Found problems: 85335
2002 May Olympiad, 4
The vertices of a regular $2002$-sided polygon are numbered $1$ through $2002$, clockwise. Given an integer $ n$, $1 \le n \le 2002$, color vertex $n$ blue, then, going clockwise, count$ n$ vertices starting at the next of $n$, and color $n$ blue. And so on, starting from the vertex that follows the last vertex that was colored, n vertices are counted, colored or uncolored, and the number $n$ is colored blue. When the vertex to be colored is already blue, the process stops. We denote $P(n)$ to the set of blue vertices obtained with this procedure when starting with vertex $n$. For example, $P(364)$ is made up of vertices $364$, $728$, $1092$, $1456$, $1820$, $182$, $546$, $910$, $1274$, $1638$, and $2002$.
Determine all integers $n$, $1 \le n \le 2002$, such that $P(n)$ has exactly $14 $ vertices,
2021 Azerbaijan Junior NMO, 5
In $\triangle ABC\ T$ is a point lies on the internal angle bisector of $B$. Let $\omega$ be circle with diameter $BT$.
$\omega$ intersects with $BA$ and $BC$ at $P$ and $Q$,respectively. A circle passes through $A$ and tangent to $\omega$ at $P$ intersects with $AC$ again at $X$ . A circle passes through $B$ and tangent to $\omega$ at $Q$ intersects with $AC$ again at $Y$ . Prove that $TX=TY$
2013 Tournament of Towns, 6
There are fi ve distinct real positive numbers. It is known that the total sum of their squares and the total sum of their pairwise products are equal.
(a) Prove that we can choose three numbers such that it would not be possible to make a triangle with sides' lengths equal to these numbers.
(b) Prove that the number of such triples is at least six (triples which consist of the same numbers in different order are considered the same).
2017 CCA Math Bonanza, T5
Twelve people go to a party. First, everybody with no friends at the party leave. Then, at the $i$-th hour, everybody with exactly $i$ friends left at the party leave. After the eleventh hour, what is the maximum number of people left? Note that friendship is mutual.
[i]2017 CCA Math Bonanza Team Round #5[/i]
2010 Today's Calculation Of Integral, 649
Let $f_n(x,\ y)=\frac{n}{r\cos \pi r+n^2r^3}\ (r=\sqrt{x^2+y^2})$,
$I_n=\int\int_{r\leq 1} f_n(x,\ y)\ dxdy\ (n\geq 2).$
Find $\lim_{n\to\infty} I_n.$
[i]2009 Tokyo Institute of Technology, Master Course in Mathematics[/i]
1955 AMC 12/AHSME, 35
Three boys agree to divide a bag of marbles in the following manner. The first boy takes one more than half the marbles. The second takes a third of the number remaining. The third boy finds that he is left with twice as many marbles as the second boy. The original number of marbles:
$ \textbf{(A)}\ \text{is none of the following} \qquad
\textbf{(B)}\ \text{cannot be determined from the given data}\\
\textbf{(C)}\ \text{is 20 or 26} \qquad
\textbf{(D)}\ \text{is 14 or 32} \qquad
\textbf{(E)}\ \text{is 8 or 38}$
Durer Math Competition CD Finals - geometry, 2013.D3
The circle circumscribed to the triangle $ABC$ is $k$. The altitude $AT$ intersects circle $k$ at $P$. The perpendicular from $P$ on line $AB$ intersects is at $R$. Prove that line $TR$ is parallel to the tangent of the circle $k$ at point $A$.
2017 AMC 12/AHSME, 11
Claire adds the degree measures of the interior angles of a convex polygon and arrives at a sum of $2017$. She then discovers that she forgot to include one angle. What is the degree measure of the forgotten angle?
$\textbf{(A)}\ 37\qquad\textbf{(B)}\ 63\qquad\textbf{(C)}\ 117\qquad\textbf{(D)}\ 143\qquad\textbf{(E)}\ 163$
2020 Brazil National Olympiad, 5
Let $ABC$ be a triangle and $M$ the midpoint of $AB$. Let circumcircles of triangles $CMO$ and $ABC$ intersect at $K$ where $O$ is the circumcenter of $ABC$. Let $P$ be the intersection of lines $OM$ and $CK$. Prove that $\angle{PAK} = \angle{MCB}$.
2021 Princeton University Math Competition, 6
Jack plays a game in which he first rolls a fair six-sided die and gets some number $n$, then, he flips a coin until he flips $n$ heads in a row and wins, or he flips $n$ tails in a row in which case he rerolls the die and tries again. What is the expected number of times Jack must flip the coin before he wins the game.
1994 Brazil National Olympiad, 5
Call a super-integer an infinite sequence of decimal digits: $\ldots d_n \ldots d_2d_1$.
(Formally speaking, it is the sequence $(d_1,d_2d_1,d_3d_2d_1,\ldots)$ )
Given two such super-integers $\ldots c_n \ldots c_2c_1$ and $\ldots d_n \ldots d_2d_1$, their product $\ldots p_n \ldots p_2p_1$ is formed by taking $p_n \ldots p_2p_1$ to be the last n digits of the product $c_n \ldots c_2c_1$ and $d_n \ldots d_2d_1$.
Can we find two non-zero super-integers with zero product?
