Found problems: 1415
2005 Iran Team Selection Test, 3
Suppose $S= \{1,2,\dots,n\}$ and $n \geq 3$. There is $f:S^k \longmapsto S$ that if $a,b \in S^k$ and $a$ and $b$ differ in all of elements then $f(a) \neq f(b)$. Prove that $f$ is a function of one of its elements.
1994 Tuymaada Olympiad, 8
Prove that in space there is a sphere containing exactly $1994$ points with integer coordinates.
2008 Polish MO Finals, 4
Each point of a plane with both coordinates being integers has been colored black or white. Show that there exists an infinite subset of colored points, whose points are in the same color, having a center of symmetry.
[EDIT: added condition about being infinite - now it makes sense]
1993 Greece National Olympiad, 2
During a recent campaign for office, a candidate made a tour of a country which we assume lies in a plane. On the first day of the tour he went east, on the second day he went north, on the third day west, on the fourth day south, on the fifth day east, etc. If the candidate went $n^2/2$ miles on the $n^{\text{th}}$ day of this tour, how many miles was he from his starting point at the end of the $40^{\text{th}}$ day?
2023 Tuymaada Olympiad, 5
A small ship sails on an infinite coordinate sea. At the moment $t$ the ship is at the point with coordinates $(f(t), g(t))$, where $f$ and $g$ are two polynomials of third degree. Yesterday at $14:00$ the ship was at the same point as at $13:00$, and at $20:00$, it was at the same point as at $19:00$. Prove that the ship sails along a straight line.
2007 Estonia National Olympiad, 3
Does there exist an equilateral triangle
(a) on a plane; (b) in a 3-dimensional space;
such that all its three vertices have integral coordinates?
2010 AIME Problems, 6
Let $ P(x)$ be a quadratic polynomial with real coefficients satisfying \[x^2 \minus{} 2x \plus{} 2 \le P(x) \le 2x^2 \minus{} 4x \plus{} 3\] for all real numbers $ x$, and suppose $ P(11) \equal{} 181$. Find $ P(16)$.
1990 IMO Longlists, 3
In coordinate plane, we call a point $(x, y)$ "lattice point" if both $x$ and $y$ are integers. Knowing that the vertices of triangle $ABC$ are all lattice points, and there exists exactly one lattice point interior to triangle $ABC$ (there might exist lattice points on the sides of $ABC$). Prove that the area of triangle $ABC$ is no larger than $\frac 92.$
2009 AMC 12/AHSME, 22
Parallelogram $ ABCD$ has area $ 1,\!000,\!000$. Vertex $ A$ is at $ (0,0)$ and all other vertices are in the first quadrant. Vertices $ B$ and $ D$ are lattice points on the lines $ y\equal{}x$ and $ y\equal{}kx$ for some integer $ k>1$, respectively. How many such parallelograms are there?
$ \textbf{(A)}\ 49\qquad
\textbf{(B)}\ 720\qquad
\textbf{(C)}\ 784\qquad
\textbf{(D)}\ 2009\qquad
\textbf{(E)}\ 2048$
2013 NIMO Problems, 4
Let $a,b,c$ be the answers to problems $4$, $5$, and $6$, respectively. In $\triangle ABC$, the measures of $\angle A$, $\angle B$, and $\angle C$ are $a$, $b$, $c$ in degrees, respectively. Let $D$ and $E$ be points on segments $AB$ and $AC$ with $\frac{AD}{BD} = \frac{AE}{CE} = 2013$. A point $P$ is selected in the interior of $\triangle ADE$, with barycentric coordinates $(x,y,z)$ with respect to $\triangle ABC$ (here $x+y+z=1$). Lines $BP$ and $CP$ meet line $DE$ at $B_1$ and $C_1$, respectively. Suppose that the radical axis of the circumcircles of $\triangle PDC_1$ and $\triangle PEB_1$ pass through point $A$. Find $100x$.
[i]Proposed by Evan Chen[/i]
2007 Today's Calculation Of Integral, 193
For $a>0$, let $l$ be the line created by rotating the tangent line to parabola $y=x^{2}$, which is tangent at point $A(a,a^{2})$, around $A$ by $-\frac{\pi}{6}$.
Let $B$ be the other intersection of $l$ and $y=x^{2}$. Also, let $C$ be $(a,0)$ and let $O$ be the origin.
(1) Find the equation of $l$.
(2) Let $S(a)$ be the area of the region bounded by $OC$, $CA$ and $y=x^{2}$. Let $T(a)$ be the area of the region bounded by $AB$ and $y=x^{2}$. Find $\lim_{a \to \infty}\frac{T(a)}{S(a)}$.
2013 Today's Calculation Of Integral, 861
Answer the questions as below.
(1) Find the local minimum of $y=x(1-x^2)e^{x^2}.$
(2) Find the total area of the part bounded the graph of the function in (1) and the $x$-axis.
2004 AMC 12/AHSME, 13
Let $ S$ be the set of points $ (a,b)$ in the coordinate plane, where each of $ a$ and $ b$ may be $ \minus{} 1$, $ 0$, or $ 1$. How many distinct lines pass through at least two members of $ S$?
