Found problems: 85335
2016 Canadian Mathematical Olympiad Qualification, 5
Consider a convex polygon $P$ with $n$ sides and perimeter $P_0$. Let the polygon $Q$, whose vertices are the midpoints of the sides of $P$, have perimeter $P_1$. Prove that $P_1 \geq \frac{P_0}{2}$.
1977 Chisinau City MO, 135
Solve the equation:
$$x=1978 - \dfrac{1977}{1978 - \dfrac{1977}{\frac{...}{...\dfrac{1977}{1978 -\dfrac{1977}{x}}}}}{}$$
1970 AMC 12/AHSME, 3
If $x=1+2^p$ and $y=1+2^{-p}$, then $y$ in terms of $x$ is:
$\textbf{(A) }\dfrac{x+1}{x-1}\qquad\textbf{(B) }\dfrac{x+2}{x-1}\qquad\textbf{(C) }\dfrac{x}{x-1}\qquad\textbf{(D) }2-x\qquad \textbf{(E) }\dfrac{x-1}{x}$
2023 Princeton University Math Competition, A2 / B4
Let $\triangle{ABC}$ be an isosceles triangle with $AB = AC =\sqrt{7}, BC=1$. Let $G$ be the centroid of $\triangle{ABC}$. Given $ j\in \{0,1,2\}$, let $T_{j}$ denote the triangle obtained by rotating $\triangle{ABC}$ about $G$ by $\frac{2\pi j}{3}$ radians. Let $\mathcal{P}$ denote the intersection of the interiors of triangles $T_0,T_1,T_2$. If $K$ denotes the area of $\mathcal{P}$, then $K^2=\frac{a}{b}$ for relatively prime positive integers $a, b$. Find $a + b$.
1995 AMC 8, 9
Three congruent circles with centers $P$, $Q$, and $R$ are tangent to the sides of rectangle $ABCD$ as shown. The circle centered at $Q$ has diameter $4$ and passes through points $P$ and $R$. The area of the rectangle is
[asy]
pair A,B,C,D,P,Q,R;
A = (0,4); B = (8,4); C = (8,0); D = (0,0);
P = (2,2); Q = (4,2); R = (6,2);
dot(A); dot(B); dot(C); dot(D); dot(P); dot(Q); dot(R);
draw(A--B--C--D--cycle);
draw(circle(P,2));
draw(circle(Q,2));
draw(circle(R,2));
label("$A$",A,NW);
label("$B$",B,NE);
label("$C$",C,SE);
label("$D$",D,SW);
label("$P$",P,W);
label("$Q$",Q,W);
label("$R$",R,W);
[/asy]
$\text{(A)}\ 16 \qquad \text{(B)}\ 24 \qquad \text{(C)}\ 32 \qquad \text{(D)}\ 64 \qquad \text{(E)}\ 128$
2021 Girls in Math at Yale, 1
Given that $2x + 7y = 3$, find $2^{6x + 21y - 4}$.
[i]Proposed by Deyuan Li[/i]
2019 China Northern MO, 6
For nonnegative real numbers $a,b,c,x,y,z$, if$a+b+c=x+y+z=1$, find the maximum value of $(a-x^2)(b-y^2)(c-z^2)$.
2010 National Olympiad First Round, 10
How many integers $n$ with $0\leq n < 840$ are there such that $840$ divides $n^8-n^4+n-1$?
$ \textbf{(A)}\ 1
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 6
\qquad\textbf{(E)}\ 8
$
2024 International Zhautykov Olympiad, 1
In an alphabet of $n$ letters, is $syllable$ is any ordered pair of two (not necessarily distinct) letters. Some syllables are considered $indecent$. A $word$ is any sequence, finite or infinite, of letters, that does not contain indecent syllables. Find the least possible number of indecent syllables for which infinite words do not exist.
2003 India Regional Mathematical Olympiad, 3
Let $a,b,c$ be three positive real numbers such that $a + b +c =1$ . prove that among the three numbers $a-ab, b - bc, c-ca$ there is one which is at most $\frac{1}{4}$ and there is one which is at least $\frac{2}{9}$.
1989 Mexico National Olympiad, 6
Determine the number of paths from $A$ to $B$ on the picture that go along gridlines only, do not pass through any point twice, and never go upwards?