(a zero super-integer has all its digits zero)
PEN P Problems, 33
Let $a_{1}, a_{2}, \cdots, a_{k}$ be relatively prime positive integers. Determine the largest integer which cannot be expressed in the form \[x_{1}a_{2}a_{3}\cdots a_{k}+x_{2}a_{1}a_{3}\cdots a_{k}+\cdots+x_{k}a_{1}a_{2}\cdots a_{k-1}\] for some nonnegative integers $x_{1}, x_{2}, \cdots, x_{k}$.
Estonia Open Senior - geometry, 2016.2.5
The circumcentre of an acute triangle $ABC$ is $O$. Line $AC$ intersects the circumcircle of $AOB$ at a point $X$, in addition to the vertex $A$. Prove that the line $XO$ is perpendicular to the line $BC$.
1987 Vietnam National Olympiad, 3
Prove that among any five distinct rays $ Ox$, $ Oy$, $ Oz$, $ Ot$, $ Or$ in space there exist two which form an angle less than or equal to $ 90^{\circ}$.
1985 Tournament Of Towns, (087) 3
A certain class of $32$ pupils is organised into $33$ clubs , so that each club contains $3$ pupils and no two clubs have the same composition. Prove that there are two clubs which have exactly one common member.
2009 Abels Math Contest (Norwegian MO) Final, 1b
Show that the sum of three consecutive perfect cubes can always be written as the difference between two perfect squares.
Russian TST 2019, P3
Four positive integers $x,y,z$ and $t$ satisfy the relations
\[ xy - zt = x + y = z + t. \]
Is it possible that both $xy$ and $zt$ are perfect squares?
1969 Miklós Schweitzer, 8
Let $ f$ and $ g$ be continuous positive functions defined on the interval $ [0, +\infty)$, and let $ E \subset[0,+\infty)$ be a set of positive measure. Prove that the range of the function defined on $ E \times E$ by the relation \[ F(x,y)= %Error. "dispalymath" is a bad command.
\int_0^xf(t)dt+ %Error. "dispalymath" is a bad command.
\int_0^y g(t)dt\] has a nonvoid interior.
[i]L. Losonczi[/i]
1970 AMC 12/AHSME, 29
It is now between $10:00$ and $11:00$ o'clock, and six minutes form now, the minute hand of the watch will be exactly opposite the place where the hour hand was three minutes ago. What is the exact time now?
$\textbf{(A) }10:05\dfrac{5}{11}\qquad\textbf{(B) }10:07\dfrac{1}{2}\qquad\textbf{(C) }10:10\qquad\textbf{(D) }10:15\qquad$
$\textbf{(E) }10:17\dfrac{1}{2}$
2007 AMC 10, 14
A triangle with side lengths in the ratio $ 3: 4: 5$ is inscribed in a circle of radius $ 3$. What is the area of the triangle?
$ \textbf{(A)}\ 8.64 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 5\pi \qquad \textbf{(D)}\ 17.28 \qquad \textbf{(E)}\ 18$
1978 IMO Longlists, 26
For every integer $d \geq 1$, let $M_d$ be the set of all positive integers that cannot be written as a sum of an arithmetic progression with difference $d$, having at least two terms and consisting of positive integers. Let $A = M_1$, $B = M_2 \setminus \{2 \}, C = M_3$. Prove that every $c \in C$ may be written in a unique way as $c = ab$ with $a \in A, b \in B.$
2013 India IMO Training Camp, 3
We define an operation $\oplus$ on the set $\{0, 1\}$ by
\[ 0 \oplus 0 = 0 \,, 0 \oplus 1 = 1 \,, 1 \oplus 0 = 1 \,, 1 \oplus 1 = 0 \,.\]
For two natural numbers $a$ and $b$, which are written in base $2$ as $a = (a_1a_2 \ldots a_k)_2$ and $b = (b_1b_2 \ldots b_k)_2$ (possibly with leading 0's), we define $a \oplus b = c$ where $c$ written in base $2$ is $(c_1c_2 \ldots c_k)_2$ with $c_i = a_i \oplus b_i$, for $1 \le i \le k$. For example, we have $7 \oplus 3 = 4$ since $ 7 = (111)_2$ and $3 = (011)_2$.
For a natural number $n$, let $f(n) = n \oplus \left[ n/2 \right]$, where $\left[ x \right]$ denotes the largest integer less than or equal to $x$. Prove that $f$ is a bijection on the set of natural numbers.
1974 Chisinau City MO, 80
Each side face of a regular hexagonal prism is colored in one of three colors (for example, red, yellow, blue), and the adjacent prism faces have different colors. In how many different ways can the edges of the prism be colored (using all three colors is optional)?
2009 AIME Problems, 6
Let $ m$ be the number of five-element subsets that can be chosen from the set of the first $ 14$ natural numbers so that at least two of the five numbers are consecutive. Find the remainder when $ m$ is divided by $ 1000$.
1992 Tournament Of Towns, (349) 1
We are given a cube with edges of length $n$ cm. At our disposal is a long piece of insulating tape of width $1$ cm. It is required to stick this tape to the cube. The tape may freely cross an edge of the cube on to a different face but it must always be parallel to an edge of the cube. It may not overhang the edge of a face or cross over a vertex. How many pieces of the tape are necessary in order to completely cover the cube? (You may assume that $n$ is an integer.)
(A Spivak)