Found problems: 85335
1991 All Soviet Union Mathematical Olympiad, 551
A sequence of positive integers is constructed as follows. If the last digit of $a_n$ is greater than $5$, then $a_{n+1}$ is $9a_n$. If the last digit of $a_n$ is $5$ or less and an has more than one digit, then $a_{n+1}$ is obtained from $a_n$ by deleting the last digit. If $a_n$ has only one digit, which is $5$ or less, then the sequence terminates. Can we choose the first member of the sequence so that it does not terminate?
2011 Grand Duchy of Lithuania, 2
Let $n \ge 2$ be a natural number and suppose that positive numbers $a_0,a_1,...,a_n$ satisfy the equality
$(a_{k-1}+a_{k})(a_{k}+a_{k+1})=a_{k-1}-a_{k+1}$ for each $k =1,2,...,n -1$. Prove that $a_n< \frac{1}{n-1}$
2017 NIMO Problems, 5
In triangle $ABC$, $AB=12$, $BC=17$, and $AC=25$. Distinct points $M$ and $N$ lie on the circumcircle of $ABC$ such that $BM=CM$ and $BN=CN$. If $AM + AN = \tfrac{a\sqrt{b}}{c}$, where $a, b, c$ are positive integers such that $\gcd(a, c) = 1$ and $b$ is not divisible by the square of a prime, compute $100a+10b+c$.
[i]Proposed by Michael Tang[/i]
2020 AMC 8 -, 21
A game board consists of $64$ squares that alternate in color between black and white. The figure below shows square $P$ in the bottom and square $Q$ in the top row. A marker is placed at $P$. A [i]step[/i] consists of moving the marker onto one of the adjoining white squares in the row above. How many $7$-step paths are there from $P$ to $Q$? (The figure shows a sample path.)
[asy]//diagram by SirCalcsALot
size(200); int[] x = {6, 5, 4, 5, 6, 5, 6}; int[] y = {1, 2, 3, 4, 5, 6, 7}; int N = 7; for (int i = 0; i < 8; ++i) { for (int j = 0; j < 8; ++j) { draw((i,j)--(i+1,j)--(i+1,j+1)--(i,j+1)--(i,j)); if ((i+j) % 2 == 0) { filldraw((i,j)--(i+1,j)--(i+1,j+1)--(i,j+1)--(i,j)--cycle,black); } } } for (int i = 0; i < N; ++i) { draw(circle((x[i],y[i])+(0.5,0.5),0.35)); } label("$P$", (5.5, 0.5)); label("$Q$", (6.5, 7.5)); [/asy]
$\textbf{(A)}\ 28 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 33 \qquad \textbf{(E)}\ 35$
2001 Brazil Team Selection Test, Problem 3
For which positive integers $n$ is there a permutation $(x_1,x_2,\ldots,x_n)$ of $1,2,\ldots,n$ such that all the differences $|x_k-k|$, $k = 1,2,\ldots,n$, are distinct?
2010 Princeton University Math Competition, 2
Calculate $\displaystyle{\sum_{n=1}^\infty\left(\lfloor\sqrt[n]{2010}\rfloor-1\right)}$ where $\lfloor x\rfloor$ is the largest integer less than or equal to $x$.
2009 Princeton University Math Competition, 5
Let $x_1,x_2,\ldots, x_{10}$ be non-negative real numbers such that $\frac{x_1}{1}+ \frac{x_2}{2} +\cdots+ \frac{x_{10}}{10}$ $\leq9$. Find the maximum possible value of $\frac{{x_1}^2}{1}+\frac{{x_2}^2}{2}+\cdots+\frac{{x_{10}}^2}{10}$.
2015 Saudi Arabia JBMO TST, 3
A natural number is called $nice$ if it doesn't contain 0 and if we add the product of its digit to the number, we obtain number with the same product of its digits. Prove that there is a nice 2015-digit number.
