Found problems: 85335
1983 IMO Longlists, 13
Let $p$ be a prime number and $a_1, a_2, \ldots, a_{(p+1)/2}$ different natural numbers less than or equal to $p.$ Prove that for each natural number $r$ less than or equal to $p$, there exist two numbers (perhaps equal) $a_i$ and $a_j$ such that
\[p \equiv a_i a_j \pmod r.\]
2000 JBMO ShortLists, 12
Consider a sequence of positive integers $x_n$ such that:
\[(\text{A})\ x_{2n+1}=4x_n+2n+2 \]
\[(\text{B})\ x_{3n+\color[rgb]{0.9529,0.0980,0.0118}2}=3x_{n+1}+6x_n \]
for all $n\ge 0$.
Prove that
\[(\text{C})\ x_{3n-1}=x_{n+2}-2x_{n+1}+10x_n \]
for all $n\ge 0$.
1995 All-Russian Olympiad Regional Round, 9.4
Every side and diagonal of a regular $12$-gon is colored in one of $12$ given colors. Can this be done in such a way that, for every three colors, there exist three vertices which are connected to each other by segments of these three colors?
2017 CHMMC (Fall), Individual
[b]p1.[/b] A dog on a $10$ meter long leash is tied to a $10$ meter long, infinitely thin section of fence. What is the minimum area over which the dog will be able to roam freely on the leash, given that we can fix the position of the leash anywhere on the fence?
[b]p2.[/b] Suppose that the equation $$\begin{tabular}{cccccc}
&\underline{C} &\underline{H} &\underline{M}& \underline{M}& \underline{C}\\
+& &\underline{H}& \underline{M}& \underline{M} & \underline{T}\\
\hline
&\underline{P} &\underline{U} &\underline{M} &\underline{A} &\underline{C}\\
\end{tabular}$$
holds true, where each letter represents a single nonnegative digit, and distinct letters represent different digits (so that $\underline{C}\, \underline{H}\, \underline{ M}\, \underline{ M}\, \underline{ C}$ and $ \underline{P}\, \underline{U}\, \underline{M}\, \underline{A}\, \underline{C}$ are both five digit positive integers, and the number $\underline{H }\, \underline{M}\, \underline{M}\, \underline{T}$ is a four digit positive integer). What is the largest possible value of the five digit positive integer$\underline{C}\, \underline{H}\, \underline{ M}\, \underline{ M}\, \underline{ C}$ ?
[b]p3.[/b] Square $ABCD$ has side length $4$, and $E$ is a point on segment $BC$ such that $CE = 1$. Let $C_1$ be the circle tangent to segments $AB$, $BE$, and $EA$, and $C_2$ be the circle tangent to segments $CD$, $DA$, and $AE$. What is the sum of the radii of circles $C_1$ and $C_2$?
[b]p4.[/b] A finite set $S$ of points in the plane is called tri-separable if for every subset $A \subseteq S$ of the points in the given set, we can find a triangle $T$ such that
(i) every point of $A$ is inside $T$ , and
(ii) every point of $S$ that is not in $A$ is outside$ T$ .
What is the smallest positive integer $n$ such that no set of $n$ distinct points is tri-separable?
[b]p5.[/b] The unit $100$-dimensional hypercube $H$ is the set of points $(x_1, x_2,..., x_{100})$ in $R^{100}$ such that $x_i \in \{0, 1\}$ for $i = 1$, $2$, $...$, $100$. We say that the center of $H$ is the point
$$\left( \frac12,\frac12, ..., \frac12 \right)$$
in $R^{100}$, all of whose coordinates are equal to $1/2$.
For any point $P \in R^{100}$ and positive real number $r$, the hypersphere centered at $P$ with radius $r$ is defined to be the set of all points in $R^{100}$ that are a distance $r$ away from $P$. Suppose we place hyperspheres of radius $1/2$ at each of the vertices of the $100$-dimensional unit hypercube $H$. What is the smallest real number $R$, such that a hypersphere of radius $R$ placed at the center of $H$ will intersect the hyperspheres at the corners of $H$?
