Found problems: 85335
2022 MIG, 7
Alice, Bob, and Charlie are each thinking of a number. Alice's number differs from Bob's number by $2$. Bob's number differs from Charlie's number by $6$. Charlie's number differs from Alice's number by $N$. What is the sum of all possible values for $N$?
$\textbf{(A) }4\qquad\textbf{(B) }6\qquad\textbf{(C) }7\qquad\textbf{(D) }12\qquad\textbf{(E) }14$
1994 Czech And Slovak Olympiad IIIA, 3
A convex $1994$-gon $M$ is given in the plane. A closed polygonal line consists of $997$ of its diagonals. Every vertex is adjacent to exactly one diagonal. Each diagonal divides $M$ into two sides, and the smaller of the numbers of edges on the two sides of $M$ is defined to be the length of the diagonal. Is it posible to have
(a) $991$ diagonals of length $3$ and $6$ of length $2$?
(b) $985$ diagonals of length $6, 4$ of length $8$, and $8$ of length $3$?
2015 All-Russian Olympiad, 5
An immortal flea jumps on whole points of the number line, beginning with $0$. The length of the first jump is $3$, the second $5$, the third $9$, and so on. The length of $k^{\text{th}}$ jump is equal to $2^k + 1$. The flea decides whether to jump left or right on its own. Is it possible that sooner or later the flee will have been on every natural point, perhaps having visited some of the points more than once?
2017 Harvard-MIT Mathematics Tournament, 10
[b]D[/b]enote $\phi=\frac{\sqrt{5}+1}{2}$ and consider the set of all finite binary strings without leading zeroes. Each string $S$ has a “base-$\phi$” value $p(S)$. For example, $p(1101)=\phi^3+\phi^2+1$. For any positive integer n, let $f(n)$ be the number of such strings S that satisfy $p(S) =\frac{\phi^{48n}-1}{\phi^{48}-1}$. The sequence of fractions $\frac{f(n+1)}{f(n)}$ approaches a real number $c$ as $n$ goes to infinity. Determine the value of $c$.
1991 Greece National Olympiad, 2
Given two circles $(C_1)$ and $(C_2)$ with centers $\displaystyle{O_1}$ and $O_2$ respectively, intersecting at points $A$ and $B$. Let $AC$ και $AD$ be the diameters of $(C_1)$ and $(C_2)$ respectively . Tangent line of circle $(C_1)$ at point $A$ intersects $(C_2)$ at point $M$ and tangent line of circle $(C_2)$ at point A intersects $(C_1)$ at point $N$. Let $P$ be a point on line $AB$ such that $AB=BP$. Prove that:
a) Points $B,C,D$ are collinear.
b) Quadrilateral $AMPN$ is cyclic.
2019 Baltic Way, 11
Let $ABC$ be a triangle with $AB = AC$. Let $M$ be the midpoint of $BC$. Let the circles with diameters $AC$ and $BM$ intersect at points $M$ and $P$. Let $MP$ intersect $AB$ at $Q$. Let $R$ be a point on $AP$ such that $QR \parallel BP$. Prove that $CP$ bisects $\angle RCB$.
2014 Peru IMO TST, 3
Let $ABC$ be an acuteangled triangle with $AB> BC$ inscribed in a circle. The perpendicular bisector of the side $AC$ cuts arc $AC,$ containing $B,$ in $Q.$ Let $M$ be a point on the segment $AB$ such that $AM = MB + BC.$ Prove that the circumcircle of the triangle $BMC$ cuts $BQ$ in its midpoint.
2008 Iran Team Selection Test, 11
$ k$ is a given natural number. Find all functions $ f: \mathbb{N}\rightarrow\mathbb{N}$ such that for each $ m,n\in\mathbb{N}$ the following holds: \[ f(m)\plus{}f(n)\mid (m\plus{}n)^k\]
2005 Croatia National Olympiad, 4
Let $P$ and $Q$ be points on the sides $BC$ and $CD$ of a convex quadrilateral $ABCD$, respectively, such that $\angle{BAP}=\angle{ DAQ}$. Prove that the triangles $ABP$ and $ADQ$ have equal area if and only if the line joining their orthocenters is perpendicular to $AC.$
2012 Hitotsubashi University Entrance Examination, 3
For constants $a,\ b,\ c,\ d$ consider a process such that the point $(p,\ q)$ is mapped onto the point $(ap+bq,\ cp+dq)$.
Note : $(a,\ b,\ c,\ d)\neq (1,\ 0,\ 0,\ 1)$. Let $k$ be non-zero constant. All points of the parabola $C: y=x^2-x+k$ are mapped onto $C$ by the process.
(1) Find $a,\ b,\ c,\ d$.
(2) Let $A'$ be the image of the point $A$ by the process. Find all values of $k$ and the coordinates of $A$ such that the tangent line of $C$ at $A$ and the tangent line of $C$ at $A'$ formed by the process are perpendicular at the origin.
2007 ITest, 16
How many lattice points lie within or on the border of the circle defined in the $xy$-plane by the equation $x^2+y^2=100$?
