Found problems: 85335
2023 India Regional Mathematical Olympiad, 2
Let $\omega$ be a semicircle with $A B$ as the bounding diameter and let $C D$ be a variable chord of the semicircle of constant length such that $C, D$ lie in the interior of the arc $A B$. Let $E$ be a point on the diameter $A B$ such that $C E$ and $D E$ are equally inclined to the line $A B$. Prove that
(a) the measure of $\angle C E D$ is a constant;
(b) the circumcircle of triangle $C E D$ passes through a fixed point.
2005 Iran MO (2nd round), 1
Let $n,p>1$ be positive integers and $p$ be prime. We know that $n|p-1$ and $p|n^3-1$. Prove that $4p-3$ is a perfect square.
1969 Czech and Slovak Olympiad III A, 2
Five different points $O,A,B,C,D$ are given in plane such that \[OA\le OB\le OC\le OD.\] Show that for area $P$ of any convex quadrilateral with vertices $A,B,C,D$ (not necessarily in this order) the inequality \[P\le \frac12(OA+OD)(OB+OC)\] holds and determine when equality occurs.
2009 Hanoi Open Mathematics Competitions, 10
Let $ABC$ be an acute-angled triangle with $AB =4$ and $CD$ be the altitude through $C$ with $CD = 3$. Find the distance between the midpoints of $AD$ and $BC$
2002 Baltic Way, 20
Does there exist an infinite non-constant arithmetic progression, each term of which is of the form $a^b$, where $a$ and $b$ are positive integers with $b\ge 2$?
2015 Peru MO (ONEM), 3
Let $a_1, a_2, . . . , a_n$ be positive integers, with $n \ge 2$, such that $$ \lfloor \sqrt{a_1 \cdot a_2\cdot\cdot\cdot a_n} \rfloor = \lfloor \sqrt{a_1} \rfloor \cdot \lfloor \sqrt{a_2} \rfloor \cdot\cdot\cdot \lfloor \sqrt{a_n} \rfloor.$$
Prove that at least $n - 1$ of these numbers are perfect squares.
Clarification: Given a real number $x$, $\lfloor x\rfloor$ denotes the largest integer that is less than or equal to $x$.
For example $\lfloor \sqrt2\rfloor$ and $\lfloor 3\rfloor =3$.
2013 Online Math Open Problems, 9
Let $AXYZB$ be a regular pentagon with area $5$ inscribed in a circle with center $O$. Let $Y'$ denote the reflection of $Y$ over $\overline{AB}$ and suppose $C$ is the center of a circle passing through $A$, $Y'$ and $B$. Compute the area of triangle $ABC$.
[i]Proposed by Evan Chen[/i]
2023 Chile Junior Math Olympiad, 6
What is the smallest positive integer that is divisible by $225$ and that has ony the numbers one and zero as digits?
1974 Czech and Slovak Olympiad III A, 2
Let a triangle $ABC$ be given. For any point $X$ of the triangle denote $m(X)=\min\{XA,XB,XC\}.$ Find all points $X$ (of triangle $ABC$) such that $m(X)$ is maximal.
2015 AMC 10, 1
What is the value of $(2^0-1+5^2+0)^{-1}\times 5$?
$\textbf{(A) }-125\qquad\textbf{(B) }-120\qquad\textbf{(C) }\dfrac15\qquad\textbf{(D) }\dfrac5{24}\qquad\textbf{(E) }25$
1954 Czech and Slovak Olympiad III A, 4
Consider a cube $ABCDA'B'C'D$ (with $AB\perp AA'\parallel BB'\parallel CC'\parallel DD$). Let $X$ be an inner point of the segment $AB$ and denote $Y$ the intersection of the edge $AD$ and the plane $B'D'X$.
(a) Let $M=B'Y\cap D'X$. Find the locus of all $M$s.
(b) Determine whether there is a quadrilateral $B'D'YX$ such that its diagonals divide each other in the ratio 1:2.
