Found problems: 85335
2018 Taiwan TST Round 1, 4
Let $n$ be a positive integer not divisible by $3$. A triangular grid of length $n$ is obtained by dissecting a regular triangle with length $n$ into $n^2$ unit regular triangles. There is an orange at each vertex of the grid, which sums up to
\[\frac{(n+1)(n+2)}{2}\]
oranges. A triple of oranges $A,B,C$ is [i]good[/i] if each $AB,AC$ is some side of some unit regular triangles, and $\angle BAC = 120^{\circ}$. Each time, Yen can take away a good triple of oranges from the grid. Determine the maximum number of oranges Yen can take.
2008 China Team Selection Test, 2
Prove that for all $ n\geq 2,$ there exists $ n$-degree polynomial $ f(x) \equal{} x^n \plus{} a_{1}x^{n \minus{} 1} \plus{} \cdots \plus{} a_{n}$ such that
(1) $ a_{1},a_{2},\cdots, a_{n}$ all are unequal to $ 0$;
(2) $ f(x)$ can't be factorized into the product of two polynomials having integer coefficients and positive degrees;
(3) for any integers $ x, |f(x)|$ isn't prime numbers.
2014 Saudi Arabia GMO TST, 3
Let $ABC$ be a triangle, $I$ its incenter, and $\omega$ a circle of center $I$. Points $A',B', C'$ are on $\omega$ such that rays $IA', IB', IC',$ starting from $I$ intersect perpendicularly sides $BC, CA, AB$, respectively. Prove that lines $AA', BB', CC'$ are concurrent.
2010 AMC 10, 16
Nondegenerate $ \triangle ABC$ has integer side lengths, $ BD$ is an angle bisector, $ AD \equal{} 3$, and $ DC \equal{} 8$. What is the smallest possible value of the perimeter?
$ \textbf{(A)}\ 30 \qquad
\textbf{(B)}\ 33 \qquad
\textbf{(C)}\ 35 \qquad
\textbf{(D)}\ 36 \qquad
\textbf{(E)}\ 37$
2000 AIME Problems, 11
The coordinates of the vertices of isosceles trapezoid $ABCD$ are all integers, with $A=(20,100)$ and $D=(21,107).$ The trapezoid has no horizontal or vertical sides, and $\overline{AB}$ and $\overline{CD}$ are the only parallel sides. The sum o f the absolute values of all possible slopes for $\overline{AB}$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
2019 ELMO Shortlist, C5
Given a permutation of $1,2,3,\dots,n$, with consecutive elements $a,b,c$ (in that order), we may perform either of the [i]moves[/i]:
[list]
[*] If $a$ is the median of $a$, $b$, and $c$, we may replace $a,b,c$ with $b,c,a$ (in that order)
[*] If $c$ is the median of $a$, $b$, and $c$, we may replace $a,b,c$ with $c,a,b$ (in that order)
[/list]
What is the least number of sets in a partition of all $n!$ permutations, such that any two permutations in the same set are obtainable from each other by a sequence of moves?
[i]Proposed by Milan Haiman[/i]
2004 Purple Comet Problems, 9
Let $M$ and $m$ be the largest and the smallest values of $x$, respectively, which satisfy $4x(x - 5) \le 375$. Find $M - m$.
1998 Swedish Mathematical Competition, 5
Show that for any $n > 5$ we can find positive integers $x_1, x_2, ... , x_n$ such that $\frac{1}{x_1} + \frac{1}{x_2} +... + \frac{1}{x_n} = \frac{1997}{1998}$. Show that in any such equation there must be two of the $n$ numbers with a common divisor ($> 1$).
2022 AMC 12/AHSME, 13
Let $\mathcal{R}$ be the region in the complex plane consisting of all complex numbers $z$ that can be written as the sum of complex numbers $z_1$ and $z_2$, where $z_1$ lies on the segment with endpoints $3$ and $4i$, and $z_2$ has magnitude at most $1$. What integer is closest to the area of $\mathcal{R}$?
$\textbf{(A) }13\qquad\textbf{(B) }14\qquad\textbf{(C) }15\qquad\textbf{(D) }16\qquad\textbf{(E) }17$
2020 LMT Fall, 13
Let set $S$ contain all positive integers that are one less than a perfect square. Find the sum of all powers of $2$ that can be expressed as the product of two (not necessarily distinct) members of $S.$
[i]Proposed by Alex Li[/i]
2016 AMC 12/AHSME, 2
The harmonic mean of two numbers can be calculated as twice their product divided by their sum. The harmonic mean of $1$ and $2016$ is closest to which integer?
$\textbf{(A)}\ 2 \qquad
\textbf{(B)}\ 45 \qquad
\textbf{(C)}\ 504 \qquad
\textbf{(D)}\ 1008 \qquad
\textbf{(E)}\ 2015 $
2019 Jozsef Wildt International Math Competition, W. 4
If $x, y, z, t > 1$ then: $$\left(\log _{zxt}x\right)^2+\left(\log _{xyt}y\right)^2+\left(\log _{xyz}z\right)^2+\left(\log _{yzt}t\right)^2>\frac{1}{4}$$
1957 Putnam, B7
Let $C$ consist of a regular polygon and its interior. Show that for each positive integer $n$, there exists a set of points $S(n)$ in the plane such that every $n$ points can be covered by $C$, but $S(n)$ cannot be covered by $C.$
2011 Mathcenter Contest + Longlist, 6 sl8
Let $x,y,z$ represent the side lengths of any triangle, and $s=\dfrac{x+y+z}{2}$ and the area of this triangle be $\sqrt{s}$ square units. Prove that $$s\Big(\frac{1}{x(s-x)^2}+\frac{1}{y(s-y)^2}+\frac{1}{z(s-z)^ 2} \Big)\ge \frac{1}{2} \Big(\frac{1}{s-x}+\frac{1}{s-y}+\frac{1}{s-z}\Big)$$
[i](Zhuge Liang)[/i]
2013 NIMO Summer Contest, 6
Let $ABC$ and $DEF$ be two triangles, such that $AB=DE=20$, $BC=EF=13$, and $\angle A = \angle D$. If $AC-DF=10$, determine the area of $\triangle ABC$.
