Found problems: 85335
2007 QEDMO 4th, 5
Let $ ABC$ be a triangle, and let $ X$, $ Y$, $ Z$ be three points on the segments $ BC$, $ CA$, $ AB$, respectively. Denote by $ X^{\prime}$, $ Y^{\prime}$, $ Z^{\prime}$ the reflections of these points $ X$, $ Y$, $ Z$ in the midpoints of the segments $ BC$, $ CA$, $ AB$, respectively. Prove that $ \left\vert XYZ\right\vert \equal{}\left\vert X^{\prime}Y^{\prime}Z^{\prime}\right\vert$.
1984 AIME Problems, 7
The function $f$ is defined on the set of integers and satisfies \[ f(n)=\begin{cases} n-3 & \text{if } n\ge 1000 \\ f(f(n+5)) & \text{if } n<1000\end{cases} \] Find $f(84)$.
MMPC Part II 1996 - 2019, 2009
[b]p1.[/b] Given a group of $n$ people. An $A$-list celebrity is one that is known by everybody else (that is, $n - 1$ of them) but does not know anybody. A $B$-list celebrity is one that is known by exactly $n - 2$ people but knows at most one person.
(a) What is the maximum number of $A$-list celebrities? You must prove that this number is attainable.
(b) What is the maximum number of $B$-list celebrities? You must prove that this number is attainable.
[b]p2.[/b] A polynomial $p(x)$ has a remainder of $2$, $-13$ and $5$ respectively when divided by $x+1$, $x-4$ and $x-2$. What is the remainder when $p(x)$ is divided by $(x + 1)(x - 4)(x - 2)$?
[b]p3.[/b] (a) Let $x$ and y be positive integers satisfying $x^2 + y = 4p$ and $y^2 + x = 2p$, where $p$ is an odd prime number. Prove: $x + y = p + 1$.
(b) Find all values of $x, y$ and $p$ that satisfy the conditions of part (a). You will need to prove that you have found all such solutions.
[b]p4.[/b] Let function $f(x, y, z)$ be defined as following:
$$f(x, y, z) = \cos^2(x - y) + \cos^2(y - z) + \cos^2(z - x), x, y, z \in R.$$
Find the minimum value and prove the result.
[b]p5.[/b] In the diagram below, $ABC$ is a triangle with side lengths $a = 5$, $b = 12$,$ c = 13$. Let $P$ and $Q$ be points on $AB$ and $AC$, respectively, chosen so that the segment $PQ$ bisects the area of $\vartriangle ABC$. Find the minimum possible value for the length $PQ$.
[img]https://cdn.artofproblemsolving.com/attachments/b/2/91a09dd3d831b299b844b07cd695ddf51cb12b.png[/img]
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url]. Thanks to gauss202 for sending the problems.
2013 Saint Petersburg Mathematical Olympiad, 2
if $a^2+b^2+c^2+d^2=1$ prove that \[ (1-a)(1-b)\ge cd. \]
A. Khrabrov
2022 Ecuador NMO (OMEC), 2
Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all real numbers $x, y$
\[f(x + y)=f(f(x)) + y + 2022\]
2004 USAMTS Problems, 2
Call a number $a-b\sqrt2$ with $a$ and $b$ both positive integers $tiny$ if it is closer to zero than any number $c-d\sqrt2$ such that $c$ and $d$ are positive integers with $c<a$ and $d<b$. Three numbers which are tiny are $1-\sqrt2$, $3-2\sqrt2$, and $7-5\sqrt2$. Without using any calculator or computer, prove whether or not each of the following is tiny:
\[(a)\ 58-41\sqrt2,\qquad\qquad (b)\ 99-70\sqrt2.\]
2012 Peru IMO TST, 4
An infinite triangular lattice is given, such that the distance between any two adjacent points is always equal to $1$.
Points $A$, $B$, and $C$ are chosen on the lattice such that they are the vertices of an equilateral triangle of side length $L$, and the sides of $ABC$ contain no points from the lattice. Prove that, inside triangle $ABC$, there are exactly $\frac{L^2-1}{2}$ points from the lattice.
