Found problems: 85335
2008 Thailand Mathematical Olympiad, 5
Let $P(x)$ be a polynomial of degree $2008$ with the following property: all roots of $P$ are real, and for all real $a$, if $P(a) = 0$ then $P(a+ 1) = 1$. Prove that P must have a repeated root.
KoMaL A Problems 2023/2024, A. 878
Let point $A$ be one of the intersections of circles $c$ and $k$. Let $X_1$ and $X_2$ be arbitrary points on circle $c$. Let $Y_i$ denote the intersection of line $AX_i$ and circle $k$ for $i=1,2$. Let $P_1$, $P_2$ and $P_3$ be arbitrary points on circle $k$, and let $O$ denote the center of circle $k$. Let $K_{ij}$ denote the center of circle $(X_iY_iP_j)$ for $i=1,2$ and $j=1,2,3$. Let $L_j$ denote the center of circle $(OK_{1j}K_{2j})$ for $j=1,2,3$. Prove that points $L_1$, $L_2$ and $L_3$ are collinear.
Proposed by [i]Vilmos Molnár-Szabó[/i], Budapest
2017 USA TSTST, 4
Find all nonnegative integer solutions to $2^a + 3^b + 5^c = n!$.
[i]Proposed by Mark Sellke[/i]
1996 French Mathematical Olympiad, Problem 5
Let $n$ be a positive integer. We say that a natural number $k$ has the property $C_n$ if there exist $2k$ distinct positive integers $a_1,b_1,\ldots,a_k,b_k$ such that the sums $a_1+b_1,\ldots,a_k+b_k$ are distinct and strictly smaller than $n$.
(a) Prove that if $k$ has the property $C_n$ then $k\le \frac{2n-3}{5}$.
(b) Prove that $5$ has the property $C_{14}$.
(c) If $(2n-3)/5$ is an integer, prove that it has the property $C_n$.
1955 Moscow Mathematical Olympiad, 305
$25$ chess players are going to participate in a chess tournament. All are on distinct skill levels, and of the two players the one who plays better always wins. What is the least number of games needed to select the two best players?
1989 Irish Math Olympiad, 1
Suppose $L$ is a fixed line, and $A$ is a fixed point not on $L$. Let $k$ be a fixed nonzero real number. For $P$ a point on $L$, let $Q$ be a point on the line $AP$ with $|AP|\cdot |AQ|=k^2$. Determine the locus of $Q$ as $P$ varies along the line $L$.
XMO (China) 2-15 - geometry, 6.5
As shown in the figure, $\odot O$ is the circumcircle of $\vartriangle ABC$, $\odot J$ is inscribed in $\odot O$ and is tangent to $AB$, $AC$ at points $D$ and E respectively, line segment $FG$ and $\odot O$ are tangent to point $A$, and $AF =AG=AD$, the circumscribed circle of $\vartriangle AFB$ intersects $\odot J$ at point $S$. Prove that the circumscribed circle of $\vartriangle ASG$ is tangent to $\odot J$.
[img]https://cdn.artofproblemsolving.com/attachments/a/a/62d44e071ea9903ebdd68b43943ba1d93b4138.png[/img]
2014 Purple Comet Problems, 14
Let $a$, $b$, $c$ be positive integers such that $abc + bc + c = 2014$. Find the minimum possible value of $a + b + c$.
2014 ASDAN Math Tournament, 7
Let $ABCD$ be a square piece of paper with side length $4$. Let $E$ be a point on $AB$ such that $AE=3$ and let $F$ be a point on $CD$ such that $DF=1$. Now, fold $AEFD$ over the line $EF$. Compute the area of the resulting shape.
2018 239 Open Mathematical Olympiad, 8-9.6
Petya wrote down 100 positive integers $n, n+1, \ldots, n+99$, and Vasya wrote down 99 positive integers $m, m-1, \ldots, m-98$. It turned out that for each of Petya's numbers, there is a number from Vasya that divides it. Prove that $m>n^3/10, 000, 000$.
[i]Proposed by Ilya Bogdanov[/i]
2014 Junior Balkan Team Selection Tests - Romania, 1
We call a composite positive integer $n$ nice if it is possible to arrange its factors that are larger than $1$ on a circle such that two neighboring numbers are not coprime. How many of the elements of the set $\{1, 2, 3, ..., 100\}$ are nice?
2020 Caucasus Mathematical Olympiad, 7
A regular triangle $ABC$ is given. Points $K$ and $N$ lie in the segment $AB$, a point $L$ lies in the segment $AC$, and a point $M$ lies in the segment $BC$ so that $CL=AK$, $CM=BN$, $ML=KN$. Prove that $KL \parallel MN$.
2008 Turkey Team Selection Test, 5
$ D$ is a point on the edge $ BC$ of triangle $ ABC$ such that $ AD\equal{}\frac{BD^2}{AB\plus{}AD}\equal{}\frac{CD^2}{AC\plus{}AD}$. $ E$ is a point such that $ D$ is on $ [AE]$ and $ CD\equal{}\frac{DE^2}{CD\plus{}CE}$. Prove that $ AE\equal{}AB\plus{}AC$.
