Found problems: 85335
2020 Austrian Junior Regional Competition, 4
Find all positive integers $a$ for which the equation $7an -3n! = 2020$ has a positive integer solution $n$.
(Richard Henner)
1977 IMO, 1
Let $a,b,A,B$ be given reals. We consider the function defined by \[ f(x) = 1 - a \cdot \cos(x) - b \cdot \sin(x) - A \cdot \cos(2x) - B \cdot \sin(2x). \] Prove that if for any real number $x$ we have $f(x) \geq 0$ then $a^2 + b^2 \leq 2$ and $A^2 + B^2 \leq 1.$
1996 Czech And Slovak Olympiad IIIA, 1
A sequence $(G_n)_{n=0}^{\infty}$ satisfies $G(0) = 0$ and $G(n) = n-G(G(n-1))$ for each $n \in N$. Show that
(a) $G(k) \ge G(k -1)$ for every $k \in N$;
(b) there is no integer $k$ for which $G(k -1) = G(k) = G(k +1)$.
2019 Taiwan TST Round 2, 1
Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}_{>0}\to \mathbb{Q}_{>0}$ satisfying $$f(x^2f(y)^2)=f(x)^2f(y)$$ for all $x,y\in\mathbb{Q}_{>0}$
2024 5th Memorial "Aleksandar Blazhevski-Cane", P5
For a given integer $k \geq 1$, find all $k$-tuples of positive integers $(n_1,n_2,...,n_k)$ with $\text{GCD}(n_1,n_2,...,n_k) = 1$ and $n_2|(n_1+1)^{n_1}-1$, $n_3|(n_2+1)^{n_2}-1$, ... , $n_1|(n_k+1)^{n_k}-1$.
[i]Authored by Pavel Dimovski[/i]
2024 Harvard-MIT Mathematics Tournament, 6
Compute the sum of all positive integers $50 \leq n \leq 100$ such that $2n+3 \nmid 2^{n!}-1$.
2005 USAMTS Problems, 3
An equilateral triangle is tiled with $n^2$ smaller congruent equilateral triangles such that there are $n$ smaller triangles along each of the sides of the original triangle. For each of the small equilateral triangles, we randomly choose a vertex $V$ of the triangle and draw an arc with that vertex as center connecting the midpoints of the two sides of the small triangle with $V$ as an endpoint. Find, with proof, the expected value of the number of full circles formed, in terms of $n.$
[img]http://s3.amazonaws.com/classroom.artofproblemsolving.com/Images/Transcripts/497b4e1ef5043a84b433a5c52c4be3ae.png[/img]
2003 AMC 8, 23
In the pattern below, the cat (denoted as a large circle in the figures below) moves clockwise through the four squares and the mouse (denoted as a dot in the figures below) moves counterclockwise through the eight exterior segments of the four squares.
[asy]defaultpen(linewidth(0.8));
size(350);
path p=unitsquare;
int i;
for(i=0; i<5; i=i+1) {
draw(shift(3i,0)*(p^^shift(1,0)*p^^shift(0,1)*p^^shift(1,1)*p));
}
path cat=Circle((0.5,0.5), 0.3);
draw(shift(0,1)*cat^^shift(4,1)*cat^^shift(7,0)*cat^^shift(9,0)*cat^^shift(12,1)*cat);
dot((1.5,0)^^(5,0.5)^^(8,1.5)^^(10.5,2)^^(12.5,2));
label("1", (1,2), N);
label("2", (4,2), N);
label("3", (7,2), N);
label("4", (10,2), N);
label("5", (13,2), N);
[/asy]
If the pattern is continued, where would the cat and mouse be after the 247th move?