2015 AIME Problems, 15
Circles $\mathcal{P}$ and $\mathcal{Q}$ have radii $1$ and $4$, respectively, and are externally tangent at point $A$. Point $B$ is on $\mathcal{P}$ and point $C$ is on $\mathcal{Q}$ so that line $BC$ is a common external tangent of the two circles. A line $\ell$ through $A$ intersects $\mathcal{P}$ again at $D$ and intersects $\mathcal{Q}$ again at $E$. Points $B$ and $C$ lie on the same side of $\ell$, and the areas of $\triangle DBA$ and $\triangle ACE$ are equal. This common area is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[asy]
import cse5;
pathpen=black; pointpen=black;
size(6cm);
pair E = IP(L((-.2476,1.9689),(0.8,1.6),-3,5.5),CR((4,4),4)), D = (-.2476,1.9689);
filldraw(D--(0.8,1.6)--(0,0)--cycle,gray(0.7));
filldraw(E--(0.8,1.6)--(4,0)--cycle,gray(0.7));
D(CR((0,1),1)); D(CR((4,4),4,150,390));
D(L(MP("D",D(D),N),MP("A",D((0.8,1.6)),NE),1,5.5));
D((-1.2,0)--MP("B",D((0,0)),S)--MP("C",D((4,0)),S)--(8,0));
D(MP("E",E,N));
[/asy]
2024 Israel Olympic Revenge, P4
Let $ABC$ be an acute triangle. Let $D$ be a point inside side $BC$. Let $E$ be the foot from $D$ to $AC$, and let $F$ be a point on $AB$ so that $FE\perp AB$. It is given that the lines $AD, BE, CF$ concur. $M_A, M_B, M_C$ are the midpoints of sides $BC, AC, AB$ respectively, and $O$ is the circumcenter of $ABC$. Moreover, we define $P=EF\cap M_AM_B, S=DE\cap M_AM_C$. Prove that $O, P, S$ are collinear.
2022 Francophone Mathematical Olympiad, 1
find all the integer $n\geq1$ such that $\lfloor\sqrt{n}\rfloor \mid n$
1987 IMO Longlists, 38
Let $S_1$ and $S_2$ be two spheres with distinct radii that touch externally. The spheres lie inside a cone $C$, and each sphere touches the cone in a full circle. Inside the cone there are $n$ additional solid spheres arranged in a ring in such a way that each solid sphere touches the cone $C$, both of the spheres $S_1$ and $S_2$ externally, as well as the two neighboring solid spheres. What are the possible values of $n$?
[i]Proposed by Iceland.[/i]
2009 China Team Selection Test, 3
Consider function $ f: R\to R$ which satisfies the conditions for any mutually distinct real numbers $ a,b,c,d$ satisfying $ \frac {a \minus{} b}{b \minus{} c} \plus{} \frac {a \minus{} d}{d \minus{} c} \equal{} 0$, $ f(a),f(b),f(c),f(d)$ are mutully different and $ \frac {f(a) \minus{} f(b)}{f(b) \minus{} f(c)} \plus{} \frac {f(a) \minus{} f(d)}{f(d) \minus{} f(c)} \equal{} 0.$ Prove that function $ f$ is linear
2007 Moldova National Olympiad, 11.2
Define $a_{n}$ as satisfying: $\left(1+\frac{1}{n}\right)^{n+a_{n}}=e$.
Find $\lim_{n\rightarrow\infty}a_{n}$.
2024 Australian Mathematical Olympiad, P2
Let $ABCD$ be a cyclic quadrilateral. Point $P$ is on line $CB$ such that $CP=CA$and $B$ lies between $C$ and $P$. Point $Q$ is on line $CD$ such that $CQ=CA$ and $D$ lies between $C$ and $Q$. Prove that the incentre of triangle $ABD$ lies on line $PQ.$
2014 China Team Selection Test, 1
Let the circumcenter of triangle $ABC$ be $O$. $H_A$ is the projection of $A$ onto $BC$. The extension of $AO$ intersects the circumcircle of $BOC$ at $A'$. The projections of $A'$ onto $AB, AC$ are $D,E$, and $O_A$ is the circumcentre of triangle $DH_AE$. Define $H_B, O_B, H_C, O_C$ similarly.
