Found problems: 85335
2013 Harvard-MIT Mathematics Tournament, 5
Rahul has ten cards face-down, which consist of five distinct pairs of matching cards. During each move of his game, Rahul chooses one card to turn face-up, looks at it, and then chooses another to turn face-up and looks at it. If the two face-up cards match, the game ends. If not, Rahul flips both cards face-down and keeps repeating this process. Initially, Rahul doesn't know which cards are which. Assuming that he has perfect memory, find the smallest number of moves after which he can guarantee that the game has ended.
MOAA Gunga Bowls, 2021.21
King William is located at $(1, 1)$ on the coordinate plane. Every day, he chooses one of the eight lattice points closest to him and moves to one of them with equal probability. When he exits the region bounded by the $x, y$ axes and $x+y = 4$, he stops moving and remains there forever. Given that after an arbitrarily large amount of time he must exit the region, the probability he ends up on $x+y = 4$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[i]Proposed by Andrew Wen[/i]
2022 Tuymaada Olympiad, 3
Bisectors of a right triangle $\triangle ABC$ with right angle $B$ meet at point $I.$ The perpendicular to $IC$ drawn from $B$ meets the line $IA$ at $D;$ the perpendicular to $IA$ drawn from $B$ meets the line $IC$ at $E.$ Prove that the circumcenter of the triangle $\triangle IDE$ lies on the line $AC.$
[i](A. Kuznetsov )[/i]
2023 AMC 10, 14
A number is chosen at random from among the first $100$ positive integers, and a positive integer divisor of that number is then chosen at random. What is the probability that the chosen divisor is divisible by $11$?
$\textbf{(A)}~\frac{4}{100}\qquad\textbf{(B)}~\frac{9}{200} \qquad \textbf{(C)}~\frac{1}{20} \qquad\textbf{(D)}~\frac{11}{200}\qquad\textbf{(E)}~\frac{3}{50}$
2015 AMC 8, 25
One-inch squares are cut from the corners of this 5 inch square. What is the area in square inches of the largest square that can be fitted into the remaining space?
$ \textbf{(A) } 9\qquad \textbf{(B) } 12\frac{1}{2}\qquad \textbf{(C) } 15\qquad \textbf{(D) } 15\frac{1}{2}\qquad \textbf{(E) } 17$
[asy]
draw((0,0)--(0,5)--(5,5)--(5,0)--cycle);
filldraw((0,4)--(1,4)--(1,5)--(0,5)--cycle, gray);
filldraw((0,0)--(1,0)--(1,1)--(0,1)--cycle, gray);
filldraw((4,0)--(4,1)--(5,1)--(5,0)--cycle, gray);
filldraw((4,4)--(4,5)--(5,5)--(5,4)--cycle, gray);
[/asy]
2021 AMC 10 Fall, 10
A school has $100$ students and $5$ teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The enrollments in the classes are $50, 20, 20, 5, $ and $5$. Let $t$ be the average value obtained if a teacher is picked at random and the number of students in their class is noted. Let $s$ be the average value obtained if a student was picked at random and the number of students in their class, including the student, is noted. What is $t-s$?
$\textbf{(A)}\ {-}18.5 \qquad\textbf{(B)}\ {-}13.5 \qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\
13.5 \qquad\textbf{(E)}\ 18.5$
2019 Thailand TSTST, 3
Find all function $f:\mathbb{Z}\to\mathbb{Z}$ satisfying
$\text{(i)}$ $f(f(m)+n)+2m=f(n)+f(3m)$ for every $m,n\in\mathbb{Z}$,
$\text{(ii)}$ there exists a $d\in\mathbb{Z}$ such that $f(d)-f(0)=2$, and
$\text{(iii)}$ $f(1)-f(0)$ is even.
1980 Yugoslav Team Selection Test, Problem 3
A sequence $(x_n)$ satisfies $x_{n+1}=\frac{x_n^2+a}{x_{n-1}}$ for all $n\in\mathbb N$. Prove that if $x_0,x_1$, and $\frac{x_0^2+x_1^2+a}{x_0x_1}$ are integers, then all the terms of sequence $(x_n)$ are integers.
2024 China Team Selection Test, 24
Let $N=10^{2024}$. $S$ is a square in the Cartesian plane with side length $N$ and the sides parallel to the coordinate axes. Inside there are $N$ points $P_1$, $P_2$, $\dots$, $P_N$ all of which have different $x$ coordinates, and the absolute value of the slope of any connected line between these points is at most $1$. Prove that there exists a line $l$ such that at least $2024$ of these points is at most distance $1$ away from $l$.
2017 NIMO Problems, 4
For how many positive integers $100 < n \le 10000$ does $\lfloor \sqrt{n-100} \rfloor$ divide $n$?
[i]Proposed by Michael Tang[/i]
1984 IMO, 1
Prove that $0\le yz+zx+xy-2xyz\le{7\over27}$, where $x,y$ and $z$ are non-negative real numbers satisfying $x+y+z=1$.
2016 Brazil National Olympiad, 6
Lei it \(ABCD\) be a non-cyclical, convex quadrilateral, with no parallel sides.
