Found problems: 85335
Let us say that a circle intersects a quadrilateral [i]properly[/i] if it intersects each of the quadrilateral’s sides at two distinct interior points. Is it true that for each convex quadrilateral there exists a circle which intersects it properly?
[i]Alexandr Perepechko[/i]
Ed and Sue bike at equal and constant rates. Similarly, they jog at equal and constant rates, and they swim at equal and constant rates. Ed covers $ 74$ kilometers after biking for $ 2$ hours, jogging for $ 3$ hours, and swimming for $ 4$ hours, while Sue covers $ 91$ kilometers after jogging for $ 2$ hours, swimming for $ 3$ hours, and biking for $ 4$ hours. Their biking, jogging, and swimming rates are all whole numbers of kilometers per hour. Find the sum of the squares of Ed's biking, jogging, and swimming rates.
Show that the equation $z^{n}+z+1=0$ has a solution with $|z|=1$ if and only if $n-2$ is divisble by $3$.
Vasya chose a certain number $x$ and calculated the following:
$a_1=1+x^2+x^3, a_2=1+x^3+x^4, a_3=1+x^4+x^5, ..., a_n=1+x^{n+1}+x^{n+2} ,...$
It turned out that $a_2^2 = a_1a_3$.
Prove that for all $n\ge 3$, the equality $a_n^2 = a_{n-1}a_{n+1}$ holds.
Numbers in an $n$ by $n$ table may be changed by adding $1$ to each number on an arbitrary closed non-selfintersecting “rook path” (a broken line with segments parallel to the borders of the table). Originally $1$’s stand on one of the diagonals, and $0S’s in the other cells of the table. Can one get (after several transformations) a table in which all numbers are equal to each other? (A “rook path” contains all cells through which it passes.)
(AA Egorov)
To the exterior of side $AB$ of square $ABCD$, we have drawn the regular triangle $ABE$. Point $A$ reflected on line $BE$ is $F$, and point $E$ reflected on line $BF$ is $G$. Let the perpendicular bisector of segment $FG$ meet segment $AD$ at $X$. Show that the circle centered at $X$ with radius $XA$ touches line$ FB$.
Let $ f(x)\equal{}x^2\plus{}3$ and $ y\equal{}g(x)$ be the equation of the line with the slope $ a$, which pass through the point $ (0,\ f(0))$ .
Find the maximum and minimum values of $ I(a)\equal{}3\int_{\minus{}1}^1 |f(x)\minus{}g(x)|\ dx$.
A tetrahedron $ABCD$ satisfies $\angle BAC=\angle CAD=\angle DAB=90^o$. Show that the areas of its faces satisfy the equation $area(BAC)^2 + area(CAD)^2 + area(DAB)^2 = area(BCD)^2$.
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Two persons, A and B, set up an incantation contest in which they spell incantations (i.e. a finite sequence of letters) alternately. They must obey the following rules:
i) Any incantation can appear no more than once;
ii) Except for the first incantation, any incantation must be obtained by permuting the letters of the last one before it, or deleting one letter from the last incantation before it;
iii)The first person who cannot spell an incantation loses the contest. Answer the following questions:
a) If A says '$STAGEPREIMO$' first, then who will win?
b) Let $M$ be the set of all possible incantations whose lengths (i.e. the numbers of letters in them) are $2009$ and containing only four letters $A,B,C,D$, each of them appearing at least once. Find the first incantation (arranged in dictionary order) in $M$ such that A has a winning strategy by starting with it.
Let $ABC$ be a triangle such that $AB = 5$, $AC = 8$, and $\angle BAC = 60^{\circ}$. Let $P$ be a point inside the triangle such that $\angle APB = \angle BPC = \angle CPA$. Lines $BP$ and $AC$ intersect at $E$, and lines $CP$ and $AB$ intersect at $F$. The circumcircles of triangles $BPF$ and $CPE$ intersect at points $P$ and $Q \neq P$. Then $QE + QF=\frac{m}{n}$, where $m$ and $n$ are positive integers with $\gcd(m,n)=1$. Compute $100m + n$.
[i]Proposed by Ankan Bhattacharya[/i]
Write some positive integers in the following table such that
$\cdot$ there is at most one number in each field
$\cdot$ each number is equal to how many numbers there are in edge-adjacent fields,
$\cdot$ edge-adjacent fields cannot have equal numbers.
What is the sum of numbers in the resulting table?
[img]https://cdn.artofproblemsolving.com/attachments/a/9/63a9c38762a4c895688fff049ed08c96b2c22c.png[/img]
There are $2008$ participants in a programming competition. In every round, all programmers are divided into two equal-sized teams. Find the minimal number of rounds after which there can be a situation in which every two programmers have been in different teams at least once.