$ \textbf{(A)}\ 8 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 27\qquad \textbf{(E)}\ 36$
1987 AMC 12/AHSME, 25
$ABC$ is a triangle: $A=(0,0)$, $B=(36,15)$ and both the coordinates of $C$ are integers. What is the minimum area $\triangle ABC$ can have?
$ \textbf{(A)}\ \frac{1}{2} \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ \frac{3}{2} \qquad\textbf{(D)}\ \frac{13}{2} \qquad\textbf{(E)}\ \text{there is no minimum} $
2012 AIME Problems, 11
A frog begins at $P_0 = (0,0)$ and makes a sequence of jumps according to the following rule: from $P_n=(x_n,y_n)$, the frog jumps to $P_{n+1}$, which may be any of the points $(x_n+7, y_n+2)$, $(x_n+2,y_n+7)$, $(x_n-5, y_n-10)$, or $(x_n-10,y_n-5)$. There are $M$ points $(x,y)$ with $|x|+|y| \le 100$ that can be reached by a sequence of such jumps. Find the remainder when $M$ is divided by $1000$.
1987 IMO Longlists, 10
In a Cartesian coordinate system, the circle $C_1$ has center $O_1(-2, 0)$ and radius $3$. Denote the point $(1, 0)$ by $A$ and the origin by $O$.Prove that there is a constant $c > 0$ such that for every $X$ that is exterior to $C1$,
\[OX- 1 \geq c \min\{AX,AX^2\}.\]
Find the largest possible $c.$
IV Soros Olympiad 1997 - 98 (Russia), 11.3
Draw on the coordinate plane the set of points whose coordinates satisfy the equation $$\sin x \cos^2 y +\sin y \cos^2 x =0$$
2014 Saudi Arabia IMO TST, 2
Define a [i]domino[/i] to be an ordered pair of [i]distinct[/i] positive integers. A [i]proper sequence[/i] of dominoes is a list of distinct dominoes in which the first coordinate of each pair after the first equals the second coordinate of the immediately preceding pair, and in which $(i, j)$ and $(j, i)$ do not [i]both[/i] appear for any $i$ and $j$. Let $D_n$ be the set of all dominoes whose coordinates are no larger than $n$. Find the length of the longest proper sequence of dominoes that can be formed using the dominoes of $D_n$.
2010 Turkey Team Selection Test, 2
For an interior point $D$ of a triangle $ABC,$ let $\Gamma_D$ denote the circle passing through the points $A, \: E, \: D, \: F$ if these points are concyclic where $BD \cap AC=\{E\}$ and $CD \cap AB=\{F\}.$ Show that all circles $\Gamma_D$ pass through a second common point different from $A$ as $D$ varies.
2016 ASMT, T3
An ellipse lies in the $xy$-plane and is tangent to both the $x$-axis and $y$-axis. Given that one of the foci is at $(9, 12)$, compute the minimum possible distance between the two foci.
2012 USAJMO, 4
Let $\alpha$ be an irrational number with $0<\alpha < 1$, and draw a circle in the plane whose circumference has length $1$. Given any integer $n\ge 3$, define a sequence of points $P_1, P_2, \ldots , P_n$ as follows. First select any point $P_1$ on the circle, and for $2\le k\le n$ define $P_k$ as the point on the circle for which the length of arc $P_{k-1}P_k$ is $\alpha$, when travelling counterclockwise around the circle from $P_{k-1}$ to $P_k$. Suppose that $P_a$ and $P_b$ are the nearest adjacent points on either side of $P_n$. Prove that $a+b\le n$.
2003 Romania National Olympiad, 1
Find the locus of the points $ M $ that are situated on the plane where a rhombus $ ABCD $ lies, and satisfy:
$$ MA\cdot MC+MB\cdot MD=AB^2 $$
[i]Ovidiu Pop[/i]
2008 Tuymaada Olympiad, 6
Let $ ABCD$ be an isosceles trapezoid with $ AD \parallel BC$. Its diagonals $ AC$ and $ BD$ intersect at point $ M$. Points $ X$ and $ Y$ on the segment $ AB$ are such that $ AX \equal{} AM$, $ BY \equal{} BM$. Let $ Z$ be the midpoint of $ XY$ and $ N$ is the point of intersection of the segments $ XD$ and $ YC$. Prove that the line $ ZN$ is parallel to the bases of the trapezoid.
[i]Author: A. Akopyan, A. Myakishev[/i]
1999 National High School Mathematics League, 14
Given $A(-2,2)$, and $B$ is a moving point on ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$. $F$ is the left focal point of the ellipse, find the coordinate of $B$ when $|AB|+\frac{5}{3}|BF|$ takes its minumum value.
2004 AIME Problems, 4
A square has sides of length $2$. Set $S$ is the set of all line segments that have length $2$ and whose endpoints are on adjacent sides of the square. The midpoints of the line segments in set $S$ enclose a region whose area to the nearest hundredth is $k$. Find $100k$.