[i]Proposed by Lewis Chen[/i]
2023 USAJMO Solutions by peace09, 6
Isosceles triangle $ABC$, with $AB=AC$, is inscribed in circle $\omega$. Let $D$ be an arbitrary point inside $BC$ such that $BD\neq DC$. Ray $AD$ intersects $\omega$ again at $E$ (other than $A$). Point $F$ (other than $E$) is chosen on $\omega$ such that $\angle DFE = 90^\circ$. Line $FE$ intersects rays $AB$ and $AC$ at points $X$ and $Y$, respectively. Prove that $\angle XDE = \angle EDY$.
[i]Proposed by Anton Trygub[/i]
2023 All-Russian Olympiad, 5
Find the largest natural number $n$ for which the product of the numbers $n, n+1, n+2, \ldots, n+20$ is divisible by the square of one of them.
2020 Israel National Olympiad, 2
202 participants arrived at a mathematical conference from three countries: Israel, Greece, and Japan.
On the first day of the conference, every pair of participants from the same country shook hands.
On the second day, every pair of participants exactly one of whom was Israeli shook hands.
On the third day, every pair of participants one of whom was Israeli and the other Greek shook hands.
In total 20200 handshakes occurred. How many Israelis participated in the conference?
VI Soros Olympiad 1999 - 2000 (Russia), 10.10
Take an arbitrary point $D$ on side $BC$ of triangle $ABC$ and draw a circle through point $D$ and the centers of the circles inscribed in triangles $ABD$ and $ACD$. Prove that all circles obtained for different points $D$ of side $BC$ have a common point.
2024 All-Russian Olympiad Regional Round, 10.3
There are $100$ white points on a circle. Asya and Borya play the following game: they alternate, starting with Asya, coloring a white point in green or blue. Asya wants to obtain as much as possible pairs of adjacent points of distinct colors, while Borya wants these pairs to be as less as possible. What is the maximal number of such pairs Asya can guarantee to obtain, no matter how Borya plays.
2021 All-Russian Olympiad, 2
Let $n$ be a natural number. An integer $a>2$ is called $n$-decomposable, if $a^n-2^n$ is divisible by all the numbers of the form $a^d+2^d$, where $d\neq n$ is a natural divisor of $n$. Find all composite $n\in \mathbb{N}$, for which there's an $n$-decomposable number.
MMPC Part II 1958 - 95, 1962
[b]p1.[/b] Consider this statement: An equilateral polygon circumscribed about a circle is also equiangular.
Decide whether this statement is a true or false proposition in euclidean geometry.
If it is true, prove it; if false, produce a counterexample.
[b]p2.[/b] Show that the fraction $\frac{x^2-3x+1}{x-3}$ has no value between $1$ and $5$, for any real value of $x$.
[b]p3.[/b] A man walked a total of $5$ hours, first along a level road and then up a hill, after which he turned around and walked back to his starting point along the same route. He walks $4$ miles per hour on the level, three miles per hour uphill, and $r$ miles per hour downhill. For what values of $r$ will this information uniquely determine his total walking distance?
[b]p4.[/b] A point $P$ is so located in the interior of a rectangle that the distance of $P$ from one comer is $5$ yards, from the opposite comer is $14$ yards, and from a third comer is $10$ yards. What is the distance from $P$ to the fourth comer?
[b]p5.[/b] Each small square in the $5$ by $5$ checkerboard shown has in it an integer according to the following rules: $\begin{tabular}{|l|l|l|l|l|}
\hline
& & & & \\ \hline
& & & & \\ \hline
& & & & \\ \hline
& & & & \\ \hline
& & & & \\ \hline \end{tabular}$
i. Each row consists of the integers $1, 2, 3, 4, 5$ in some order.
ii. Тhе order of the integers down the first column has the same as the order of the integers, from left to right, across the first row and similarly fог any other column and the corresponding row.
Prove that the diagonal squares running from the upper left to the lower right contain the numbers $1, 2, 3, 4, 5$ in some order.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2009 Princeton University Math Competition, 8
Find the number of functions $f:\mathbb{Z}\mapsto\mathbb{Z}$ for which $f(h+k)+f(hk)=f(h)f(k)+1$, for all integers $h$ and $k$.
2008 Junior Balkan Team Selection Tests - Romania, 3
Solve in prime numbers $ 2p^q \minus{} q^p \equal{} 7$.
2010 National Olympiad First Round, 4
How many positive integers less than $2010$ are there such that the sum of factorials of its digits is equal to itself?
$ \textbf{(A)}\ 5
\qquad\textbf{(B)}\ 4
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 2
\qquad\textbf{(E)}\ \text{None}
$