Found problems: 85335
2023 AMC 12/AHSME, 24
Let $K$ be the number of sequences $A_1$, $A_2$, $\dots$, $A_n$ such that $n$ is a positive integer less than or equal to $10$, each $A_i$ is a subset of $\{1, 2, 3, \dots, 10\}$, and $A_{i-1}$ is a subset of $A_i$ for each $i$ between $2$ and $n$, inclusive. For example, $\{\}$, $\{5, 7\}$, $\{2, 5, 7\}$, $\{2, 5, 7\}$, $\{2, 5, 6, 7, 9\}$ is one such sequence, with $n = 5$. What is the remainder when $K$ is divided by $10$?
$\textbf{(A) } 1 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 9$
2015 NIMO Summer Contest, 15
Suppose $x$ and $y$ are real numbers such that \[x^2+xy+y^2=2\qquad\text{and}\qquad x^2-y^2=\sqrt5.\] The sum of all possible distinct values of $|x|$ can be written in the form $\textstyle\sum_{i=1}^n\sqrt{a_i}$, where each of the $a_i$ is a rational number. If $\textstyle\sum_{i=1}^na_i=\frac mn$ where $m$ and $n$ are positive realtively prime integers, what is $100m+n$?
[i] Proposed by David Altizio [/i]
1972 All Soviet Union Mathematical Olympiad, 159
Given a rectangle $ABCD$, points $M$ -- the midpoint of $[AD]$ side, $N$ -- the midpoint of $[BC]$ side. Let us take a point $P$ on the extension of the $[DC]$ segment over the point $D$. Let us denote the intersection point of lines $(PM)$ and $(AC)$ as $Q$. Prove that the $\angle QNM= \angle MNP$
2023 CMIMC Team, 9
A positive integer $N$ is a [i]triple-double[/i] if there exists non-negative integers $a$, $b$, $c$ such that $2^a + 2^b + 2^c = N$. How many three-digit numbers are triple-doubles?
[i]Proposed by Giacomo Rizzo[/i]
2011 Romania National Olympiad, 4
Let be a natural number $ n. $ Prove that there exists a number $ k\in\{ 0,1,2,\ldots n \} $ such that the floor of $ 2^{n+k}\sqrt 2 $ is even.
2017 ELMO Shortlist, 2
Let $ABC$ be a scalene triangle with $\angle A = 60^{\circ}$. Let $E$ and $F$ be the feet of the angle bisectors of $\angle ABC$ and $\angle ACB$, respectively, and let $I$ be the incenter of $\triangle ABC$. Let $P,Q$ be distinct points such that $\triangle PEF$ and $\triangle QEF$ are equilateral. If $O$ is the circumcenter of of $\triangle APQ$, show that $\overline{OI}\perp \overline{BC}$.
[i]Proposed by Vincent Huang
2021 Regional Competition For Advanced Students, 4
Determine all triples $(x, y, z)$ of positive integers satisfying $x | (y + 1)$, $y | (z + 1)$ and $z | (x + 1)$.
(Walther Janous)
2015 Postal Coaching, Problem 5
Let $S$ be a set of in $3-$ space such that each of the points in $S$ has integer coordinates $(x,y,z)$ with $1 \le x,y,z \le n $. Suppose the pairwise distances between these points are all distinct. Prove that
$$|S| \le min \{(n+2)\sqrt{\frac{n}{3}},n\sqrt{6} \}$$
2005 Thailand Mathematical Olympiad, 4
Triangle $\vartriangle ABC$ is inscribed in the circle with diameter $BC$. If $AB = 3$, $AC = 4$, and $O$ is the incenter of $\vartriangle ABC$, then find $BO \cdot OC$.
2013 BMT Spring, 7
Denote by $S(a,b)$ the set of integers $k$ that can be represented as $k=a\cdot m+b\cdot n$, for some non-negative integers $m$ and $n$. So, for example, $S(2,4)=\{0,2,4,6,\ldots\}$. Then, find the sum of all possible positive integer values of $x$ such that $S(18,32)$ is a subset of $S(3,x)$.
2002 Baltic Way, 8
Let $P$ be a set of $n\ge 3$ points in the plane, no three of which are on a line. How many possibilities are there to choose a set $T$ of $\binom{n-1}{2}$ triangles, whose vertices are all in $P$, such that each triangle in $T$ has a side that is not a side of any other triangle in $T$?
2011 National Olympiad First Round, 33
What is the largest volume of a sphere which touches to a unit sphere internally and touches externally to a regular tetrahedron whose corners are over the unit sphere?