$\textbf{(A) }1\hspace{14em}\textbf{(B) }2\hspace{14em}\textbf{(C) }4$
$\textbf{(D) }5\hspace{14em}\textbf{(E) }41\hspace{13.5em}\textbf{(F) }42$
$\textbf{(G) }69\hspace{13.5em}\textbf{(H) }76\hspace{13.4em}\textbf{(I) }130$
$\textbf{(J) }133\hspace{13.3em}\textbf{(K) }233\hspace{12.8em}\textbf{(L) }311$
$\textbf{(M) }317\hspace{12.7em}\textbf{(N) }420\hspace{12.9em}\textbf{(O) }520$
$\textbf{(P) }2007$
1993 Greece National Olympiad, 14
A rectangle that is inscribed in a larger rectangle (with one vertex on each side) is called [i]unstuck[/i] if it is possible to rotate (however slightly) the smaller rectangle about its center within the confines of the larger. Of all the rectangles that can be inscribed unstuck in a 6 by 8 rectangle, the smallest perimeter has the form $\sqrt{N}$, for a positive integer $N$. Find $N$.
2004 South East Mathematical Olympiad, 8
Determine the number of ordered quadruples $(x, y, z, u)$ of integers, such that
\[\dfrac{x-y}{x+y}+\dfrac{y-z}{y+z}+\dfrac{z-u}{z+u}>0 \textrm{ and } 1\le x,y,z,u\le 10.\]
2012 CHMMC Fall, 2
Consider a triangle $ABC$ with points $D$ on $AB$, $E$ on $BC$, and let $F$ be the intersection of $AE$ and $CD$. Suppose $AD = 1$, $DB = 2$,$BE = 1$,$EC = 3$, and $CA = 5$. Find the value of the area of $ECF$ minus the area of $ADF$.
2021 Tuymaada Olympiad, 8
An acute triangle $ABC$ is given, $AC \not= BC$. The altitudes drawn from $A$ and $B$ meet at $H$ and intersect the external bisector of the angle $C$ at $Y$ and $X$ respectively. The external bisector of the angle $AHB$ meets the segments $AX$ and $BY$ at $P$ and $Q$ respectively. If $PX = QY$, prove that $AP + BQ \ge 2CH$.
1995 All-Russian Olympiad, 5
The sequence $a_1, a_2, ...$ of natural numbers satisfies $GCD(a_i, a_j)=GCD(i, j)$ for all $i \neq j$. Prove that $a_i=i$ for all $i$.
PEN J Problems, 15
Determine all positive integers for which $d(n)=\frac{n}{3}$ holds.
2012 ELMO Shortlist, 5
Let $n>2$ be a positive integer and let $p$ be a prime. Suppose that the nonzero integers are colored in $n$ colors. Let $a_1,a_2,\ldots,a_{n}$ be integers such that for all $1\le i\le n$, $p^i\nmid a_i$ and $p^{i-1}\mid a_i$. In terms of $n$, $p$, and $\{a_i\}_{i=1}^{n}$, determine if there must exist integers $x_1,x_2,\ldots,x_{n}$ of the same color such that $a_1x_1+a_2x_2+\cdots+a_{n}x_{n}=0$.
[i]Ravi Jagadeesan.[/i]
2015 Costa Rica - Final Round, 6
Let $\vartriangle ABC$ be a triangle with circumcenter $O$. Let $ P$ and $Q$ be internal points on the sides $AB$ and $AC$ respectively such that $\angle POB = \angle ABC$ and $\angle QOC = \angle ACB$. Show that the reflection of line $BC$ over line $PQ$ is tangent to the circumcircle of triangle $\vartriangle APQ$.
2017 China Team Selection Test, 2
Let $x>1$ ,$n$ be positive integer. Prove that$$\sum_{k=1}^{n}\frac{\{kx \}}{[kx]}<\sum_{k=1}^{n}\frac{1}{2k-1}$$
Where $[kx ]$ be the integer part of $kx$ ,$\{kx \}$ be the decimal part of $kx$.
2022 New Zealand MO, 6
Let a positive integer $n$ be given. Determine, in terms of $n$, the least positive integer $k$ such that among any $k$ positive integers, it is always possible to select a positive even number of them having sum divisible by $n$.
1955 Putnam, A5
If a parabola is given in the plane, find a geometric construction (ruler and compass) for the focus.
Russian TST 2014, P2
Let $p,q$ and $s{}$ be prime numbers such that $2^sq =p^y-1$ where $y > 1.$ Find all possible values of $p.$
2023 Oral Moscow Geometry Olympiad, 5
Altitudes $BB_1$ and $CC_1$ of acute triangle $ABC$ intersect at $H$, and $\angle A = 60^{o}$, $AB < AC$. The median $AM$ intersects the circumcircle of $ABC$ at point $K$; $L$ is the midpoint of the arc $BC$ of the circumcircle that does not contain point $A$; lines $B_1C_1$ and $BC$ intersect at point $E$. Prove that $\angle EHL = \angle ABK$.
2012 Israel National Olympiad, 2
In some foreign country, there is a secret object, guarded by seven guards. Each guard has a guarding shift of 7 consecutive hours every day, in fixed hours. There is always at least one guard guarding the secret object at any given time.
Prove that one of the guards can be fired, and there will still be at least one guard guarding at any given time (without changing the schedule of the other guards).