Found problems: 85335
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 $n \geq 2.$
2021 Azerbaijan EGMO TST, 2
Let $\omega$ be a circle with center $O,$ and let $A$ be a point with tangents $AP$ and $AQ$ to the circle. Denote by $K$ the intersection point of $AO$ and $PQ.$ $l_1$ and $l_2$ are two lines passing through $A$ that intersect $\omega.$ Call $B$ the intersection point of $l_1$ with $\omega,$ which is located nearer to $A$ on $l_1.$ Call $C$ the intersection point of $l_2$ with $\omega,$ which is located further to $A$ on $l_2.$ Prove that $\angle PAB = \angle QAC$ if and only if the points $B, K, C$ are on line.
2017 HMNT, 9
Find the minimum value of $\sqrt{58-42x}+\sqrt{149-140\sqrt{1-x^2}}$ where $-1 \le x \le 1$.
2012 Junior Balkan Team Selection Tests - Romania, 2
Consider a semicircle of center $O$ and diameter $[AB]$, and let $C$ be an arbitrary point on the segment $(OB)$. The perpendicular to the line $AB$ through $C$ intersects the semicircle in $D$. A circle centered in $P$ is tangent to the arc $BD$ in $F$ and to the segments $[AB]$ and $[CD]$ in $G$ and $E$, respectively. Prove that the triangle $ADG$ is isosceles.
2023 China Second Round, 2
if a,b∈R+,$a^{\log b}=2$,$a^{\log a}b^{\log b}=5$,find out $(ab)^{\log ab}$
2010 AMC 12/AHSME, 15
For how many ordered triples $ (x,y,z)$ of nonnegative integers less than $ 20$ are there exactly two distinct elements in the set $ \{i^x,(1 \plus{} i)^y,z\}$, where $ i \equal{} \sqrt { \minus{} 1}$?
$ \textbf{(A)}\ 149 \qquad
\textbf{(B)}\ 205 \qquad
\textbf{(C)}\ 215 \qquad
\textbf{(D)}\ 225 \qquad
\textbf{(E)}\ 235$
2020 AMC 12/AHSME, 3
A driver travels for $2$ hours at $60$ miles per hour, during which her car gets $30$ miles per gallon of gasoline. She is paid $\$0.50$ per mile, and her only expense is gasoline at $\$2.00$ per gallon. What is her net rate of pay, in dollars per hour, after this expense?
$\textbf{(A) }20 \qquad\textbf{(B) }22 \qquad\textbf{(C) }24 \qquad\textbf{(D) } 25\qquad\textbf{(E) } 26$
2001 China Team Selection Test, 1
In an acute-angled triangle $\triangle ABC$, construct $\triangle ACD$ and $\triangle BCE$ externally on sides $CA$ and $CB$ respectively, such that $AD=CD$. Let $M$ be the midpoint of $AB$, and connect $DM$ and $EM$. Given that $DM$ is perpendicular to $EM$, set $\frac{AC}{BC} =u$ and $\frac{DM}{EM}=v$. Express $\frac{DC}{EC}$ in terms of $u$ and $v$.
2015 Vietnam National Olympiad, 1
Let ${\left\{ {f(x)} \right\}}$ be a sequence of polynomial, where ${f_0}(x) = 2$, ${f_1}(x) = 3x$, and
${f_n}(x) = 3x{f_{n - 1}}(x) + (1 - x - 2{x^2}){f_{n - 2}}(x)$ $(n \ge 2)$
Determine the value of $n$ such that ${f_n}(x)$ is divisible by $x^3-x^2+x$.
1996 Bosnia and Herzegovina Team Selection Test, 2
$a)$ Let $m$ and $n$ be positive integers. If $m>1$ prove that $ n \mid \phi(m^n-1)$ where $\phi$ is Euler function
$b)$ Prove that number of elements in sequence $1,2,...,n$ $(n \in \mathbb{N})$, which greatest common divisor with $n$ is $d$, is $\phi\left(\frac{n}{d}\right)$