Found problems: 85335
2021 Chile National Olympiad, 1
Consider the sequence of numbers defined by $a_1 = 7$, $a_2 = 7^7$ , $ ...$ , $a_n = 7^{a_{n-1}}$ for $n \ge 2$. Determine the last digit of the decimal representation of $a_{2021}$.
2024 Singapore Junior Maths Olympiad, Q3
Seven triangles of area $7$ lie in a square of area $27$. Prove that among the $7$ triangles there are $2$ that intersect in a region of area not less than $1$.
2013 JBMO Shortlist, 2
$\boxed{\text{A2}}$ Find the maximum value of $|\sqrt{x^2+4x+8}-\sqrt{x^2+8x+17}|$ where $x$ is a real number.
2018 Romanian Master of Mathematics, 4
Let $a,b,c,d$ be positive integers such that $ad \neq bc$ and $gcd(a,b,c,d)=1$. Let $S$ be the set of values attained by $\gcd(an+b,cn+d)$ as $n$ runs through the positive integers. Show that $S$ is the set of all positive divisors of some positive integer.
2019 Saint Petersburg Mathematical Olympiad, 7
Let $\omega$ and $O$ be respectively the circumcircle and the circumcenter of a triangle $ABC$. The line $AO$ intersects $\omega$ second time at $A'$. $M_B$ and $M_C$ are the midpoints of $AC$ and $AB$, respectively. The lines $A'M_B$ and $A'M_C$ intersect $\omega$ secondly at points $B'$ and $C$, and also intersect $BC$ at points $D_B$ and $D_C$, respectively. The circumcircles of $CD_BB'$ and $BD_CC'$ intersect at points $P$ and $Q$.
Prove that $O$, $P$, $Q$ are collinear.
[i] (М. Германсков)[/i]
[hide=Thanks]Thanks to the user Vlados021 for translating the problem.[/hide]
2012 Purple Comet Problems, 20
Square $ABCD$ has side length $68$. Let $E$ be the midpoint of segment $\overline{CD}$, and let $F$ be the point on segment $\overline{AB}$ a distance $17$ from point $A$. Point $G$ is on segment $\overline{EF}$ so that $\overline{EF}$ is perpendicular to segment $\overline{GD}$. The length of segment $\overline{BG}$ can be written as $m\sqrt{n}$ where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.
1990 Irish Math Olympiad, 2
A sequence of primes $a_n$ is defined as follows: $a_1 = 2$, and, for all $n \geq 2$,$
a_n$ is the largest prime divisor of $a_1a_2...a_{n-1} + 1$. Prove that $a_n \neq 5$
for all n.
I'm presuming it must involve proving it's never equal to 0 mod 5, but I don't know what to do. Thanks
2009 Junior Balkan Team Selection Tests - Moldova, 3
Let $ABC$ be a triangle with $\angle BCA=20.$ Let points $D\in(BC), F\in(AC)$ be such that $CD=DF=FB=BA.$ Find $\angle ADF.$
2021 Indonesia TST, G
Let $P$ be a point in the plane of $\triangle ABC$, and $\gamma$ a line passing through $P$. Let $A', B', C'$ be the points where the reflections of lines $PA, PB, PC$ with respect to $\gamma$ intersect lines $BC, AC, AB$ respectively. Prove that $A', B', C'$ are collinear.
2008 Romania National Olympiad, 4
On the sides of triangle $ ABC$ we consider points $ C_1,C_2 \in (AB), B_1,B_2 \in (AC), A_1,A_2 \in (BC)$ such that triangles $ A_1,B_1,C_1$ and $ A_2B_2C_2$ have a common centroid.
Prove that sets $ [A_1,B_1]\cap [A_2B_2], [B_1C_1]\cap[B_2C_2], [C_1A_1]\cap [C_2A_2]$ are not empty.
2020 Korea Junior Math Olympiad, 5
Let $a, b, c, d, e$ be real numbers satisfying the following conditions.
\[a \le b \le c \le d \le e, \quad a+e=1, \quad b+c+d=3, \quad a^2+b^2+c^2+d^2+e^2=14\]Determine the maximum possible value of $ae$.
2020 USMCA, 23
Let $f_n$ be a sequence defined by $f_0=2020$ and
\[f_{n+1} = \frac{f_n + 2020}{2020f_n + 1}\]
for all $n \geq 0$. Determine $f_{2020}$.
2019 Durer Math Competition Finals, 3
For each integer $n$ ($n \ge 2$), let $f(n)$ denote the sum of all positive integers that are at most $n$ and not relatively prime to $n$.
Prove that $f(n+p) \neq f(n)$ for each such $n$ and every prime $p$.