[b]p6.[/b] Greg has a $9\times 9$ grid of unit squares. In each square of the grid, he writes down a single nonzero digit. Let $N$ be the number of ways Greg can write down these digits, so that each of the nine nine-digit numbers formed by the rows of the grid (reading the digits in a row left to right) and each of the nine nine-digit numbers formed by the columns (reading the digits in a column top to bottom) are multiples of $3$. What is the number of positive integer divisors of $N$?
[b]p7.[/b] Find the largest positive integer $n$ for which there exists positive integers $x$, $y$, and $z$ satisfying
$$n \cdot gcd(x, y, z) = gcd(x + 2y, y + 2z, z + 2x).$$
[b]p8.[/b] Suppose $ABCDEFGH$ is a cube of side length $1$, one of whose faces is the unit square $ABCD$. Point $X$ is the center of square $ABCD$, and $P$ and $Q$ are two other points allowed to range on the surface of cube $ABCDEFHG$. Find the largest possible volume of tetrahedron $AXPQ$.
[b]p9.[/b] Deep writes down the numbers $1, 2, 3, ... , 8$ on a blackboard. Each minute after writing down the numbers, he uniformly at random picks some number $m$ written on the blackboard, erases that number from the blackboard, and increases the values of all the other numbers on the blackboard by $m$. After seven minutes, Deep is left with only one number on the black board. What is the expected value of the number Deep ends up with after seven minutes?
[b]p10.[/b] Find the number of ordered tuples $(x_1, x_2, x_3, x_4, x_5)$ of positive integers such that $x_k \le 6$ for each index $k = 1$, $2$, $... $,$ 5$, and the sum $$x_1 + x_2 +... + x_5$$ is $1$ more than an integer multiple of $7$.
[b]p11.[/b] The equation $$\left( x- \sqrt[3]{13}\right)\left( x- \sqrt[3]{53}\right)\left( x- \sqrt[3]{103}\right)=\frac13$$ has three distinct real solutions $r$, $s$, and $t$ for $x$. Calculate the value of $$r^3 + s^3 + t^3.$$
[b]p12.[/b] Suppose $a$, $b$, and $c$ are real numbers such that
$$\frac{ac}{a + b}+\frac{ba}{b + c}+\frac{cb}{c + a}= -9$$
and
$$\frac{bc}{a + b}+\frac{ca}{b+c}+\frac{ab}{c + a}= 10.$$
Compute the value of
$$\frac{b}{a + b}+\frac{c}{b + c}+\frac{a}{c + a}.$$
[b]p13.[/b] The complex numbers $w$ and $z$ satisfy the equations $|w| = 5$, $|z| = 13$, and $$52w - 20z = 3(4 + 7i).$$ Find the value of the product $wz$.
[b]p14.[/b] For $i = 1, 2, 3, 4$, we choose a real number $x_i$ uniformly at random from the closed interval $[0, i]$. What is the probability that $x_1 < x_2 < x_3 < x_4$ ?
[b]p15.[/b] The terms of the infinite sequence of rational numbers $a_0$, $a_1$, $a_2$, $...$ satisfy the equation $$a_{n+1} + a_{n-2} = a_na_{n-1}$$ for all integers $n\ge 2$. Moreover, the values of the initial terms of the sequence are $a_0 =\frac52$, $a_1 = 2$ and} $a_2 =\frac52.$ Call a nonnegative integer $m$ lucky if when we write $a_m =\frac{p}{q}$ for some relatively prime positive integers $p$ and $q$, the integer $p + q$ is divisible by $13$. What is the $101^{st}$ smallest lucky number?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1981 Romania Team Selection Tests, 1.
Let $P(X)=aX^3-\frac16 X$ where $a\in\mathbb{R}$.
[b]1)[/b] Determine $a$ such that for every $\alpha\in\mathbb{Z}$ we have $P(\alpha)\in\mathbb{Z}$.
