Found problems: 85335
On the plane three different points $P, Q$, and $R$ are chosen. It is known that however one chooses another point $X$ on the plane, the point $P$ is always either closer to $X$ than the point $Q$ or closer to $X$ than the point $R$. Prove that the point $P$ lies on the line segment $QR$.
Find the smallest positive integer $n$ or show no such $n$ exists, with the following property: there are infinitely many distinct $n$-tuples of positive rational numbers $(a_1, a_2, \ldots, a_n)$ such that both
$$a_1+a_2+\dots +a_n \quad \text{and} \quad \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n}$$
are integers.
Prove that for all primes $p>3$, $\binom{2p}{p}-2$ is divisible by $p^3$
A weighted complete graph with distinct positive wights is given such that in every triangle is [i]degenerate [/i] that is wight of an edge is equal to sum of two other. Prove that one can assign values to the vertexes of this graph such that the wight of each edge is the difference between two assigned values of the endpoints.
[i]Proposed by Morteza Saghafian [/i]
Consider a triangle $ABC$ and two congruent triangles $A_1B_1C_1$ and $A_2B_2C_2$ which are respectively similar to $ABC$ and inscribed in it: $A_i,B_i,C_i$ are located on the sides of $ABC$ in such a way that the points $A_i$ are on the side opposite to $A$, the points $B_i$ are on the side opposite to $B$, and the points $C_i$ are on the side opposite to $C$ (and the angle at A are equal to angles at $A_i$ etc.).
The circumcircles of $A_1B_1C_1$ and $A_2B_2C_2$ intersect at points $P$ and $Q$. Prove that the line $PQ$ passes through the orthocenter of $ABC$.
The coordinate of an object is given as a function of time by $x = 8t - 3t^2$, where $x$ is in meters and $t$ is in seconds. Its average velocity over the interval from $ t = 1$ to $t = 2 \text{ s}$ is
$ \textbf{(A)}\ -2\text{ m/s}\qquad\textbf{(B)}\ -1\text{ m/s}\qquad\textbf{(C)}\ -0.5\text{ m/s}\qquad\textbf{(D)}\ 0.5\text{ m/s}\qquad\textbf{(E)}\ 1\text{ m/s} $
[b]a)[/b] Find an example of an infinite list of numbers of the form $a + n \cdot d$, with $n \geqslant 0$, where $a$ and $d$ are positive integers, such that no number in the list is equal to the $k$-th power of an integer, for all $k = 2, 3, 4, \dots$
[b]b)[/b] Find an example of an infinite list of numbers of the form $a + n \cdot d$, with $n \geqslant 0$, where $a$ and $d$ are positive integers, such that no number in the list is equal to the square of an integer, but the list contains infinitely many numbers that are equal to the cubes of positive integers.
Suppose a function $f(x)$ is continuous on $[-1,\ 1]$ and satisfies the condition :
1) $f(-1)\geq f(1).$
2) $x+f(x)$ is non decreasing function.
3) $\int_{-1}^ 1 f(x)\ dx=0.$
Show that $\int_{-1}^1 f(x)^2dx\leq \frac 23.$
Let $r, s \geq 1$ be integers and $a_{0}, a_{1}, . . . , a_{r-1}, b_{0}, b_{1}, . . . , b_{s-1} $ be real non-negative numbers such that $(a_0+a_1x+a_2x^2+. . .+a_{r-1}x^{r-1}+x^r)(b_0+b_1x+b_2x^2+. . .+b_{s-1}x^{s-1}+x^s) =1 +x+x^2+. . .+x^{r+s-1}+x^{r+s}$.
Prove that each $a_i$ and each $b_j$ equals either $0$ or $1$.
Steve is piling $m\geq 1$ indistinguishable stones on the squares of an $n\times n$ grid. Each square can have an arbitrarily high pile of stones. After he finished piling his stones in some manner, he can then perform [i]stone moves[/i], defined as follows. Consider any four grid squares, which are corners of a rectangle, i.e. in positions $(i, k), (i, l), (j, k), (j, l)$ for some $1\leq i, j, k, l\leq n$, such that $i<j$ and $k<l$. A stone move consists of either removing one stone from each of $(i, k)$ and $(j, l)$ and moving them to $(i, l)$ and $(j, k)$ respectively, or removing one stone from each of $(i, l)$ and $(j, k)$ and moving them to $(i, k)$ and $(j, l)$ respectively.
Two ways of piling the stones are equivalent if they can be obtained from one another by a sequence of stone moves.
How many different non-equivalent ways can Steve pile the stones on the grid?
Find the number of divisors of $20^{22}$ that are perfect squares.
Let $a, b$ and $c$ be positive real numbers such that $$a^2+ab+ac-bc = 0.$$
a) Show that if two of the numbers $a, b$ and $c$ are equal, then at least one of the numbers $a, b$ and $c$ is irrational.
b) Show that there exist infinitely many triples $(m, n, p)$ of positive integers such that $$m^2 + mn + mp -np = 0.$$
Let $m,\ n$ be non negative integers.
