Found problems: 85335
2025 Junior Macedonian Mathematical Olympiad, 3
Is there an infinite sequence of prime numbers $p_1, p_2, ..., p_n, ...,$ such that for every $i \in \mathbb{N}, p_{i + 1} \in \{2p_i - 1, 2p_i + 1\}$ is satisfied? Explain the answer.
2021 Israel TST, 4
Let $r$ be a positive integer and let $a_r$ be the number of solutions to the equation $3^x-2^y=r$ ,such that $0\leq x,y\leq 5781$ are integers. What is the maximal value of $a_r$?
2018 Moldova EGMO TST, 2
Let $S$ = { $x_1$ , $x_2$ } be the solutions of the equation $x^2-2*a*x -1 = 0 $ , where $a$ is a positive integer.Prove that for any $ n \in\mathbb{N} $ the expression $ E=\frac{1}{8}$($x_1^{2n}-x_2^{2n}$)($x_1^{4n}-x_2^{4n}$) is a product of consecutive numbers.
2014 Contests, 3
(i) $ABC$ is a triangle with a right angle at $A$, and $P$ is a point on the hypotenuse $BC$.
The line $AP$ produced beyond $P$ meets the line through $B$ which is perpendicular to $BC$ at $U$.
Prove that $BU = BA$ if, and only if, $CP = CA$.
(ii) $A$ is a point on the semicircle $CB$, and points $X$ and $Y$ are on the line segment $BC$.
The line $AX$, produced beyond $X$, meets the line through $B$ which is perpendicular to $BC$ at $U$.
Also the line $AY$, produced beyond $Y$, meets the line through $C$ which is perpendicular to $BC$ at $V$.
Given that $BY = BA$ and $CX = CA$, determine the angle $\angle VAU$.
2012 Thailand Mathematical Olympiad, 7
Let $a, b, m$ be integers such that gcd $(a, b) = 1$ and $5 | ma^2 + b^2$ . Show that there exists an integer $n$ such that $5 | m - n^2$.
2001 Tournament Of Towns, 2
Let $n\ge3$ be an integer. A circle is divided into $2n$ arcs by $2n$ points. Each arc has one of three possible lengths, and no two adjacent arcs have the same lengths. The $2n$ points are colored alternately red and blue. Prove that the $n$-gon with red vertices and the $n$-gon with blue vertices have the same perimeter and the same area.
2014 ISI Entrance Examination, 6
Define $\mathcal{A}=\{(x,y)|x=u+v,y=v, u^2+v^2\le 1\}$. Find the length of the longest segment that is contained in $\mathcal{A}$.
2002 France Team Selection Test, 1
There are three colleges in a town. Each college has $n$ students. Any student of any college knows $n+1$ students of the other two colleges. Prove that it is possible to choose a student from each of the three colleges so that all three students would know each other.
2010 Dutch IMO TST, 4
Let $ABCD$ be a cyclic quadrilateral satisfying $\angle ABD = \angle DBC$. Let $E$ be the intersection of the diagonals $AC$ and $BD$. Let $M$ be the midpoint of $AE$, and $N$ be the midpoint of $DC$. Show that $MBCN$ is a cyclic quadrilateral.
1961 Miklós Schweitzer, 1
[b]1.[/b] Let $a$ ( $\neq e$, the unit element) be an element of finite order of a group $G$ and let $t$ ($\geq 2$) be a positive integer. Show: if the complex $A= \{ e,a,a^2, \dots , a^{t-1} \} $ is not a group, then for every positive integer $k$( $2 \leq k \leq t$) the complex $B= \{ e. a^k, a^{2k}, \dots , a^{(t-1)k} \} $ differs from $A$. [b](A. 16)[/b]
1998 VJIMC, Problem 4-M
A function $f:\mathbb R\to\mathbb R$ has the property that for every
$x,y\in\mathbb R$ there exists a real number $t$ (depending on $x$ and $y$) such
that $0<t<1$ and
$$f(tx+(1-t)y)=tf(x)+(1-t)f(y).$$
Does it imply that
$$f\left(\frac{x+y}2\right)=\frac{f(x)+f(y)}2$$
for every $x,y\in\mathbb R$?