2015 Saudi Arabia GMO TST, 1
Let be given the sequence $(x_n)$ defined by $x_1 = 1$ and $x_{n+1} = 3x_n + \lfloor x_n \sqrt5 \rfloor$ for all $n = 1,2,3,...,$ where $\lfloor x \rfloor$ denotes the greatest integer that does not exceed $x$. Prove that for any positive integer $n$ we have $$x_nx_{n+2} - x^2_{n+1} = 4^{n-1}$$
Trần Nam Dũng
2017 Morocco TST-, 4
Two circles $ G_1$ and $ G_2$ intersect at two points $ M$ and $ N$. Let $ AB$ be the line tangent to these circles at $ A$ and $ B$, respectively, so that $ M$ lies closer to $ AB$ than $ N$. Let $ CD$ be the line parallel to $ AB$ and passing through the point $ M$, with $ C$ on $ G_1$ and $ D$ on $ G_2$. Lines $ AC$ and $ BD$ meet at $ E$; lines $ AN$ and $ CD$ meet at $ P$; lines $ BN$ and $ CD$ meet at $ Q$. Show that $ EP \equal{} EQ$.
2014 Contests, 1
Prove that for positive reals $a$,$b$,$c$ so that $a+b+c+abc=4$, \[\left (1+\dfrac{a}{b}+ca \right )\left (1+\dfrac{b}{c}+ab \right)\left (1+\dfrac{c}{a}+bc \right) \ge 27\] holds.
2014 Sharygin Geometry Olympiad, 3
An acute angle $A$ and a point $E$ inside it are given. Construct points $B, C$ on the sides of the angle such that $E$ is the center of the Euler circle of triangle $ABC$.
(E. Diomidov)
2011 District Olympiad, 4
Find all the functions $f:[0,1]\rightarrow \mathbb{R}$ for which we have:
\[|x-y|^2\le |f(x)-f(y)|\le |x-y|,\]
for all $x,y\in [0,1]$.
2022 Utah Mathematical Olympiad, 3
Find all sequences $a_1, a_2, a_3, \dots$ of real numbers such that for all positive integers $m,n\ge 1$, we have
\begin{align*}
a_{m+n} &= a_m+a_n - mn \text{ and} \\
a_{mn} &= m^2a_n + n^2a_m + 2a_ma_n. \\
\end{align*}
2022 Irish Math Olympiad, 5
5. Let $\triangle$ABC be a triangle with circumcentre [i]O[/i]. The perpendicular line from [i]O[/i] to [i]BC[/i] intersects line [i]BC[/i] at [i]M[/i] and line [i]AC[/i] at [i]P[/i], and the perpendicular line from [i]O[/i] to [i]AC[/i] intersects line [i]AC[/i] at [i]N[/i] and line [i]BC[/i] at [i]Q[/i]. Let [i]D[/i] be the intersection point of lines [i]PQ[/i] and [i]MN[/i]. construct the parallelogram [i]PCQJ[/i] with [i]PJ[/i] || [i]CQ[/i] and [i]QJ[/i] || [i]CP[/i].
Prove the following:
a) The points [i]A[/i], [i]B[/i], [i]O[/i], [i]P[/i], [i]Q[/i], [i]J[/i] are all on the same circle.
b) line [i]OD[/i] is perpendicular to line [i]CJ[/i].
2019 Auckland Mathematical Olympiad, 2
Prove that among any $43$ positive integers there exist two $a$ and $b$ such that $a^2 - b^2$ is divisible by $100$.
2014 Romania Team Selection Test, 1
Let $\triangle ABC$ be an acute triangle of circumcentre $O$. Let the tangents to the circumcircle of $\triangle ABC$ in points $B$ and $C$ meet at point $P$. The circle of centre $P$ and radius $PB=PC$ meets the internal angle bisector of $\angle BAC$ inside $\triangle ABC$ at point $S$, and $OS \cap BC = D$. The projections of $S$ on $AC$ and $AB$ respectively are $E$ and $F$. Prove that $AD$, $BE$ and $CF$ are concurrent.