2014 Indonesia MO Shortlist, C1
Is it possible to fill a $3 \times 3$ grid with each of the numbers $1,2,\ldots,9$ once each such that the sum of any two numbers sharing a side is prime?
2014-2015 SDML (High School), 12
Which of the following polynomials with integer coefficients has $\sin\left(10^{\circ}\right)$ as a root?
$\text{(A) }4x^3-4x+1\qquad\text{(B) }6x^3-8x^2+1\qquad\text{(C) }4x^3+4x-1\qquad\text{(D) }8x^3+6x-1\qquad\text{(E) }8x^3-6x+1$
1990 Greece National Olympiad, 4
Froa nay real $x$, we denote $[x]$, the integer part of $x$ and with $\{x\}$ the fractional part of $x$, such that $x=[x]+\{x\}$.
a) Find at least one real $x$ such that$\{x\}+\left\{\frac{1}{x}\right\}=1$
b) Find all rationals $x$ such that $\{x\}+\left\{\frac{1}{x}\right\}=1$
2006 Cezar Ivănescu, 3
[b]a)[/b] Prove that the function $ f:\mathbb{Z}_{\ge 0}\longrightarrow\mathbb{Z}_{\ge 0} , $ given as $ f(n)=n+(-1)^n $ is bijective.
[b]b)[/b] Find all surjective functions $ g:\mathbb{Z}_{\ge 0}\longrightarrow\mathbb{Z}_{\ge 0} $ that have the property that $ g(n)\ge n+(-1)^n , $ for any nonnegative integer.
2019 HMNT, 7
Consider sequences $a$ of the form $a = (a_1, a_2, ... , a_{20})$ such that each term $a_i$ is either $0$ or $1$. For each such sequence $a$, we can produce a sequence $b = (b_1, b_2, ..., b_{20})$, where $$b_i\begin{cases} a_i + a_{i+1} & i = 1 \\ a_{i-1} + a_i + a_{i+1} & 1 < i < 20\\ a_{i-1} + a_i &i = 20 \end{cases}$$
2015 District Olympiad, 1
Let $ f:[0,1]\longrightarrow [0,1] $ a function with the property that, for all $ y\in [0,1] $ and $ \varepsilon >0, $ there exists a $ x\in [0,1] $ such that $ |f(x)-y|<\varepsilon . $
[b]a)[/b] Prove that if $ \left. f\right|_{[0,1]} $ is continuos, then $ f $ is surjective.
[b]b)[/b] Give an example of a function with the given property, but which isn´t surjective.
1964 AMC 12/AHSME, 26
In a ten-mile race First beats Second by $2$ miles and First beats Third by $4$ miles. If the runners maintain constant speeds throughout the race, by how many miles does Second beat Third?
$ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 2\frac{1}{4}\qquad\textbf{(C)}\ 2\frac{1}{2}\qquad\textbf{(D)}\ 2\frac{3}{4}\qquad\textbf{(E)}\ 3 $
1995 Miklós Schweitzer, 1
Prove that a harmonic function that is not identically zero in the plane cannot vanish on a two-dimensional positive-measure set.
2018 Yasinsky Geometry Olympiad, 6
Let $O$ and $I$ be the centers of the circumscribed and inscribed circle the acute-angled triangle $ABC$, respectively. It is known that line $OI$ is parallel to the side $BC$ of this triangle. Line $MI$, where $M$ is the midpoint of $BC$, intersects the altitude $AH$ at the point $T$. Find the length of the segment $IT$, if the radius of the circle inscribed in the triangle $ABC$ is equal to $r$.
(Grigory Filippovsky)
2004 Estonia National Olympiad, 2
Albert and Brita play a game with a bar of $19$ adjacent squares. Initially, there is a button on the middle square of the bar. At every turn Albert mentions one positive integer less than $5$, and Brita moves button a number of squares in the direction of her choice - while doing so however, Brita must not move the button more than twice in one direction order. Prove that Albert can choose the numbers so that by the $19$th turn, Brita to be forced to move the button out of the bar.
2005 South East Mathematical Olympiad, 6
Let $P(A)$ be the arithmetic-means of all elements of set $A = \{ a_1, a_2, \ldots, a_n \}$, namely $P(A) = \frac{1}{n} \sum^{n}_{i=1}a_i$. We denote $B$ "balanced subset" of $A$, if $B$ is a non-empty subset of $A$ and $P(B) = P(A)$. Let set $M = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \}$.
Find the number of all "balanced subset" of $M$.
1995 USAMO, 1
Let $\, p \,$ be an odd prime. The sequence $(a_n)_{n \geq 0}$ is defined as follows: $\, a_0 = 0,$ $a_1 = 1, \, \ldots, \, a_{p-2} = p-2 \,$ and, for all $\, n \geq p-1, \,$ $\, a_n \,$ is the least positive integer that does not form an arithmetic sequence of length $\, p \,$ with any of the preceding terms. Prove that, for all $\, n, \,$ $\, a_n \,$ is the number obtained by writing $\, n \,$ in base $\, p-1 \,$ and reading the result in base $\, p$.