Found problems: 85335
For which numbers $n$ is it possible to put marks on a stick such that all distances $1$ cm, $2$ cm, . . . , $n$ cm each appear exactly once as the distance between two of the marks, and no other distance appears as such a distance?
Triangle $ABC$ has positive integer side lengths with $AB=AC$. Let $I$ be the intersection of the bisectors of $\angle B$ and $\angle C$. Suppose $BI=8$. Find the smallest possible perimeter of $\triangle ABC$.
Let $ABCD$ be a cyclic quadrilateral, $O$ be its circumcenter, $P$ be a common points of its diagonals, and $M , N$ be the midpoints of $AB$ and $CD$ respectively. A circle $OPM$ meets for the second time segments $AP$ and $BP$ at points $A_1$ and $B_1$ respectively and a circle $OPN$ meets for the second time segments $CP$ and $DP$ at points $C_1$ and $D_1$ respectively. Prove that the areas of quadrilaterals $AA_1B_1B$ and $CC_1D_1D$ are equal.
Let $ f: (1,\infty)\to\mathbb{R}$ be a differentiable function such that
\[ f'(x)\equal{}\frac{x^2\minus{}\left(f(x)\right)^2}{x^2\left(\left(f(x)\right)^2\plus{}1\right)}\quad\text{for all }x>1.\]
Prove that $ \displaystyle\lim_{x\to\infty}f(x)\equal{}\infty.$
An unfair coin lands on heads with a probability of $\tfrac{1}{4}$. When tossed $n$ times, the probability of exactly two heads is the same as the probability of exactly three heads. What is the value of $n$?
$ \textbf {(A) } 5 \qquad \textbf {(B) } 8 \qquad \textbf {(C) } 10 \qquad \textbf {(D) } 11 \qquad \textbf {(E) } 13 $
Let $k\ge 1$ be an integer and $\mathsf S$ be a family of 2-element subsets of the index set $\{1,\ldots,2k\}$ with the following property: if $\mathsf M_1,\ldots,\mathsf M_{2k}$ are arbitrary sets such that \[\mathsf M_i\cap\mathsf M_j\neq\emptyset\quad\Leftrightarrow\quad\{i,j\}\in\mathsf S,\] then the union $\mathsf M_1\cup\ldots\cup\mathsf M_{2k}$ contains at least $k^2$ elements. Show that there is a suitable family $\mathsf S$ for any integer $k\ge1.$
Of all positive integral solutions $(x,y,z)$ to the equation \[x^3+y^3+z^3-3xyz=607,\] compute the minimum possible value of $x+2y+3z.$
[i]Individual #7[/i]
Prove that the equation
$$ (x - a) (x - c) + 2 (x - b) (x - d) = 0,$$
in which $ a < b < c < d $, has two real roots.
For each positive integer $n$, let $M(n)$ be the $n\times n$ matrix whose $(i,j)$ entry is equal to $1$ if $i+1$ is divisible by $j$, and equal to $0$ otherwise. Prove that $M(n)$ is invertible if and only if $n+1$ is square-free. (An integer is [i]square-free[/i] if it is not divisible by a square of an integer larger than $1$.)
For a positive integer $n>2$, consider the $n-1$ fractions $$\dfrac21, \dfrac32, \cdots, \dfrac{n}{n-1}$$ The product of these fractions equals $n$, but if you reciprocate (i.e. turn upside down) some of the fractions, the product will change. Can you make the product equal 1? Find all values of $n$ for which this is possible and prove that you have found them all.
Let $A$ be the set of all polynomials $f(x)$ of order $3$ with integer coefficients and cubic coefficient $1$, so that for every $f(x)$ there exists a prime number $p$ which does not divide $2004$ and a number $q$ which is coprime to $p$ and $2004$, so that $f(p)=2004$ and $f(q)=0$.
Prove that there exists a infinite subset $B\subset A$, so that the function graphs of the members of $B$ are identical except of translations
What is the graph of $y^4+1=x^4+2y^2$ in the coordinate plane?
$ \textbf{(A)}\ \textbf{Two intersecting parabolas} \qquad
\textbf{(B)}\ \textbf{Two nonintersecting parabolas} \qquad
\textbf{(C)}\ \textbf{Two intersecting circles} \qquad
\textbf{(D)}\ \textbf{A circle and a hyperbola} \qquad
\textbf{(E)}\ \textbf{A circle and two parabolas}$
Three different points $A,B$ and $C$ lie on a circle with center $M$ so that $| AB | = | BC |$. Point $D$ is inside the circle in such a way that $\vartriangle BCD$ is equilateral. Let $F$ be the second intersection of $AD$ with the circle . Prove that $| F D | = | FM |$.