[img]https://cdn.artofproblemsolving.com/attachments/0/2/87868e24a48a2e130fb5039daeb85af42f4b9d.png[/img]
2021 China Team Selection Test, 4
Let $f(x),g(x)$ be two polynomials with integer coefficients. It is known that for infinitely many prime $p$, there exist integer $m_p$ such that
$$f(a) \equiv g(a+m_p) \pmod p$$
holds for all $a \in \mathbb{Z}.$ Prove that there exists a rational number $r$ such that
$$f(x)=g(x+r).$$
2023 ELMO Shortlist, C5
Define the [i]mexth[/i] of \(k\) sets as the \(k\)th smallest positive integer that none of them contain, if it exists. Does there exist a family \(\mathcal F\) of sets of positive integers such that [list] [*]for any nonempty finite subset \(\mathcal G\) of \(\mathcal F\), the mexth of \(\mathcal G\) exists, and [*]for any positive integer \(n\), there is exactly one nonempty finite subset \(\mathcal G\) of \(\mathcal F\) such that \(n\) is the mexth of \(\mathcal G\). [/list]
[i]Proposed by Espen Slettnes[/i]
1990 AMC 12/AHSME, 19
For how many integers $N$ between $1$ and $1990$ is the improper fraction $\frac{N^2+7}{N+4}$ not in lowest terms?
$\text{(A)} \ 0 \qquad \text{(B)} \ 86 \qquad \text{(C)} \ 90 \qquad \text{(D)} \ 104 \qquad \text{(E)} \ 105$
2022 Estonia Team Selection Test, 3
Determine all tuples of integers $(a,b,c)$ such that:
$$(a-b)^3(a+b)^2 = c^2 + 2(a-b) + 1$$
2003 JHMMC 8, 12
Compute $\frac{664.02}{9.3}$.
1992 IMTS, 4
An international firm has 250 employees, each of whom speaks several languages. For each pair of employees, $(A,B)$, there is a language spoken by $A$ and not $B$, and there is another language spoken by $B$ but not $A$. At least how many languages must be spoken at the firm?
2013 Online Math Open Problems, 18
Given an $n\times n$ grid of dots, let $f(n)$ be the largest number of segments between adjacent dots which can be drawn such that (i) at most one segment is drawn between each pair of dots, and (ii) each dot has $1$ or $3$ segments coming from it. (For example, $f(4)=16$.) Compute $f(2000)$.
[i]Proposed by David Stoner[/i]
2016 Peru MO (ONEM), 3
Find all functions $f\colon \mathbb{R}\to\mathbb{R}$ such that
\[f(x + y) + f(x + z) - f(x)f(y + z) \ge 1\]
for all $x,y,z \in \mathbb{R}$
1994 AMC 8, 13
The number halfway between $\dfrac{1}{6}$ and $\dfrac{1}{4}$ is
$\text{(A)}\ \dfrac{1}{10} \qquad \text{(B)}\ \dfrac{1}{5} \qquad \text{(C)}\ \dfrac{5}{24} \qquad \text{(D)}\ \dfrac{7}{24} \qquad \text{(E)}\ \dfrac{5}{12}$
2023 Polish Junior MO Second Round, 1.
On the sides $AB$ and $BC$ of triangle $ABC$, there are points $D$ and $E$, respectively, such that \[\angle ADC=\angle BDE\quad\text{and}\quad \angle BCD=\angle AED.\] Prove that $AE=BE$.
2001 China Team Selection Test, 3
For a positive integer \( n \geq 6 \), find the smallest integer \( S(n) \) such that any graph with \( n \) vertices and at least \( S(n) \) edges must contain at least two disjoint cycles (cycles with no common vertices).
2018 Malaysia National Olympiad, A4
Given a circle with diameter $AB$. Points $C$ and $D$ are selected on the circumference of the circle such that the chord $CD$ intersects $AB$ inside the circle, at point $P$. The ratio of the arc length $\overarc {AC}$ to the arc length $\overarc {BD}$ is $4 : 1$ , while the ratio of the arc length $\overarc{AD}$ to the arc length $\overarc {BC}$ is $3 : 2$ . Find $\angle{APC}$ , in degrees.
1985 IMO Longlists, 53
For each $P$ inside the triangle $ABC$, let $A(P), B(P)$, and $C(P)$ be the points of intersection of the lines $AP, BP$, and $CP$ with the sides opposite to $A, B$, and $C$, respectively. Determine $P$ in such a way that the area of the triangle $A(P)B(P)C(P)$ is as large as possible.
2007 Kurschak Competition, 3
Prove that any finite set $H$ of lattice points on the plane has a subset $K$ with the following properties:
[list]
[*]any vertical or horizontal line in the plane cuts $K$ in at most $2$ points,
[*]any point of $H\setminus K$ is contained by a segment with endpoints from $K$.[/list]