,q}$ are in such form.
Prove that sum of $12$ consecutive integers cannot be a square. Give an example of $11$ consecutive integers whose sum is a perfect square.
In the rectangle $ABCD$, point $M$ is the midpoint of the side $BC$. The points $P$ and $Q$ lie on the diagonal $AC$ such that $\angle DPC = \angle DQM = 90^o$. Prove that $Q$ is the midpoint of the segment $AP$.
$(a)$ Prove that for $a, b, c, d \in\mathbb{R}, m \in [1,+\infty)$ with $am + b =-cm + d = m$,
\[(i)\sqrt{a^2 + b^2}+\sqrt{c^2 + d^2}+\sqrt{(a-c)^2 + (b-d)^2}\ge \frac{4m^2}{1+m^2},\text{ and}\]
\[(ii) 2 \le \frac{4m^2}{1+m^2} < 4.\]
$(b)$ Express $a, b, c, d$ as functions of $m$ so that there is equality in $(i).$
Let $a,b\in \mathbb{C}$. Prove that $\left| az+b\bar{z} \right|\le 1$, for every $z\in \mathbb{C}$, with $\left| z \right|=1$, if and only if $\left| a \right|+\left| b \right|\le 1$.
For any natural number $n{}$ and positive integer $k{}$, we say that $n{}$ is $k-good$ if there exist non-negative integers $a_1,\ldots, a_k$ such that $$n=a_1^2+a_2^4+a_3^8+\ldots+a_k^{2^k}.$$ Is there a positive integer $k{}$ for which every natural number is $k-good$?
Let $ a,b,c$ be the three sides of a triangle. Determine all possible values of $ \frac {a^2 \plus{} b^2 \plus{} c^2}{ab \plus{} bc \plus{} ca}$
Any circle in the $xy$-plane is "represented" by a point on the vertical line through the center of the circle and at a distance "above" the plane of the circle equal to the radius of the circle.
Show that the locus of the representations of all the circles which cut a fixed circle at a constant angle is a portion of a one-sheeted hyperboloid.
By consideration of a suitable family of circles in the plane, demonstrate the existence of two families of rulings on the hyperboloid.
To $m$ ounces of a $m\%$ solution of acid, $x$ ounces of water are added to yield a $(m-10)\%$ solution. If $m>25$, then $x$ is
$\textbf{(A)}\ \frac{10m}{m-10} \qquad
\textbf{(B)}\ \frac{5m}{m-10} \qquad
\textbf{(C)}\ \frac{m}{m-10} \qquad
\textbf{(D)}\ \frac{5m}{m-20} \\
\textbf{(E)}\ \text{not determined by the given information}$
Let $X,Y,Z,T$ be four points in the plane. The segments $[XY]$ and $[ZT]$ are said to be [i]connected[/i], if there is some point $O$ in the plane such that the triangles $OXY$ and $OZT$ are right-angled at $O$ and isosceles.
Let $ABCDEF$ be a convex hexagon such that the pairs of segments $[AB],[CE],$ and $[BD],[EF]$ are [i]connected[/i]. Show that the points $A,C,D$ and $F$ are the vertices of a parallelogram and $[BC]$ and $[AE]$ are [i]connected[/i].
In the sequence of the natural (i.e. positive integers) numbers every member from the third equals the absolute value of the difference of the two previous. What is the maximal possible length of such a sequence, if every member is less or equal to $1967$?
Each interior point of an equilateral triangle with sides equal to $1$ lies in one of six circles of the same radius $ r $. Prove that $ r \geq \frac {{\sqrt 3}} {{10}} $.
Let $M$ be the point of intersection of diagonals $AC$ and $BD$ of the convex quadrilateral $ABCD$. Let $K$ be the point of intersection of the extension of side $AB$ (beyond$A$) with the bisector of the angle $ACD$. Let $L$ be the intersection of $KC$ and $BD$. If $MA \cdot CD = MB \cdot LD$, prove that the angle $BKC$ is equal to the angle $CDB$.
Right triangle $ABC$ has side lengths $BC=6$, $AC=8$, and $AB=10$. A circle centered at $O$ is tangent to line $BC$ at $B$ and passes through $A$. A circle centered at $P$ is tangent to line $AC$ at $A$ and passes through $B$. What is $OP$?
$\textbf{(A)} ~\frac{23}{8}\qquad\textbf{(B)} ~\frac{29}{10}\qquad\textbf{(C)} ~\frac{35}{12}\qquad\textbf{(D)} ~\frac{73}{25}\qquad\textbf{(E)} ~3$
Find all triples $(a,b,c)$ of real numbers for which there exists a function $f:\mathbb{Z}^{+}\rightarrow\mathbb{Z}^{+}$ satisfying $af(n)+bf(n+1)+cf(n+2)<0$ for every $n\in\mathbb{Z}^{+}$ ($\mathbb{Z}^{+}$ denotes the set of positive integers).
Proposed by [i]András Imolay[/i], Budapest
The function $ f(x,y,z)\equal{}\frac{x^2\plus{}y^2\plus{}z^2}{x\plus{}y\plus{}z}$ is defined for every $ x,y,z \in R$ whose sum is not 0. Find a point $ (x_0,y_0,z_0)$ such that $ 0 < x_0^2\plus{}y_0^2\plus{}z_0^2 < \frac{1}{1999}$ and $ 1.999 < f(x_0,y_0,z_0) < 2$.
Let $F$ be the set of all $n-tuples$ $(A_1,A_2,…,A_n)$ such that each $A_i$ is a subset of ${1,2,…,2019}$. Let $\mid{A}\mid$ denote the number of elements o the set $A$ . Find
$\sum_{(A_1,…,A_n)\in{F}}^{}\mid{A_1\cup{A_2}\cup...\cup{A_n}}\mid$
Let $p\geq 3$ a prime number, $a$ and $b$ integers such that $\gcd(a, b)=1$. Let $n$ a natural number such that $p$ divides $a^{2^n}+b^{2^n}$, prove that $2^{n+1}$ divides $p-1$.
Consider the sequence defined recursively by $(x_1,y_1)=(0,0)$,
$(x_{n+1},y_{n+1})=\left(\left(1-\frac{2}{n}\right)x_n-\frac{1}{n}y_n+\frac{4}{n},\left(1-\frac{1}{n}\right)y_n-\frac{1}{n}x_n+\frac{3}{n}\right)$.
Find $\lim_{n\to \infty}(x_n,y_n)$.
How many integers between $1000$ and $9999$ have four distinct digits?
$\textbf{(A) }3024\qquad\textbf{(B) }4536\qquad\textbf{(C) }5040\qquad\textbf{(D) }6480\qquad \textbf{(E) }6561$
Let $b,m,n\in\mathbb{N}$ with $b>1$ and $m\not=n$. Suppose that $b^{m}-1$ and $b^{n}-1$ have the same set of prime divisors. Show that $b+1$ must be a power of $2$.
Find the smallest constant $C > 0$ for which the following statement holds: among any five positive real numbers $a_1,a_2,a_3,a_4,a_5$ (not necessarily distinct), one can always choose distinct subscripts $i,j,k,l$ such that
\[ \left| \frac{a_i}{a_j} - \frac {a_k}{a_l} \right| \le C. \]