[b]2)[/b] Show that if $a$ is irrational then for every $0<u<v<1$ there exists $n\in\mathbb{Z}$ such that
\[u<P(n)-\lfloor P(n)\rfloor <v.\]
Generalize the problem!
2012 Olympic Revenge, 2
We define $(x_1, x_2, \ldots , x_n) \Delta (y_1, y_2, \ldots , y_n) = \left( \sum_{i=1}^{n}x_iy_{2-i}, \sum_{i=1}^{n}x_iy_{3-i}, \ldots , \sum_{i=1}^{n}x_iy_{n+1-i} \right)$, where the indices are taken modulo $n$.
Besides this, if $v$ is a vector, we define $v^k = v$, if $k=1$, or $v^k = v \Delta v^{k-1}$, otherwise.
Prove that, if $(x_1, x_2, \ldots , x_n)^k = (0, 0, \ldots , 0)$, for some natural number $k$, then $x_1 = x_2 = \ldots = x_n = 0$.
1991 Denmark MO - Mohr Contest, 5
Show that no matter how $15$ points are plotted within a circle of radius $2$ (circle border included), there will be a circle with radius $1$ (circle border including) which contains at least three of the $15$ points.
Geometry Mathley 2011-12, 3.1
$AB,AC$ are tangent to a circle $(O)$, $B,C$ are the points of tangency. $Q$ is a point iside the angle $BAC$, on the ray $AQ$, take a point $P$ suc that $OP$ is perpendicular to $AQ$. The line $OP$ meets the circumcircles triangles $BPQ$ and $CPQ$ at $I, J$. Prove that $OI = OJ$.
Hồ Quang Vinh
2021 JHMT HS, 3
Let $ABCDEF$ be a convex hexagon such that $AB=CD=EF=20, \ BC=DE=FA=21,$ and $\angle A=\angle C=\angle E=90^{\circ}.$ The area of $ABCDEF$ can then be expressed in the form $a+\tfrac{b\sqrt{c}}{d},$ where $a,\ b,\ c,$ and $d$ are positive integers, $b$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a+b+c+d.$
2014 Iran Geometry Olympiad (senior), 2:
In the Quadrilateral $ABCD$ we have $ \measuredangle B=\measuredangle D = 60^\circ $.$M$ is midpoint of side $AD$.The line through $M$ parallel to $CD$ meets $BC$ at $P$.Point $X$ lying on $CD$ such that $BX=MX$.Prove that $AB=BP$ if and only if $\measuredangle MXB=60^\circ$.
Author: Davoud Vakili, Iran
2015 AMC 12/AHSME, 19
In $\triangle{ABC}$, $\angle{C} = 90^{\circ}$ and $AB = 12$. Squares $ABXY$ and $ACWZ$ are constructed outside of the triangle. The points $X, Y, Z$, and $W$ lie on a circle. What is the perimeter of the triangle?
$ \textbf{(A)}\ 12+9\sqrt{3}\qquad\textbf{(B)}\ 18+6\sqrt{3}\qquad\textbf{(C)}\ 12+12\sqrt{2}\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 32 $
2023 SG Originals, Q3
Let $\vartriangle ABC$ be a triangle with orthocenter $H$, and let $M$, $N$ be the midpoints of $BC$ and $AH$ respectively. Suppose $Q$ is a point on $(ABC)$ such that $\angle AQH = 90^o$. Show that $MN$, the circumcircle of $QNH$, and the $A$-symmedian concur.
Note: the $A$-symmedian is the reflection of line $AM$ in the bisector of angle $\angle BAC$.
2017 Costa Rica - Final Round, A1
Let $P (x)$ be a polynomial of degree $2n$, such that $P (k) =\frac{k}{k + 1}$ for $k = 0,...,2n$. Determine $P (2n + 1)$.
2017 CMIMC Individual Finals, 1
Let $\tau(n)$ denote the number of positive integer divisors of $n$. For example, $\tau(4) = 3$. Find the sum of all positive integers $n$ such that $2 \tau(n) = n$.