$\textbf{(A)}\ \frac13 \qquad\textbf{(B)}\ \frac14 \qquad\textbf{(C)}\ \frac12\left ( 1 - \frac1{\sqrt3} \right ) \qquad\textbf{(D)}\ \frac12\left ( \frac{2\sqrt2}{\sqrt3} - 1 \right ) \qquad\textbf{(E)}\ \text{None}$
1998 Slovenia Team Selection Test, 5
On a line $p$ which does not meet a circle $K$ with center $O$, point $P$ is taken such that $OP \perp p$. Let $X \ne P$ be an arbitrary point on $p$. The tangents from $X$ to $K$ touch it at $A$ and $B$. Denote by $C$ and $D$ the orthogonal projections of $P$ on $AX$ and $BX$ respectively.
(a) Prove that the intersection point $Y$ of $AB$ and $OP$ is independent of the location of $X$.
(b) Lines $CD$ and $OP$ meet at $Z$. Prove that $Z$ is the midpoint of $P$.
2008 German National Olympiad, 1
Find all real numbers $ x$ such that \[ \sqrt{x\plus{}1}\plus{}\sqrt{x\plus{}3} \equal{} \sqrt{2x\minus{}1}\plus{}\sqrt{2x\plus{}1}.\]
2012 NIMO Problems, 3
The expression $\circ \ 1\ \circ \ 2 \ \circ 3 \ \circ \dots \circ \ 2012$ is written on a blackboard. Catherine places a $+$ sign or a $-$ sign into each blank. She then evaluates the expression, and finds the remainder when it is divided by 2012. How many possible values are there for this remainder?
[i]Proposed by Aaron Lin[/i]
1956 AMC 12/AHSME, 48
If $ p$ is a positive integer, then $ \frac {3p \plus{} 25}{2p \minus{} 5}$ can be a positive integer, if and only if $ p$ is:
$ \textbf{(A)}\ \text{at least }3 \qquad\textbf{(B)}\ \text{at least }3\text{ and no more than }35 \qquad\textbf{(C)}\ \text{no more than }35$
$ \textbf{(D)}\ \text{equal to }35 \qquad\textbf{(E)}\ \text{equal to }3\text{ or }35$
2006 Silk Road, 1
Found all functions $f: \mathbb{R} \to \mathbb{R}$, such that for any $x,y \in \mathbb{R}$,
\[f(x^2+xy+f(y))=f^2(x)+xf(y)+y.\]
2010 Purple Comet Problems, 1
If $125 + n + 135 + 2n + 145 = 900,$ find $n.$
2023 Taiwan TST Round 1, 5
Find all $f:\mathbb{N}\to\mathbb{N}$ satisfying that for all $m,n\in\mathbb{N}$, the nonnegative integer $|f(m+n)-f(m)|$ is a divisor of $f(n)$.
[i]
Proposed by usjl[/i]
1949 Moscow Mathematical Olympiad, 160
Prove that for any triangle the circumscribed circle divides the line segment connecting the center of its inscribed circle with the center of one of the exscribed circles in halves.
1985 IMO Longlists, 3
A function f has the following property: If $k > 1, j > 1$, and $\gcd(k, j) = m$, then $f(kj) = f(m) (f\left(\frac km\right) + f\left(\frac jm\right))$. What values can $f(1984)$ and $f(1985)$ take?
PEN S Problems, 28
Let $A$ be the set of the $16$ first positive integers. Find the least positive integer $k$ satisfying the condition: In every $k$-subset of $A$, there exist two distinct $a, b \in A$ such that $a^2 + b^2$ is prime.
2018 Hanoi Open Mathematics Competitions, 4
A pyramid of non-negative integers is constructed as follows
(a) The first row consists of only $0$,
(b) The second row consists of $1$ and $1$,
(c) The $n^{th}$ (for $n > 2$) is an array of $n$ integers among which the left most and right most elements are equal to $n - 1$ and the interior numbers are equal to the sum of two adjacent numbers from the $(n - 1)^{th}$ row (see Figure).
Let $S_n$ be the sum of numbers in row $n^{th}$. Determine the remainder when dividing $S_{2018}$ by $2018$:
A. $2$ B. $4$ C. $6$ D. $11$ E. $17$
2006 AMC 10, 21
How many four-digit positive integers have at least one digit that is a 2 or a 3?
$ \textbf{(A) } 2439 \qquad \textbf{(B) } 4096 \qquad \textbf{(C) } 4903 \qquad \textbf{(D) } 4904 \qquad \textbf{(E) } 5416$
2015 IFYM, Sozopol, 8
Let $\mathbb{N} = \{1, 2, 3, \ldots\}$ be the set of positive integers. Find all functions $f$, defined on $\mathbb{N}$ and taking values in $\mathbb{N}$, such that $(n-1)^2< f(n)f(f(n)) < n^2+n$ for every positive integer $n$.