$\textbf{(A)}$
[asy]defaultpen(linewidth(0.8));
size(60);
path p=unitsquare;
int i=0;
draw(shift(3i,0)*(p^^shift(1,0)*p^^shift(0,1)*p^^shift(1,1)*p));
path cat=Circle((0.5,0.5), 0.3);
draw(shift(1,0)*cat);
dot((0,0.5));
[/asy]
$\textbf{(B)}$
[asy]defaultpen(linewidth(0.8));
size(60);
path p=unitsquare;
int i=0;
draw(shift(3i,0)*(p^^shift(1,0)*p^^shift(0,1)*p^^shift(1,1)*p));
path cat=Circle((0.5,0.5), 0.3);
draw(shift(1,1)*cat);
dot((0,0.5));
[/asy]
$\textbf{(C)}$
[asy]defaultpen(linewidth(0.8));
size(60);
path p=unitsquare;
int i=0;
draw(shift(3i,0)*(p^^shift(1,0)*p^^shift(0,1)*p^^shift(1,1)*p));
path cat=Circle((0.5,0.5), 0.3);
draw(shift(1,0)*cat);
dot((0,1.5));
[/asy]
$\textbf{(D)}$
[asy]defaultpen(linewidth(0.8));
size(60);
path p=unitsquare;
int i=0;
draw(shift(3i,0)*(p^^shift(1,0)*p^^shift(0,1)*p^^shift(1,1)*p));
path cat=Circle((0.5,0.5), 0.3);
draw(shift(0,0)*cat);
dot((0,1.5));
[/asy]
$\textbf{(E)}$
[asy]defaultpen(linewidth(0.8));
size(60);
path p=unitsquare;
int i=0;
draw(shift(3i,0)*(p^^shift(1,0)*p^^shift(0,1)*p^^shift(1,1)*p));
path cat=Circle((0.5,0.5), 0.3);
draw(shift(0,1)*cat);
dot((1.5,0));
[/asy]
2005 Postal Coaching, 17
Let $A',\,B',\,C'$ be points, in which excircles touch corresponding sides of triangle $ABC$. Circumcircles of triangles $A'B'C,\,AB'C',\,A'BC'$ intersect a circumcircle of $ABC$ in points $C_1\ne C,\,A_1\ne A,\,B_1\ne B$ respectively. Prove that a triangle $A_1B_1C_1$ is similar to a triangle, formed by points, in which incircle of $ABC$ touches its sides.
2002 District Olympiad, 3
Let be two real numbers $ a,b, $ that satisfy $ 3^a+13^b=17^a $ and $ 5^a+7^b=11^b. $
Show that $ a<b. $
2012 Indonesia TST, 3
Let $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n$ be positive reals such that
\[a_1 + b_1 = a_2 + b_2 = \ldots + a_n + b_n\]
and
\[\sqrt[n]{\dfrac{a_1a_2\ldots a_n}{b_1b_2\ldots b_n}} \ge n.\]
Prove that
\[\sqrt[n]{\dfrac{a_1a_2\ldots a_n}{b_1b_2\ldots b_n}} \ge \dfrac{a_1+a_2+\ldots+a_n}{b_1+b_2+\ldots+b_n}.\]
1954 Moscow Mathematical Olympiad, 283
Consider five segments $AB_1, AB_2, AB_3, AB_4, AB_5$. From each point $B_i$ there can exit either $5$ segments or no segments at all, so that the endpoints of any two segments of the resulting graph (system of segments) do not coincide. Can the number of free endpoints of the segments thus constructed be equal to $1001$? (A free endpoint is an endpoint from which no segment begins.)
2017 Canada National Olympiad, 4
Let $ABCD$ be a parallelogram. Points $P$ and $Q$ lie inside $ABCD$ such that $\bigtriangleup ABP$ and $\bigtriangleup{BCQ}$ are equilateral. Prove that the intersection of the line through $P$ perpendicular to $PD$ and the line through $Q$ perpendicular to $DQ$ lies on the altitude from $B$ in $\bigtriangleup{ABC}$.
2014 Harvard-MIT Mathematics Tournament, 31
Compute \[\sum_{k=1}^{1007}\left(\cos\left(\dfrac{\pi k}{1007}\right)\right)^{2014}.\]
VI Soros Olympiad 1999 - 2000 (Russia), 10.9
Given an acute-angled triangle $ABC$, in which $P$, $M$, $N$ are the midpoints of the sides $AB$, $BC$, $AC$, respectively. A point $H$ is taken inside the triangle and perpendiculars $HK$, $HS$, $HQ$ are lowered from it to the sides $AB$, $BC$, $AC$, respectively ($K \in AB$, $S \in BC$, $Q \in AC$). It turned out that $MK = MQ$, $NS = NK$, $PS=PQ$. Prove that $H$ is the point of intersection of the altitudes of triangle $ABC$.