Calculate
\[\sum_{k=0}^{n}(-1)^{k}\frac{n+m+1}{k+m+1}\ nC_{k}. \]
where $_{i}C_{j}$ is a binomial coefficient which means $\frac{i\cdot (i-1)\cdots(i-j+1)}{j\cdot (j-1)\cdots 2\cdot 1}$.
Tim is participating in the following three math contests. On each contest his score is the number of correct answers.
$\bullet$ The Local Area Inspirational Math Exam consists of 15 problems.
$\bullet$ The Further Away Regional Math League has 10 problems.
$\bullet$ The Distance-Optimized Math Open has 50 problems.
For every positive integer $n$, Tim knows the answer to the $n$th problems on each contest (which are pairwise distinct), if they exist; however, these answers have been randomly permuted so that he does not know which answer corresponds to which contest. Unaware of the shuffling, he competes with his modified answers. Compute the expected value of the sum of his scores on all three contests.
[i]Proposed by Evan Chen[/i]
Katie has a fair $2019$-sided die with sides labeled $1, 2,..., 2019$. After each roll, she replaces her $n$-sided die with an $(n+1)$-sided die having the $n$ sides of her previous die and an additional side with the number she just rolled. What is the probability that Katie's $2019^{th}$ roll is a $ 2019$?
It is known that for all $x$ such that $|x| < 1$, the following inequality holds $$ax^2+bx+c\le \frac{1}{\sqrt{1-x^2}}$$Find the greatest value of $a + 2c$.
How many ways are there to write $657$ as a sum of powers of two where each power of two is used at
most twice in the sum? For example, $256+256+128+16+1$ is a valid sum.
Let $ \mathcal P$ be a parabola, and let $ V_1$ and $ F_1$ be its vertex and focus, respectively. Let $ A$ and $ B$ be points on $ \mathcal P$ so that $ \angle AV_1 B \equal{} 90^\circ$. Let $ \mathcal Q$ be the locus of the midpoint of $ AB$. It turns out that $ \mathcal Q$ is also a parabola, and let $ V_2$ and $ F_2$ denote its vertex and focus, respectively. Determine the ratio $ F_1F_2/V_1V_2$.
Let $A$, $B$, and $C$ be three points such that $B$ is the midpoint of segment $AC$ and let $P$ be a point such that $<PBC=60$. Equilateral triangle $PCQ$ is constructed such that $B$ and $Q$ are on different half=planes with respect to $PC$, and the equilateral triangle $APR$ is constructed in such a way that $B$ and $R$ are in the same half-plane with respect to $AP$. Let $X$ be the point of intersection of the lines $BQ$ and $PC$, and let $Y$ be the point of intersection of the lines $BR$ and $AP$. Prove that $XY$ and $AC$ are parallel.
On plane circles $\omega_1, \omega_2, \omega_3$ with centers $O_1,O_2,O_3$ are given such that $\omega_1$ is externally tangent $\omega_2$ and $\omega_3$ at points $P, Q$ respectively. On $\omega_1$ point $C$ is chosen arbitrarily. Line $CP$ intersects $\omega_2$ at $B$, line $CQ$ intersects $\omega_3$ at $A$. Point $O$ is the circumcenter of $ABC$.
Prove that the locus of points $O$ (when $C$ changes) is a circle, the center of which lies on the circumcircle of $O_1O_2O_3$
Let $a,b$ be the roots of the equation $x^2-px+q = 0$ and let $c, d$ be the roots of the equation $x^2 - rx + s = 0$, where $p, q, r,s$ are some positive real numbers. Suppose that $M =\frac{2(abc + bcd + cda + dab)}{p^2 + q^2 + r^2 + s^2}$ is an integer. Determine $a, b, c, d$.
Let $ X$ the set of all sequences $ \{a_1, a_2,\ldots , a_{2000}\}$, such that each of the first 1000 terms is 0, 1 or 2, and each of the remaining terms is 0 or 1. The [i]distance[/i] between two members $ a$ and $ b$ of $ X$ is defined as the number of $ i$ for which $ a_i$ and $ b_i$ are different.
Find the number of functions $ f : X \to X$ which preserve the distance.
Let $k$ be a positive integer. Consider $k$ not necessarily distinct prime numbers such that their product is ten times their sum. What are these primes and what is the value of $k$?
Let $N$ be the set of those positive integers $n$ for which $n\mid k^k-1$ implies $n\mid k-1$ for every positive integer $k$. Prove that if $n_1,n_2\in N$, then their greatest common divisor is also in $N$.
Does there exist a nonnegative integer $a$ for which the equation
\[\left\lfloor\frac{m}{1}\right\rfloor + \left\lfloor\frac{m}{2}\right\rfloor + \left\lfloor\frac{m}{3}\right\rfloor + \cdots + \left\lfloor\frac{m}{m}\right\rfloor = n^2 + a\]
has more than one million different solutions $(m, n)$ where $m$ and $n$ are positive integers?
[i]The expression $\lfloor x\rfloor$ denotes the integer part (or floor) of the real number $x$. Thus $\lfloor\sqrt{2}\rfloor = 1, \lfloor\pi\rfloor =\lfloor 22/7 \rfloor = 3, \lfloor 42\rfloor = 42,$ and $\lfloor 0 \rfloor = 0$.[/i]