1950 AMC 12/AHSME, 46
In triangle $ABC$, $AB=12$, $AC=7$, and $BC=10$. If sides $AB$ and $AC$ are doubled while $BC$ remains the same, then:
$\textbf{(A)}\ \text{The area is doubled} \qquad\\
\textbf{(B)}\ \text{The altitude is doubled} \qquad\\
\textbf{(C)}\ \text{The area is four times the original area} \qquad\\
\textbf{(D)}\ \text{The median is unchanged} \qquad\\
\textbf{(E)}\ \text{The area of the triangle is 0}$
2022 BMT, Tie 1
Let $ABCDEF GH$ be a unit cube such that $ABCD$ is one face of the cube and $\overline{AE}$, $\overline{BF}$, $\overline{CG}$, and $\overline{DH}$ are all edges of the cube. Points $I, J, K$, and $L$ are the respective midpoints of $\overline{AF}$, $\overline{BG}$, $\overline{CH}$, and $\overline{DE}$. The inscribed circle of $IJKL$ is the largest cross-section of some sphere. Compute the volume of this sphere.
2020 MBMT, 28
Consider the system of equations $$a + 2b + 3c + \ldots + 26z = 2020$$ $$b + 2c + 3d + \ldots + 26a = 2019$$ $$\vdots$$ $$y + 2z + 3a + \ldots + 26x = 1996$$ $$z + 2a + 3b + \ldots + 26y = 1995$$ where each equation is a rearrangement of the first equation with the variables cycling and the coefficients staying in place. Find the value of $$z + 2y + 3x + \dots + 26a.$$
[i]Proposed by Joshua Hsieh[/i]
2024 Mexican Girls' Contest, 1
Let \( x \) be a real number. Determine the solution to the following equation:
\[
\frac{x^2 + 1}{1} + \frac{x^2 + 2}{2} + \frac{x^2 + 3}{3} + \ldots + \frac{x^2 + 2024}{2024} = 2024
\]
1954 Poland - Second Round, 3
Given: point $ A $, line $ p $, and circle $ k $. Construct a triangle $ ABC $ with angles $ A = 60^\circ $, $ B = 90^\circ $, whose vertex $ B $ lies on line $ p $, and vertex $ C $ - on circle $ k $.
1997 All-Russian Olympiad Regional Round, 10.3
Natural numbers $m$ and $n$ are given. Prove that the number $2^n-1$ is divisible by the number $(2^m -1)^2$ if and only if the number $n$ is divisible by the number $m(2^m-1)$.
2014 Greece Team Selection Test, 1
Let $(x_{n}) \ n\geq 1$ be a sequence of real numbers with $x_{1}=1$ satisfying $2x_{n+1}=3x_{n}+\sqrt{5x_{n}^{2}-4}$
a) Prove that the sequence consists only of natural numbers.
b) Check if there are terms of the sequence divisible by $2011$.
2023 Putnam, B4
For a nonnegative integer $n$ and a strictly increasing sequence of real numbers $t_0, t_1, \ldots, t_n$, let $f(t)$ be the corresponding real-valued function defined for $t \geq t_0$ by the following properties:
(a) $f(t)$ is continuous for $t \geq t_0$, and is twice differentiable for all $t>t_0$ other than $t_1, \ldots, t_n$;
(b) $f\left(t_0\right)=1 / 2$;
(c) $\lim _{t \rightarrow t_k^{+}} f^{\prime}(t)=0$ for $0 \leq k \leq n$;
(d) For $0 \leq k \leq n-1$, we have $f^{\prime \prime}(t)=k+1$ when $t_k<t<t_{k+1}$, and $f^{\prime \prime}(t)=n+1$ when $t>t_n$.
Considering all choices of $n$ and $t_0, t_1, \ldots, t_n$ such that $t_k \geq t_{k-1}+1$ for $1 \leq k \leq n$, what is the least possible value of $T$ for which $f\left(t_0+T\right)=2023$?
2000 Harvard-MIT Mathematics Tournament, 19
Define $a*b=\frac{a-b}{1-ab}$. What is $(1*(2*(3*\cdots (n*(n+1))\cdots )))$?
2018 Romanian Masters in Mathematics, 3
Ann and Bob play a game on the edges of an infinite square grid, playing in turns. Ann plays the first move. A move consists of orienting any edge that has not yet been given an orientation. Bob wins if at any point a cycle has been created. Does Bob have a winning strategy?
2016 IFYM, Sozopol, 7
Is the following set of prime numbers $p$ finite or infinite, where each $p$ [b]doesn't[/b] divide the numbers that can be expressed as $n^{2016}+2016^{2016}$ for $n\in \mathbb{N}$, if:
a) $p=4k+3$;
b) $p=4k+1$?
2022 Regional Competition For Advanced Students, 3
Let $ABC$ denote a triangle with $AC\ne BC$. Let $I$ and $U$ denote the incenter and circumcenter of the triangle $ABC$, respectively. The incircle touches $BC$ and $AC$ in the points $D$ and E, respectively. The circumcircles of the triangles $ABC$ and $CDE$ intersect in the two points $C$ and $P$. Prove that the common point $S$ of the lines $CU$ and $P I$ lies on the circumcircle of the triangle $ABC$.
[i](Karl Czakler)[/i]