2015 Indonesia MO Shortlist, A7
Suppose $P(n) $ is a nonconstant polynomial where all of its coefficients are nonnegative integers such that
\[ \sum_{i=1}^n P(i) | nP(n+1) \]
for every $n \in \mathbb{N}$.
Prove that there exists an integer $k \ge 0$ such that
\[ P(n) = \binom{n+k}{n-1} P(1) \]
for every $n \in \mathbb{N}$.
2018 Brazil National Olympiad, 1
We say that a polygon $P$ is [i]inscribed[/i] in another polygon $Q$ when all vertices of $P$ belong to perimeter of $Q$. We also say in this case that $Q$ is [i]circumscribed[/i] to $P$. Given a triangle $T$, let $l$ be the maximum value of the side of a square inscribed in $T$ and $L$ be the minimum value of the side of a square circumscribed to $T$. Prove that for every triangle $T$ the inequality $L/l \ge 2$ holds and find all the triangles $T$ for which the equality occurs.
2020 Brazil National Olympiad, 6
Let $f (x) = 2x^2 + x - 1$, $f^0(x) = x$ and $f^{n + 1}(x) = f (f^n(x))$ for all real $x$ and $n \ge 0$ integer .
(a) Determine the number of real distinct solutions of the equation of $f^3(x) = x$.
(b) Determine, for each integer $n \ge 0$, the number of real distinct solutions of the equation $f^n(x) = 0$.
2022 AMC 8 -, 7
When the World Wide Web first became popular in the $1990$s, download speeds reached a maximum of about $56$ kilobits per second. Approximately how many minutes would the download of a $4.2$-megabyte song have taken at that speed? (Note that there are $8000$ kilobits in a megabyte.)
$\textbf{(A)} ~0.6\qquad\textbf{(B)} ~10\qquad\textbf{(C)} ~1800\qquad\textbf{(D)} ~7200\qquad\textbf{(E)} ~36000\qquad$
VI Soros Olympiad 1999 - 2000 (Russia), 11.4
For prime numbers $p$ and $q$, natural numbers $n$, $k$, $r$, the equality $p^{2k}+q^{2n}=r^2$ holds. Prove that the number $r$ is prime.
MMPC Part II 1958 - 95, 1967
[b]p1.[/b] Consider the system of simultaneous equations
$$(x+y)(x+z)=a^2$$
$$(x+y)(y+z)=b^2$$
$$(x+z)(y+z)=c^2$$
, where $abc \ne 0$. Find all solutions $(x,y,z)$ in terms of $a$,$b$, and $c$.
[b]p2.[/b] Shown in the figure is a triangle $PQR$ upon whose sides squares of areas $13$, $25$, and $36$ sq. units have been constructed. Find the area of the hexagon $ABCDEF$ .
[img]https://cdn.artofproblemsolving.com/attachments/b/6/ab80f528a2691b07430d407ff19b60082c51a1.png[/img]
[b]p3.[/b] Suppose $p,q$, and $r$ are positive integers no two of which have a common factor larger than $1$. Suppose $P,Q$, and $R$ are positive integers such that $\frac{P}{p}+\frac{Q}{q}+\frac{R}{r}$ is an integer. Prove that each of $P/p$, $Q/q$, and $R/r$ is an integer.
[b]p4.[/b] An isosceles tetrahedron is a tetrahedron in which opposite edges are congruent. Prove that all face angles of an isosceles tetrahedron are acute angles.
[img]https://cdn.artofproblemsolving.com/attachments/7/7/62c6544b7c3651bba8a9d210cd0535e82a65bd.png[/img]
[b]p5.[/b] Suppose that $p_1$, $p_2$, $p_3$ and $p_4$ are the centers of four non-overlapping circles of radius $1$ in a plane and that, $p$ is any point in that plane. Prove that $$\overline{p_1p}^2+\overline{p_2p}^2+\overline{p_3p}^2+\overline{p_4p}^2 \ge 6.$$
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2018 Purple Comet Problems, 17
Let $a, b, c$, and $d$ be real numbers such that $a^2 + b^2 + c^2 + d^2 = 3a + 8b + 24c + 37d = 2018$. Evaluate $3b + 8c + 24d + 37a$.
