Found problems: 85335
2002 Federal Math Competition of S&M, Problem 4
Each of the $15$ coaches ranked the $50$ selected football players on the places from $1$ to $50$. For each football player, the highest and lowest obtained ranks differ by at most $5$. For each of the players, the sum of the ranks he obtained is computed, and the sums are denoted by $S_1\le S_2\le\ldots\le S_{50}$. Find the largest possible value of $S_1$.
2024 LMT Fall, C2
Eminem is trying to find the real Slim Shady in a row of $2025$ indistinguishable Slim Shady clones, one of which is the real Slim Shady. Eminem randomly guesses, and if he guesses wrong, a new clone joins the row and all the clones randomly rearrange themselves. He keeps guessing as more identical clones are added, trying to find the real Slim Shady. Find the probability that he will eventually find him within $15$ guesses.
2008 AMC 10, 7
An equilateral triangle of side length $ 10$ is completely filled in by non-overlapping equilateral triangles of side length $ 1$. How many small triangles are required?
$ \textbf{(A)}\ 10 \qquad
\textbf{(B)}\ 25 \qquad
\textbf{(C)}\ 100 \qquad
\textbf{(D)}\ 250 \qquad
\textbf{(E)}\ 1000$
2017 Baltic Way, 7
Each edge of a complete graph on $30$ vertices is coloured either red or blue. It is allowed to choose a non-monochromatic triangle and change the colour of the two edges of the same colour to make the triangle monochromatic.
Prove that by using this operation repeatedly it is possible to make the entire graph monochromatic.
(A complete graph is a graph where any two vertices are connected by an edge.)
1975 Chisinau City MO, 92
Solve in natural numbers the equation $x^2-y^2=105$.
2003 China National Olympiad, 2
Determine the maximal size of the set $S$ such that:
i) all elements of $S$ are natural numbers not exceeding $100$;
ii) for any two elements $a,b$ in $S$, there exists $c$ in $S$ such that $(a,c)=(b,c)=1$;
iii) for any two elements $a,b$ in $S$, there exists $d$ in $S$ such that $(a,d)>1,(b,d)>1$.
[i]Yao Jiangang[/i]
2007 Brazil National Olympiad, 1
Let $ f(x) \equal{} x^2 \plus{} 2007x \plus{} 1$. Prove that for every positive integer $ n$, the equation $ \underbrace{f(f(\ldots(f}_{n\ {\rm times}}(x))\ldots)) \equal{} 0$ has at least one real solution.
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 $