[i]Author: Cosmin Pohoata[/i]
2018 India National Olympiad, 4
Find all polynomials with real coefficients $P(x)$ such that $P(x^2+x+1)$ divides $P(x^3-1)$.
2021 Junior Balkan Team Selection Tests - Romania, P1
Let $a,b,c>0$ be real numbers with the property that $a+b+c=1$. Prove that \[\frac{1}{a+bc}+\frac{1}{b+ca}+\frac{1}{c+ab}\geq\frac{7}{1+abc}.\]
2023 MIG, 13
Five cards numbered $1,2,3,4,$ and $5$ are given to Paige, Quincy, Ronald, Selena, and Terrence. Paige, Quincy, and Ronald have the following conversation:
[list=disc]
[*]Paige: My number is between is between Selena's number and Quincy's number.
[*]Quincy: My number is between Ronald's number and Terrence's number.
[*]Ronald: My number is between Paige's number and Quincy's number.
[/list]
Who received the card numbered $3$?
$\textbf{(A) } \text{Paige}\qquad\textbf{(B) } \text{Quincy}\qquad\textbf{(C) } \text{Ronald}\qquad\textbf{(D) } \text{Selena}\qquad\textbf{(E) } \text{Terrence}$
2011 District Olympiad, 1
a) Prove that $\{x+y\}-\{y\}$ can only be equal to $\{x\}$ or $\{x\}-1$ for any $x,y\in \mathbb{R}$.
b) Let $\alpha\in \mathbb{R}\backslash \mathbb{Q}$. Denote $a_n=\{n\alpha\}$ for all $n\in \mathbb{N}^*$ and define the sequence $(x_n)_{n\ge 1}$ by
\[x_n=(a_2-a_1)(a_3-a_2)\cdot \ldots \cdot (a_{n+1}-a_n)\]
Prove that the sequence $(x_n)_{n\ge 1}$ is convergent and find it's limit.
2022 Abelkonkurransen Finale, 3
Nils has an $M \times N$ board where $M$ and $N$ are positive integers, and a tile shaped as shown below. What is the smallest number of squares that Nils must color, so that it is impossible to place the tile on the board without covering a colored square? The tile can be freely rotated and mirrored, but it must completely cover four squares.
[asy]
usepackage("tikz");
label("%
\begin{tikzpicture}
\draw[step=1cm,color=black] (0,0) grid (2,1);
\draw[step=1cm,color=black] (1,1) grid (3,2);
\fill [yellow] (0,0) rectangle (2,1);
\fill [yellow] (1,1) rectangle (3,2);
\draw[step=1cm,color=black] (0,0) grid (2,1);
\draw[step=1cm,color=black] (1,1) grid (3,2);
\end{tikzpicture}
");
[/asy]
2013 Canada National Olympiad, 2
The sequence $a_1, a_2, \dots, a_n$ consists of the numbers $1, 2, \dots, n$ in some order. For which positive integers $n$ is it possible that the $n+1$ numbers $0, a_1, a_1+a_2, a_1+a_2+a_3,\dots, a_1 + a_2 +\cdots + a_n$ all have di fferent remainders when divided by $n + 1$?
2005 District Olympiad, 4
Let $\{a_k\}_{k\geq 1}$ be a sequence of non-negative integers, such that $a_k \geq a_{2k} + a_{2k+1}$, for all $k\geq 1$.
a) Prove that for all positive integers $n\geq 1$ there exist $n$ consecutive terms equal with 0 in the sequence $\{a_k\}_k$;
b) State an example of sequence with the property in the hypothesis which contains an infinite number of non-zero terms.
1996 AMC 12/AHSME, 16
A fair standard six-sided dice is tossed three times. Given that the sum of the first two tosses equal the third, what is the probability that at least one $2$ is tossed?
$\displaystyle \textbf{(A)} \ \frac{1}{6} \qquad \textbf{(B)} \ \frac{91}{216} \qquad \textbf{(C)} \ \frac{1}{2} \qquad \textbf{(D)} \ \frac{8}{15} \qquad \textbf{(E)} \ \frac{7}{12}$