2008 Greece Team Selection Test, 4
Given is the equation $x^2+y^2-axy+2=0$ where $a$ is a positive integral parameter.
$i.$Show that,for $a\neq 4$ there exist no pairs $(x,y)$ of positive integers satisfying the equation.
$ii.$ Show that,for $a=4$ there exist infinite pairs $(x,y)$ of positive integers satisfying the equation,and determine those pairs.
LMT Team Rounds 2021+, 1
Kevin writes the multiples of three from $1$ to $100$ on the whiteboard. How many digits does he write?
2003 Putnam, 2
Let $a_1, a_2, \cdots , a_n$ and $b_1, b_2,\cdots, b_n$ be nonnegative real numbers. Show that \[(a_1a_2 \cdots a_n)^{1/n}+ (b_1b_2 \cdots b_n)^{1/n} \le ((a_1 + b_1)(a_2 + b_2) \cdots (a_n + b_n))^{1/n}\]
2021 AMC 10 Fall, 16
Five balls are arranged around a circle. Chris chooses two adjacent balls at random and interchanges them. Then Silva does the same, with her choice of adjacent balls to interchange being independent of Chris's. What is the expected number of balls that occupy their original positions after these two successive transpositions?
$(\textbf{A})\: 1.6\qquad(\textbf{B}) \: 1.8\qquad(\textbf{C}) \: 2.0\qquad(\textbf{D}) \: 2.2\qquad(\textbf{E}) \: 2.4$
2019 New Zealand MO, 8
Suppose that $x_1, x_2, x_3, . . . x_n$ are real numbers between $0$ and $ 1$ with sum $s$. Prove that $$\prod_{i=1}^{n} \frac{x_i}{s + 1 - x_i} + \prod_{i=1}^{n} (1 - x_i) \le 1.$$
1988 AMC 8, 20
The glass gauge on a cylindrical coffee maker shows that there are 45 cups left when the coffee maker is $36\%$ full. How many cups of coffee does it hold when it is full?
[asy]
draw((5,0)..(0,-1.3)..(-5,0));
draw((5,0)--(5,10)); draw((-5,0)--(-5,10));
draw(ellipse((0,10),5,1.3));
draw(circle((.3,1.3),.4));
draw((-.1,1.7)--(-.1,7.9)--(.7,7.9)--(.7,1.7)--cycle);
fill((-.1,1.7)--(-.1,4)--(.7,4)--(.7,1.7)--cycle,black);
draw((-2,11.3)--(2,11.3)..(2.6,11.9)..(2,12.2)--(-2,12.2)..(-2.6,11.9)..cycle);[/asy]
$ \text{(A)}\ 80\qquad\text{(B)}\ 100\qquad\text{(C)}\ 125\qquad\text{(D)}\ 130\qquad\text{(E)}\ 262 $
2018 Estonia Team Selection Test, 1
There are distinct points $O, A, B, K_1, . . . , K_n, L_1, . . . , L_n$ on a plane such that no three points are collinear. The open line segments $K_1L_1, . . . , K_nL_n$ are coloured red, other points on the plane are left uncoloured. An allowed path from point $O$ to point $X$ is a polygonal chain with first and last vertices at points $O$ and $X$, containing no red points. For example, for $n = 1$, and $K_1 = (-1, 0)$, $L_1 = (1, 0)$, $O = (0,-1)$, and $X = (0,1)$, $OK_1X$ and $OL_1X$ are examples of allowed paths from $O$ to $X$, there are no shorter allowed paths. Find the least positive integer n such that it is possible that the first vertex that is not $O$ on any shortest possible allowed path from $O$ to $A$ is closer to $B$ than to $A$, and the first vertex that is not $O$ on any shortest possible allowed path from $O$ to $B$ is closer to $A$ than to $B$.