2023 Rioplatense Mathematical Olympiad, 4
Luffy is playing with some magic boxes and a machine. Each box has a value(number) inside. Opening a box, Luffy sees the value, adds the value to his score(if the box value is negative, Luffy loses points) and destroys the box. Putting a box of value $X$ in the machine, this box vanishes and it is replaced by two new boxes of values $X+1$ and $X-1$(it's [b]not[/b] known which one has the respective value, but he can identify the new boxes). At the beginning, Luffy has $0$ points and has a box whose value is known(it is zero).
a) Prove that Luffy can reach at least $1000$ points
b) Is it possible that Luffy reaches at least $1000000$ points, [b]without[/b] have less than $-42$ points in any moment?
1967 Miklós Schweitzer, 10
Let $ \sigma(S_n,k)$ denote the sum of the $ k$th powers of the lengths of the sides of the convex $ n$-gon $ S_n$ inscribed in a unit circle. Show that for any natural number greater than $ 2$ there exists a real number $ k_0$ between $ 1$ and $ 2$ such that $ \sigma(S_n,k_0)$ attains its maximum for the regular $ n$-gon.
[i]L. Fejes Toth[/i]
2006 Sharygin Geometry Olympiad, 8
The segment $AB$ divides the square into two parts, in each of which a circle can be inscribed.
The radii of these circles are equal to $r_1$ and $r_2$ respectively, where $r_1> r_2$.
Find the length of $AB$.
1945 Moscow Mathematical Olympiad, 101
The side $AD$ of a parallelogram $ABCD$ is divided into $n$ equal segments. The nearest to $A$ division point $P$ is connected with $B$. Prove that line $BP$ intersects the diagonal $AC$ at point $Q$ such that $AQ = \frac{AC}{n + 1}$
2001 AMC 8, 6
Six trees are equally spaced along one side of a straight road. The distance from the first tree to the fourth is 60 feet. What is the distance in feet between the first and last trees?
$ \text{(A)}\ 90\qquad\text{(B)}\ 100\qquad\text{(C)}\ 105\qquad\text{(D)}\ 120\qquad\text{(E)}\ 140 $
2015 China Western Mathematical Olympiad, 5
Let $a,b,c,d$ are lengths of the sides of a convex quadrangle with the area equal to $S$, set $S =\{x_1, x_2,x_3,x_4\}$ consists of permutations $x_i$ of $(a, b, c, d)$. Prove that \[S \leq \frac{1}{2}(x_1x_2+x_3x_4).\]
1995 Bulgaria National Olympiad, 2
Let triangle ABC has semiperimeter $ p$. E,F are located on AB such that $ CE\equal{}CF\equal{}p$. Prove that the C-excircle of triangle ABC touches the circumcircle (EFC).
2019 Belarus Team Selection Test, 5.3
A polygon (not necessarily convex) on the coordinate plane is called [i]plump[/i] if it satisfies the following conditions:
$\bullet$ coordinates of vertices are integers;
$\bullet$ each side forms an angle of $0^\circ$, $90^\circ$, or $45^\circ$ with the abscissa axis;
$\bullet$ internal angles belong to the interval $[90^\circ, 270^\circ]$.
Prove that if a square of each side length of a plump polygon is even, then such a polygon can be cut into several convex plump polygons.
[i](A. Yuran)[/i]
2005 MOP Homework, 7
Let $n$ be a natural number and $f_1$, $f_2$, ..., $f_n$ be polynomials with integers coeffcients. Show that there exists a polynomial $g(x)$ which can be factored (with at least two terms of degree at least $1$) over the integers such that $f_i(x)+g(x)$ cannot be factored (with at least two terms of degree at least $1$ over the integers for every $i$.
2014 China Western Mathematical Olympiad, 2
Let $ AB$ be the diameter of semicircle $O$ ,
$C, D $ be points on the arc $AB$,
$P, Q$ be respectively the circumcenter of $\triangle OAC $ and $\triangle OBD $ .
Prove that:$CP\cdot CQ=DP \cdot DQ$.[asy]
import cse5; import olympiad; unitsize(3.5cm); dotfactor=4; pathpen=black;
real h=sqrt(55/64);
pair A=(-1,0), O=origin, B=(1,0),C=shift(-3/8,h)*O,D=shift(4/5,3/5)*O,P=circumcenter(O,A,C), Q=circumcenter(O,D,B);
D(arc(O,1,0,180),darkgreen);
D(MP("A",A,W)--MP("C",C,N)--MP("P",P,SE)--MP("D",D,E)--MP("Q",Q,E)--C--MP("O",O,S)--D--MP("B",B,E)--cycle,deepblue);
D(O);
[/asy]