Found problems: 85335
2001 All-Russian Olympiad Regional Round, 9.3
In parallelogram $ABCD$, points $M$ and $N$ are selected on sides $AB$ and $BC$ respectively so that $AM = NC$, $Q$ is the intersection point of segments $AN$ and $CM$. Prove that $DQ$ is the bisector of angle $D$.
2018 Bulgaria JBMO TST, 1
In the quadrilateral $ABCD$, we have $\measuredangle BAD = 100^{\circ}$, $\measuredangle BCD = 130^{\circ}$, and $AB=AD=1$ centimeter. Find the length of diagonal $AC$.
2017 QEDMO 15th, 5
For which natural numbers $n$ can the polynomial $f (x) = x^n + x^{n-1} +...+ x + 1$ as write $f (x) = g (h (x))$, where $g$ and $h$ should be real polynomials of degrees greater than $1$?
1962 AMC 12/AHSME, 3
The first three terms of an arithmetic progression are $ x \minus{} 1, x \plus{} 1, 2x \plus{} 3,$ in the order shown. The value of $ x$ is:
$ \textbf{(A)}\ \minus{} 2 \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ \text{undetermined}$
2020 BMT Fall, 4
Alice is standing on the circumference of a large circular room of radius $10$. There is a circular pillar in the center of the room of radius $5$ that blocks Alice’s view. The total area in the room Alice can see can be expressed in the form $\frac{m\pi}{n} +p\sqrt{q}$, where $m$ and $n$ are relatively prime positive integers and $p$ and $q$ are integers such that $q$ is square-free. Compute $m + n + p + q$. (Note that the pillar is not included in the total area of the room.)
[img]https://cdn.artofproblemsolving.com/attachments/1/9/a744291a61df286735d63d8eb09e25d4627852.png[/img]
2019 Hong Kong TST, 1
Determine all sequences $p_1, p_2, \dots $ of prime numbers for which there exists an integer $k$ such that the recurrence relation
\[ p_{n+2} = p_{n+1} + p_n + k \]
holds for all positive integers $n$.
1998 All-Russian Olympiad Regional Round, 9.2
Two circles intersect at points $P$ and $Q$. The straight line intersects these circles at points $A$, $B$, $C$, $D$, as shown in fig. . Prove that $\angle APB = \angle CQD$.
[img]https://cdn.artofproblemsolving.com/attachments/1/a/a581e11be68bbb628db5b5b8e75c7ff6e196c5.png[/img]
2010 Contests, 2
Let $n$ be a positive integer. Find the number of sequences $x_{1},x_{2},\ldots x_{2n-1},x_{2n}$, where $x_{i}\in\{-1,1\}$ for each $i$, satisfying the following condition: for any integer $k$ and $m$ such that $1\le k\le m\le n$ then the following inequality holds \[\left|\sum_{i=2k-1}^{2m}x_{i}\right|\le\ 2\]
1987 Canada National Olympiad, 4
On a large, flat field $n$ people are positioned so that for each person the distances to all the other people are different. Each person holds a water pistol and at a given signal fires and hits the person who is closest. When $n$ is odd show that there is at least one person left dry. Is this always true when $n$ is even?
2014 ASDAN Math Tournament, 16
Compute the number of geometric sequences of length $3$ where each number is a positive integer no larger than $10$.
2021 China Team Selection Test, 6
Proof that there exist constant $\lambda$, so that for any positive integer $m(\ge 2)$, and any lattice triangle $T$ in the Cartesian coordinate plane, if $T$ contains exactly one $m$-lattice point in its interior(not containing boundary), then $T$ has area $\le \lambda m^3$.
PS. lattice triangles are triangles whose vertex are lattice points; $m$-lattice points are lattice points whose both coordinates are divisible by $m$.
2021 Indonesia TST, N
Bamicin is initially at $(20, 20)$ in a cartesian plane. Every minute, if Bamicin is at point $P$, Bamicin can move to a lattice point exactly $37$ units from $P$. Determine all lattice points Bamicin can visit.
1971 IMO Shortlist, 11
The matrix
\[A=\begin{pmatrix} a_{11} & \ldots & a_{1n} \\ \vdots & \ldots & \vdots \\ a_{n1} & \ldots & a_{nn} \end{pmatrix}\]
satisfies the inequality $\sum_{j=1}^n |a_{j1}x_1 + \cdots+ a_{jn}x_n| \leq M$ for each choice of numbers $x_i$ equal to $\pm 1$. Show that
\[|a_{11} + a_{22} + \cdots+ a_{nn}| \leq M.\]
2024 Nepal TST, P2
Let $f: \mathbb{N} \to \mathbb{N}$ be an arbitrary function. Prove that there exist two positive integers $x$ and $y$ which satisfy $f(x+y) \le f(2x+f(y))$.
[i](Proposed by David Anghel, Romania)[/i]
1989 Turkey Team Selection Test, 1
Let $\mathbb{Z}^+$ denote the set of positive integers. Find all functions $f: \mathbb{Z}^+ \times \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ such that
[list=i]
[*] $f(m,m)=m$
[*] $f(m,k) = f(k,m)$
[*] $f(m, m+k) = f(m,k)$[/list] , for each $m,k \in \mathbb{Z}^+$.
