Found problems: 85335
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).
2014 Stanford Mathematics Tournament, 8
$O$ is a circle with radius $1$. $A$ and $B$ are fixed points on the circle such that $AB =\sqrt2$. Let C be any point on the circle, and let $M$ and $N$ be the midpoints of $AC$ and $BC$, respectively. As $C$ travels around circle $O$, find the area of the locus of points on $MN$.
2021 CMIMC, 2.6 1.2
In convex quadrilateral $ABCD$, $\angle ADC = 90^\circ + \angle BAC$. Given that $AB = BC = 17$, and $CD = 16$, what is the maximum possible area of the quadrilateral?
[i]Proposed by Thomas Lam[/i]
2017 Harvard-MIT Mathematics Tournament, 30
Consider an equilateral triangular grid $G$ with $20$ points on a side, where each row consists of points spaced $1$ unit apart. More specifically, there is a single point in the first row, two points in the second row, ..., and $20$ points in the last row, for a total of $210$ points. Let $S$ be a closed non-self-intersecting polygon which has $210$ vertices, using each point in $G$ exactly once. Find the sum of all possible values of the area of $S$.
2014 239 Open Mathematical Olympiad, 4
The median $CM$ of the triangle $ABC$ is equal to the bisector $BL$, also $\angle BAC=2\angle ACM$. prove that the triangle is right.
2000 Portugal MO, 4
Calculates the sum of all numbers that can be formed using each of the odd digits once, that is, the numbers $13579$, $13597$, ..., $97531$.