The lines \(AB\) and \(CD\) meet in \(E\).
Let it \(M \not= E\) be the intersection of circumcircles of \(ADE\) and \(BCE\).
The internal angle bisectors of \(ABCD\) form an convex, cyclical quadrilateral with circumcenter \(I\).
The external angle bisectors of \(ABCD\) form an convex, cyclical quadrilateral with circumcenter \(J\).
Show that \(I,J,M\) are colinear.
2021/2022 Tournament of Towns, P3
The hypotenuse of a right triangle has length 1. Consider the line passing through the points of tangency of the incircle with the legs of the triangle. The circumcircle of the triangle cuts out a segment of this line. What is the possible length of this segment?
[i]Maxim Volchkevich[/i]
1990 National High School Mathematics League, 7
If $n\in\mathbb{Z_+}$, positive real numbers $a+b=2$, then the minumum value of $\frac{1}{1+a^n}+\frac{1}{1+b^n}$ is________.
2006 AMC 12/AHSME, 8
The lines $ x \equal{} \frac 14y \plus{} a$ and $ y \equal{} \frac 14x \plus{} b$ intersect at the point $ (1,2)$. What is $ a \plus{} b$?
$ \textbf{(A) } 0 \qquad \textbf{(B) } \frac 34 \qquad \textbf{(C) } 1 \qquad \textbf{(D) } 2 \qquad \textbf{(E) } \frac 94$
1982 IMO Longlists, 11
A rectangular pool table has a hole at each of three of its corners. The lengths of sides of the table are the real numbers $a$ and $b$. A billiard ball is shot from the fourth corner along its angle bisector. The ball falls in one of the holes. What should the relation between $a$ and $b$ be for this to happen?
1998 National Olympiad First Round, 7
Find the minimal value of integer $ n$ that guarantees:
Among $ n$ sets, there exits at least three sets such that any of them does not include any other; or there exits at least three sets such that any two of them includes the other.
$\textbf{(A)}\ 4 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ 8$
2007 Italy TST, 3
Find all $f: R \longrightarrow R$ such that
\[f(xy+f(x))=xf(y)+f(x)\]
for every pair of real numbers $x,y$.
1972 Polish MO Finals, 1
Polynomials $u_i(x) = a_ix+b_i$ ($a_i,b_i \in R$, $ i = 1,2,3$) satisfy
$$u_1(x)^n +u_2(x)^n = u_3(x)^n$$
for some integer $n \ge 2.$
Prove that there exist real numbers $A$,$B$,$c_1$,$c_2$,$c_3$ such that $u_i(x) = c_i(Ax+B)$ for $i = 1,2,3$.
2006 IMO Shortlist, 4
Let $P(x)$ be a polynomial of degree $n > 1$ with integer coefficients and let $k$ be a positive integer. Consider the polynomial $Q(x) = P(P(\ldots P(P(x)) \ldots ))$, where $P$ occurs $k$ times. Prove that there are at most $n$ integers $t$ such that $Q(t) = t$.
1949 Moscow Mathematical Olympiad, 158
a) Prove that $x^2 + y^2 + z^2 = 2xyz$ for integer $x, y, z$ only if $x = y = z = 0$.
b) Find integers $x, y, z, u$ such that $x^2 + y^2 + z^2 + u^2 = 2xyzu$.
2015 Math Prize for Girls Problems, 14
Let $C$ be a three-dimensional cube with edge length 1. There are 8 equilateral triangles whose vertices are vertices of $C$. The 8 planes that contain these 8 equilateral triangles divide $C$ into several nonoverlapping regions. Find the volume of the region that contains the center of $C$.
2016 CIIM, Problem 1
Find all functions $f:(0,+\infty) \to (0,+\infty)$ that satisfy
$(i)$ $f(xf(y))=yf(x), \forall x,y > 0,$
$(ii)$ $\displaystyle\lim_{x\to+\infty} f(x) = 0.$
2017 AMC 12/AHSME, 15
Let $ABC$ be an equilateral triangle. Extend side $\overline{AB}$ beyond $B$ to a point $B'$ so that $BB' = 3AB$. Similarly, extend side $\overline{BC}$ beyond $C$ to a point $C'$ so that $CC' = 3BC$, and extend side $\overline{CA}$ beyond $A$ to a point $A'$ so that $AA' = 3CA$. What is the ratio of the area of $\triangle A'B'C'$ to the area of $\triangle ABC$?
$\textbf{(A) }9:1\qquad\textbf{(B) }16:1\qquad\textbf{(C) }25:1\qquad\textbf{(D) }36:1\qquad\textbf{(E) }37:1$
2017 NIMO Summer Contest, 4
The square $BCDE$ is inscribed in circle $\omega$ with center $O$. Point $A$ is the reflection of $O$ over $B$. A "hook" is drawn consisting of segment $AB$ and the major arc $\widehat{BE}$ of $\omega$ (passing through $C$ and $D$). Assume $BCDE$ has area $200$. To the nearest integer, what is the length of the hook?
[i]Proposed by Evan Chen[/i]