Joe has a collection of 23 coins, consisting of 5-cent coins, 10-cent coins, and 25-cent coins. He has 3 more 10-cent coins than 5-cent coins, and the total value of his collection is 320 cents. How many more 25-cent coins does Joe have than 5-cent coins?
$\textbf{(A) } 0 \qquad \textbf{(B) } 1 \qquad \textbf{(C) } 2 \qquad \textbf{(D) } 3 \qquad \textbf{(E) } 4 $
Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.
Determine what day of the week day was: June $6$, $1944$.
Note: Leap years are those that are multiples of $4$ and do not end in $00$ or that are multiples of $400$, for example $1812$, $1816$, $1820$, $1600$, $2000$, but $1800$, $1810$, $2100$ are not leaps.
Giving the answer without any mathematical justification will not award points.
Sum of $m$ pairwise different positive even numbers and $n$ pairwise different positive odd numbers is equal to $1987$. Find, with proof, the maximum value of $3m+4n$.
There are 36 contestants in the CMU Puyo-Puyo Tournament, each with distinct skill levels.
The tournament works as follows:
First, all $\binom{36}{2}$ pairings of players are written down on slips of paper and are placed in a hat.
Next, a slip of paper is drawn from the hat, and those two players play a match. It is guaranteed that the player with a higher skill level will always win the match.
We continue drawing slips (without replacement) and playing matches until the results of the match completely determine the order of skill levels of all 36 contestants (i.e. there is only one possible ordering of skill levels consistent with the match results), at which point the tournament immediately finishes.
What is the expected value of the number of matches played before the stopping point is reached?
[i]Proposed by Dilhan Salgado[/i]
In the country of Princetonia, there are an infinite number of cities, connected by roads. For every two distinct cities, there is a unique sequence of roads that leads from one city to the other. Moreover, there are exactly three roads from every city. On a sunny morning in early July, n tourists have arrived at the capital of Princetonia. They repeat the following process every day: in every city that contains three or more tourists, three tourists are picked and one moves to each of the three cities connected to the original one by roads. If there are $2$ or fewer tourists in the city, they do nothing. After some time, all tourists will settle and there will be no more changing cities. For how many values of n from $1$ to $2020$ will the tourists end in a configuration in which no two of them are in the same city?
A whole number larger than $2$ leaves a remainder of $2$ when divided by each of the numbers $3, 4, 5$ and $6$. The smallest such number lies between which two numbers?
$\textbf{(A)}\ 40\text{ and }49\qquad
\textbf{(B)}\ 60\text{ and }79\qquad
\textbf{(C)}\ 100\text{ and }129\qquad
\textbf{(D)}\ 210\text{ and }249\qquad
\textbf{(E)}\ 320\text{ and }369$
All squares of a $ 20\times 20$ table are empty. Misha* and Sasha** in turn put chips in free squares (Misha* begins). The player after whose move there are four chips on the intersection of two rows and two columns wins. Which of the players has a winning strategy?
[i]Proposed by A. Golovanov[/i]
[b]US Name Conversions: [/b]
[i]Misha*: Naoki
Sasha**: Richard[/i]
Show that for any $n \in N$ the polynomial $f(x) = (x^2 +x)^{2^n}+1$ is irreducible over $Z[x]$.
Prove that for each positive integer $k$ there exists a number base $b$ along with $k$ triples of Fibonacci numbers $(F_u,F_v,F_w)$ such that when they are written in base $b$, their concatenation is also a Fibonacci number written in base $b$. (Fibonacci numbers are defined by $F_1 = F_2 = 1$ and $F_{n+2} = F_{n+1} + F_n$ for all positive integers $n$.)
[i]Proposed by Serbia[/i]
If $\sum_{i=1}^n\cos^{-1}(\alpha_i)=0$, then find $\sum_{i=1}^n\alpha_i$.
$\textbf{(A)}~\frac n2$
$\textbf{(B)}~n$
$\textbf{(C)}~n\pi$
$\textbf{(D)}~\frac{n\pi}2$
Let the set $A=${$ 1,2,3, \dots ,48n+24$ } , where $ n \in \mathbb {N^*}$ . Prove that there exist a subset $B $ of $A $ with $24n+12$ elements with the property : the sum of the squares of the elements of the set $B $ is equal to the sum of the squares of the elements of the set $A$ \ $B $ .