2012 CHKMO, 2
Among the coordinates $(x,y)$ $(1\leq x,y\leq 101)$, choose some points such that there does not exist $4$ points which form a isoceles trapezium with its base parallel to either the $x$ or $y$ axis(including rectangles). Find the maximum number of coordinate points that can be chosen.
1972 Polish MO Finals, 6
Prove that the sum of digits of the number $1972^n$ is not bounded from above when $n$ tends to infinity.
2016 ASMT, 5
Plane $A$ passes through the points $(1, 0, 0)$, $(0, 1, 0)$, and $(0, 0, 1)$. Plane $B$ is parallel to plane $A$, but passes through the point $(1, 0, 1)$. Find the distance between planes $A$ and $B$.
2021 Caucasus Mathematical Olympiad, 5
A triangle $\Delta$ with sidelengths $a\leq b\leq c$ is given. It appears that it is impossible to construct a triangle from three segments whose lengths are equal to the altitudes of $\Delta$. Prove that $b^2>ac$.
2009 Bosnia And Herzegovina - Regional Olympiad, 4
Let $x$ and $y$ be positive integers such that $\frac{x^2-1}{y+1}+\frac{y^2-1}{x+1}$ is integer. Prove that numbers $\frac{x^2-1}{y+1}$ and $\frac{y^2-1}{x+1}$ are integers
2020 Jozsef Wildt International Math Competition, W19
Prove the inequality
$$\prod_{k=2}^n\left(1+\frac{k^{p-1}}{1^p+2^p+\ldots+k^p}\right)<e^{\frac{p-1}2}$$
[i]Proposed by Arkady Alt[/i]
1992 Hungary-Israel Binational, 2
A set $S$ consists of $1992$ positive integers among whose units digits all $10$ digits occur. Show that there is such a set $S$ having no nonempty subset $S_{1}$ whose sum of elements is divisible by $2000$.
2015 Vietnam Team selection test, Problem 5
Let $ABC$ be a triangle with an interior point $P$ such that $\angle APB = \angle APC = \alpha$ and $\alpha > 180^o-\angle BAC$. The circumcircle of triangle $APB$ cuts $AC$ at $E$, the circumcircle of triangle $APC$ cuts $AB$ at $F$. Let $Q$ be the point in the triangle $AEF$ such that $\angle AQE = \angle AQF =\alpha$. Let $D$ be the symmetric point of $Q$ wrt $EF$. Angle bisector of $\angle EDF$ cuts $AP$ at $T$.
a) Prove that $\angle DET = \angle ABC, \angle DFT = \angle ACB$.
b) Straight line $PA$ cuts straight lines $DE, DF$ at $M, N$ respectively. Denote $I, J$ the incenters of the triangles $PEM, PFN$, and $K$ the circumcenter of the triangle $DIJ$. Straight line $DT$ cut $(K)$ at $H$. Prove that $HK$ passes through the incenter of the triangle $DMN$.
2022 Saudi Arabia BMO + EGMO TST, p2
Determine if there exist functions $f, g : R \to R$ satisfying for every $x \in R$ the following equations $f(g(x)) = x^3$ and $g(f(x)) = x^2$.
2008 Tournament Of Towns, 3
Alice and Brian are playing a game on a $1\times (N + 2)$ board. To start the game, Alice places a checker on any of the $N$ interior squares. In each move, Brian chooses a positive integer $n$. Alice must move the checker to the $n$-th square on the left or the right of its current position. If the checker moves off the board, Alice wins. If it lands on either of the end squares, Brian wins. If it lands on another interior square, the game proceeds to the next move. For which values of $N$ does Brian have a strategy which allows him to win the game in a finite number of moves?
2006 Bulgaria National Olympiad, 1
Consider the set $A=\{1,2,3\ldots ,2^n\}, n\ge 2$. Find the number of subsets $B$ of $A$ such that for any two elements of $A$ whose sum is a power of $2$ exactly one of them is in $B$.
[i]Aleksandar Ivanov[/i]