1983 IMO Longlists, 16
Suppose that ${x_1, x_2, \dots , x_n}$ are positive integers for which $x_1 + x_2 + \cdots+ x_n = 2(n + 1)$. Show that there exists an integer $r$ with $0 \leq r \leq n - 1$ for which the following $n - 1$ inequalities hold:
\[x_{r+1} + \cdots + x_{r+i} \leq 2i+ 1, \qquad \qquad \forall i, 1 \leq i \leq n - r; \]
\[x_{r+1} + \cdots + x_n + x_1 + \cdots+ x_i \leq 2(n - r + i) + 1, \qquad \qquad \forall i, 1 \leq i \leq r - 1.\]
Prove that if all the inequalities are strict, then $r$ is unique and that otherwise there are exactly two such $r.$
2023 Mexican Girls' Contest, 3
Find all triples $(a,b,c)$ of real numbers all different from zero that satisfies:
\begin{eqnarray} a^4+b^2c^2=16a\nonumber \\ b^4+c^2a^2=16b \nonumber\\ c^4+a^2b^2=16c \nonumber \end{eqnarray}
2015 Princeton University Math Competition, A7
The lattice points $(i, j)$ for integers $0 \le i, j \le 3$ are each being painted orange or black. Suppose a coloring is good if for every set of integers $x_1, x_2, y_1, y_2$ such that $0 \le x_1 < x_2 \le 3$ and $0 \le y_1 < y_2 \le 3$, the points $(x_1, y_1),(x_1, y_2),(x_2, y_1),(x_2, y_2)$ are not all the same color. How many good colorings are possible?
1997 Korea National Olympiad, 1
Let $f(n)$ be the number of ways to express positive integer $n$ as a sum of positive odd integers.
Compute $f(n).$
(If the order of odd numbers are different, then it is considered as different expression.)
1996 Romania National Olympiad, 3
Let $A$ be a commutative ring with $0 \neq 1$ such that for any $x \in A \setminus \{0\}$ there exist positive integers $m,n$ such that $(x^m+1)^n=x.$ Prove that any endomorphism of $A$ is an automorphism.
2018 CCA Math Bonanza, L3.3
On January 15 in the stormy town of Stormville, there is a $50\%$ chance of rain. Every day, the probability of it raining has a $50\%$ chance of being $\frac{2017}{2016}$ times that of the previous day (or $100\%$ if this new quantity is over $100\%$) and a $50\%$ chance of being $\frac{1007}{2016}$ times that of the previous day. What is the probability that it rains on January 20?
[i]2018 CCA Math Bonanza Lightning Round #3.3[/i]
2013 AMC 12/AHSME, 7
Jo and Blair take turns counting from $1$ to one more than the last number said by the other person. Jo starts by saying "$1$", so Blair follows by saying "$1$, $2$". Jo then says "$1$, $2$, $3$", and so on. What is the $53$rd number said?
$ \textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }8 $
2010 ISI B.Stat Entrance Exam, 10
There are $100$ people in a queue waiting to enter a hall. The hall has exactly $100$ seats numbered from $1$ to $100$. The first person in the queue enters the hall, chooses any seat and sits there. The $n$-th person in the queue, where $n$ can be $2, . . . , 100$, enters the hall after $(n-1)$-th person is seated. He sits in seat number $n$ if he finds it vacant; otherwise he takes any unoccupied seat. Find the total number of ways in which $100$ seats can be filled up, provided the $100$-th person occupies seat number $100$.
2012 Oral Moscow Geometry Olympiad, 4
In triangle $ABC$, point $I$ is the center of the inscribed circle points, points $I_A$ and $I_C$ are the centers of the excircles, tangent to sides $BC$ and $AB$, respectively. Point $O$ is the center of the circumscribed circle of triangle $II_AI_C$. Prove that $OI \perp AC$
2011 Federal Competition For Advanced Students, Part 1, 2
For a positive integer $k$ and real numbers $x$ and $y$, let
\[f_k(x,y)=(x+y)-\left(x^{2k+1}+y^{2k+1}\right)\mbox{.}\]
If $x^2+y^2=1$, then determine the maximal possible value $c_k$ of $f_k(x,y)$.
1982 IMO Shortlist, 16
Prove that if $n$ is a positive integer such that the equation \[ x^3-3xy^2+y^3=n \] has a solution in integers $x,y$, then it has at least three such solutions. Show that the equation has no solutions in integers for $n=2891$.
2017 Caucasus Mathematical Olympiad, 5
In a football tournament $20$ teams participated, each pair of teams played exactly one game. For the win the team is awarded $3$ points, for the draw -- $1$ point, for the lose no points awarded. The total number of points of all teams in the tournament is $554$. Prove that there exist $7$ teams each having at least one draw.
2014 BMT Spring, 8
Annisa has $n$ distinct textbooks, where $n > 6$. She has a different ways to pick a group of $4$ books, b different ways to pick $5$ books and c different ways to pick $6$ books. If Annisa buys two more (distinct) textbooks, how many ways will she be able to pick a group of $6$ books?
1987 IMO Longlists, 65
The [i]runs[/i] of a decimal number are its increasing or decreasing blocks of digits. Thus $024379$ has three [i]runs[/i] : $024, 43$, and $379$. Determine the average number of runs for a decimal number in the set $\{d_1d_2 \cdots d_n | d_k \neq d_{k+1}, k = 1, 2,\cdots, n - 1\}$, where $n \geq 2.$