2022 Adygea Teachers' Geometry Olympiad, 1
In triangle $ABC$, $\angle A = 60^o$,$ \angle B = 45^o$. On the sides $AC$ and $BC$ points $M$ and $N$ are taken, respectively, so that the straight line $MN$ cuts off a triangle similar to this one. Find the ratio of $MN$ to $AB$ if it is known that $CM : AM = 2:1$.
2023 AMC 12/AHSME, 22
Let $f$ be the unique function defined on the positive integers such that \[\sum_{d\mid n}d\cdot f\left(\frac{n}{d}\right)=1\] for all positive integers $n$, where the sum is taken over all positive divisors of $n$. What is $f(2023)$?
$\textbf{(A)}~-1536\qquad\textbf{(B)}~96\qquad\textbf{(C)}~108\qquad\textbf{(D)}~116\qquad\textbf{(E)}~144$
2022 Stanford Mathematics Tournament, 10
Let $ABCDEFGH$ be a regular octagon with side length $\sqrt{60}$. Let $\mathcal{K}$ denote the locus of all points $K$ such that the circumcircles (possibly degenerate) of triangles $HAK$ and $DCK$ are tangent. Find the area of the region that $\mathcal{K}$ encloses.
2024 Malaysian IMO Training Camp, 2
Let $k>1$. Fix a circle $\omega$ with center $O$ and radius $r$, and fix a point $A$ with $OA=kr$.
Let $AB$, $AC$ be tangents to $\omega$. Choose a variable point $P$ on the minor arc $BC$ in $\omega$. Lines $AB$ and $CP$ intersect at $X$ and lines $AC$ and $BP$ intersect at $Y$. The circles $(BPX)$ and $(CPY)$ meet at another point $Z$.
Prove that the line $PZ$ always passes through a fixed point except for one value of $k>1$, and determine this value.
[i]Proposed by Ivan Chan Kai Chin[/i]
1998 AMC 8, 14
An Annville Junior High School, $30\%$ of the students in the Math Club are in the Science Club, and $80\%$ of the students in the Science Club are in the Math Club. There are $15$ students in the Science Club. How many students are in the Math Club?
$ \text{(A)}\ 12\qquad\text{(B)}\ 15\qquad\text{(C)}\ 30\qquad\text{(D)}\ 36\qquad\text{(E)}\ 40 $
2011 Today's Calculation Of Integral, 750
Let $a_n\ (n\geq 1)$ be the value for which $\int_x^{2x} e^{-t^n}dt\ (x\geq 0)$ is maximal. Find $\lim_{n\to\infty} \ln a_n.$
1982 Swedish Mathematical Competition, 3
Show that there is a point $P$ inside the quadrilateral $ABCD$ such that the triangles $PAB$, $PBC$, $PCD$, $PDA$ have equal area. Show that $P$ must lie on one of the diagonals.
2020-2021 Fall SDPC, 1
In the following grid below, each row and column contains the numbers $1,2,3,4,5$ exactly once. Furthermore, each of the three sections have the same sum. Find, with proof, all possible ways to fill the grid in.
[asy]
unitsize(0.5 cm);
draw((0,0)--(5,0)--(5,5)--(0,5)--(0,0),linewidth(3)); draw((1,0)--(1,1)--(0,1)); draw((2,0)--(2,3)--(0,3)); draw((3,0)--(3,2)--(0,2)); draw((2,5)--(2,4)--(5,4)); draw((3,5)--(3,3)--(5,3)); draw((4,5)--(4,2)--(5,2)); draw((4,0)--(4,1)); draw((1,5)--(1,4));
draw((0,4)--(1,4)--(1,1)--(5,1),linewidth(3)); draw((1,4)--(2,4)--(2,3)--(3,3)--(3,2)--(4,2)--(4,1),linewidth(3));
[/asy]
2014 NIMO Problems, 1
Let $ABC$ be a triangle with $AB=13$, $BC=14$, and $CA=15$. Let $D$ be the point inside triangle $ABC$ with the property that $\overline{BD} \perp \overline{CD}$ and $\overline{AD} \perp \overline{BC}$. Then the length $AD$ can be expressed in the form $m-\sqrt{n}$, where $m$ and $n$ are positive integers. Find $100m+n$.
[i]Proposed by Michael Ren[/i]
2024 Romania Team Selection Tests, P4
Let $m$ and $n$ be positive integers greater than $1$. In each unit square of an $m\times n$ grid lies a coin with its tail side up. A [i]move[/i] consists of the following steps.
[list=1]
[*]select a $2\times 2$ square in the grid;
[*]flip the coins in the top-left and bottom-right unit squares;
[*]flip the coin in either the top-right or bottom-left unit square.
[/list]
Determine all pairs $(m,n)$ for which it is possible that every coin shows head-side up after a finite number of moves.
[i]Thanasin Nampaisarn, Thailand[/i]