Prove: $H_AO_A, H_BO_B, H_CO_C$ are concurrent
2003 AMC 10, 1
Which of the following is the same as
\[ \frac{2\minus{}4\plus{}6\minus{}8\plus{}10\minus{}12\plus{}14}{3\minus{}6\plus{}9\minus{}12\plus{}15\minus{}18\plus{}21}?
\]$ \textbf{(A)}\ \minus{}1 \qquad
\textbf{(B)}\ \minus{}\frac23 \qquad
\textbf{(C)}\ \frac23 \qquad
\textbf{(D)}\ 1 \qquad
\textbf{(E)}\ \frac{14}{3}$
2015 Irish Math Olympiad, 5
Suppose a doubly infinite sequence of real numbers $. . . , a_{-2}, a_{-1}, a_0, a_1, a_2, . . .$ has the property that $$a_{n+3} =\frac{a_n + a_{n+1} + a_{n+2}}{3},$$ for all integers $n .$ Show that if this sequence is bounded (i.e., if there exists a number $R$ such that $|a_n| \leq R$ for all $n$), then $a_n$ has the same value for all $n.$
2014 Contests, 1
In a non-obtuse triangle $ABC$, prove that
\[ \frac{\sin A \sin B}{\sin C} + \frac{\sin B \sin C}{\sin A} + \frac{\sin C \sin A}{ \sin B} \ge \frac 52. \][i]Proposed by Ryan Alweiss[/i]
2025 Philippine MO, P3
Let $d$ be a positive integer. Define the sequence $a_1, a_2, a_3, \dots$ such that \[\begin{cases} a_1 = 1 \\ a_{n+1} = n\left\lfloor\frac{a_n}{n}\right\rfloor + d, \quad n \ge 1.\end{cases}\] Prove that there exists a positive integer $M$ such that $a_M, a_{M+1}, a_{M+2}, \dots$ is an arithmetic sequence.
1998 Singapore MO Open, 2
Let $N$ be the set of natural numbers, and let $f: N \to N$ be a function satisfying $f(x) + f(x + 2) < 2 f(x + 1)$ for any $x \in N$. Prove that there exists a straight line in the $xy$-plane which contains infinitely many points with coordinates $(n,f(n))$.
2024 Germany Team Selection Test, 1
For positive integers $n$ and $k \geq 2$, define $E_k(n)$ as the greatest exponent $r$ such that $k^r$ divides $n!$. Prove that there are infinitely many $n$ such that $E_{10}(n) > E_9(n)$ and infinitely many $m$ such that $E_{10}(m) < E_9(m)$.
1947 Moscow Mathematical Olympiad, 132
Given line $AB$ and point $M$. Find all lines in space passing through $M$ at distance $d$.
2022 Princeton University Math Competition, A2 / B4
Let $P(x,y)$ be a polynomial with real coefficients in the variables $x,y$ that is not identically zero. Suppose that $P(\lfloor 2a \rfloor, \lfloor 3a\rfloor) = 0$ for all real numbers $a.$ If $P$ has the minimum possible degree and the coefficient of the monomial $y$ is $4,$ find the coefficient of $x^2y^2$ in $P.$
(The [i]degree[/i] of a monomial $x^my^n$ is $m + n.$ The [i]degree[/i] of a polynomial $P(x,y)$ is then the maximum degree of any of its monomials.)
2015 Iran MO (2nd Round), 2
A circle is divided into $2n$ equal by $2n$ points. Ali draws $n+1$ arcs, of length $1,2,\ldots,n+1$. Prove that we can find two arcs, such that one of them is inside in the other one.
2004 JBMO Shortlist, 5
Let $ABC$ be an isosceles triangle with $AC=BC$, let $M$ be the midpoint of its side $AC$, and let $Z$ be the line through $C$ perpendicular to $AB$. The circle through the points $B$, $C$, and $M$ intersects the line $Z$ at the points $C$ and $Q$. Find the radius of the circumcircle of the triangle $ABC$ in terms of $m = CQ$.