The sum of the solutions to the equation $$x^{\log_2 x} =\frac{64}{x}$$ can be written as$ \frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Azar, Carl, Jon, and Sergey are the four players left in a singles tennis tournament. They are randomly assigned opponents in the semifinal matches, and the winners of those matches play each other in the final match to determine the winner of the tournament. When Azar plays Carl, Azar will win the match with probability $\frac23$. When either Azar or Carl plays either Jon or Sergey, Azar or Carl will win the match with probability $\frac34$. Assume that outcomes of different matches are independent. The probability that Carl will win the tournament is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
A bag contains four pieces of paper, each labeled with one of the digits $1$, $2$, $3$ or $4$, with no repeats. Three of these pieces are drawn, one at a time without replacement, to construct a three-digit number. What is the probability that the three-digit number is a multiple of $3$?
$\textbf{(A)}\ \frac{1}{4} \qquad
\textbf{(B)}\ \frac{1}{3} \qquad
\textbf{(C)}\ \frac{1}{2} \qquad
\textbf{(D)}\ \frac{2}{3} \qquad
\textbf{(E)}\ \frac{3}{4}$
Each point in the plane with integer coordinates is colored red or blue such that the following two properties hold.
For any two red points, the line segment joining them does not contain any blue points.
For any two blue points that are distance $2$ apart, the midpoint of the line segment joining them is blue.
Prove that if three red points are the vertices of a triangle, then the interior of the triangle does not contain any blue points.
Let the function $f:R \to R$ satisfies the following conditions:
1) for all $x, y\in R$, $ f(x +y) = f(x) +f(y)$
2)$ f(1)=1$
3) for all $x \ne 0$ , $ f(1/x) =\frac{f(x)}{x^2}$
Prove that for all $x \in R$, $f(x) = x$.
If four times the reciprocal of the circumference of a circle equals the diameter of the circle, then the area of the circle is
$\textbf{(A) }\frac{1}{\pi^2}\qquad\textbf{(B) }\frac{1}{\pi}\qquad\textbf{(C) }1\qquad\textbf{(D) }\pi\qquad \textbf{(E) }\pi^2$
Find all functions $f:\mathbb N\to\mathbb N$ satisfying
$$\operatorname{lcm}(f(x),y)\gcd(f(x),f(y))=f(x)f(f(y))$$
for all $x,y\in\mathbb N$.
Find all points inside a given equilateral triangle such that the distances from it to three sides of the given triangle are the side lengths of a triangle.
Consider a convex pentagon $ABCDE$ and a variable point $X$ on its side $CD$.
Suppose that points $K, L$ lie on the segment $AX$ such that $AB = BK$ and $AE = EL$ and that
the circumcircles of triangles $CXK$ and $DXL$ intersect for the second time at $Y$ . As $X$ varies,
prove that all such lines $XY$ pass through a fixed point, or they are all parallel.
[i]Proposed by Josef Tkadlec - Czech Republic[/i]
What is the smallest positive integer $t$ such that there exist integers $x_{1},x_{2}, \cdots, x_{t}$ with \[{x_{1}}^{3}+{x_{2}}^{3}+\cdots+{x_{t}}^{3}=2002^{2002}\;\;?\]
Melanie computes the mean $\mu$, the median $M$, and the modes of the $365$ values that are the dates in the months of $2019$. Thus her data consist of $12$ $1\text{s}$, $12$ $2\text{s}$, . . . , $12$ $28\text{s}$, $11$ $29\text{s}$, $11$ $30\text{s}$, and $7$ $31\text{s}$. Let $d$ be the median of the modes. Which of the following statements is true?
$\textbf{(A) } \mu < d < M \qquad\textbf{(B) } M < d < \mu \qquad\textbf{(C) } d = M =\mu \qquad\textbf{(D) } d < M < \mu \qquad\textbf{(E) } d < \mu < M$
[b]interesting sequence[/b]
$n$ is a natural number and $x_1,x_2,...$ is a sequence of numbers $1$ and $-1$ with these properties:
it is periodic and its least period number is $2^n-1$. (it means that for every natural number $j$ we have $x_{j+2^n-1}=x_j$ and $2^n-1$ is the least number with this property.)
There exist distinct integers $0\le t_1<t_2<...<t_k<n$ such that for every natural number $j$ we have
\[x_{j+n}=x_{j+t_1}\times x_{j+t_2}\times ... \times x_{j+t_k}\]
Prove that for every natural number $s$ that $s<2^n-1$ we have
\[\sum_{i=1}^{2^n-1}x_ix_{i+s}=-1\]
Time allowed for this question was 1 hours and 15 minutes.
Joy has $30$ thin rods, one each of every integer length from $1$ cm through $30$ cm. She places the rods with lengths $3$ cm, $7$ cm, and $15$ cm on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How many of the remaining rods can she choose as the fourth rod?
$\textbf{(A) }16\qquad\textbf{(B) }17\qquad\textbf{(C) }18\qquad\textbf{(D) }19\qquad\textbf{(E) }20$