Found problems: 85335
2011 USA TSTST, 4
Acute triangle $ABC$ is inscribed in circle $\omega$. Let $H$ and $O$ denote its orthocenter and circumcenter, respectively. Let $M$ and $N$ be the midpoints of sides $AB$ and $AC$, respectively. Rays $MH$ and $NH$ meet $\omega$ at $P$ and $Q$, respectively. Lines $MN$ and $PQ$ meet at $R$. Prove that $OA\perp RA$.
2019 USA TSTST, 3
On an infinite square grid we place finitely many [i]cars[/i], which each occupy a single cell and face in one of the four cardinal directions. Cars may never occupy the same cell. It is given that the cell immediately in front of each car is empty, and moreover no two cars face towards each other (no right-facing car is to the left of a left-facing car within a row, etc.). In a [i]move[/i], one chooses a car and shifts it one cell forward to a vacant cell. Prove that there exists an infinite sequence of valid moves using each car infinitely many times.
[i]Nikolai Beluhov[/i]
2004 IMO, 2
Find all polynomials $f$ with real coefficients such that for all reals $a,b,c$ such that $ab+bc+ca = 0$ we have the following relations
\[ f(a-b) + f(b-c) + f(c-a) = 2f(a+b+c). \]
1962 German National Olympiad, 2
Let $u, v$ and$ w$ be any positive numbers smaller than $1$. Prove that among the numbers $u(1 -v)$, $v(1 -w)$, $w(1 - u)$ there is always at least one value not greater than $\frac14$ .
2001 Iran MO (3rd Round), 2
Does there exist a sequence $ \{b_{i}\}_{i=1}^\infty$ of positive real numbers such that for each natural $ m$: \[ b_{m}+b_{2m}+b_{3m}+\dots=\frac1m\]
2021 Baltic Way, 20
Let $n\ge 2$ be an integer. Given numbers $a_1, a_2, \ldots, a_n \in \{1,2,3,\ldots,2n\}$ such that $\operatorname{lcm}(a_i,a_j)>2n$ for all $1\le i<j\le n$, prove that
$$a_1a_2\ldots a_n \mid (n+1)(n+2)\ldots (2n-1)(2n).$$
2020 Novosibirsk Oral Olympiad in Geometry, 4
The altitudes $AN$ and $BM$ are drawn in triangle $ABC$. Prove that the perpendicular bisector to the segment $NM$ divides the segment $AB$ in half.
2022 Taiwan Mathematics Olympiad, 3
Find all functions $f,g:\mathbb{R}^2\to\mathbb{R}$ satisfying that
\[|f(a,b)-f(c,d)|+|g(a,b)-g(c,d)|=|a-c|+|b-d|\]
for all real numbers $a,b,c,d$.
[i]Proposed by usjl[/i]
2017 Moscow Mathematical Olympiad, 5
$8$ points lie on the faces of unit cube and form another cube. What can be length of edge of this cube?
1996 AMC 8, 20
Suppose there is a special key on a calculator that replaces the number $x$ currently displayed with the number given by the formula $\frac{1}{1-x}$. For example, if the calculator is displaying $2$ and the special key is pressed, then the calculator will display $-1$ since $\frac{1}{1-2}=-1$. Now suppose that the calculator is displaying $5$. After the special key is pressed 100 times in a row, the calculator will display
$\text{(A)}\ -0.25 \qquad \text{(B)}\ 0 \qquad \text{(C)}\ 0.8 \qquad \text{(D)}\ 1.25 \qquad \text{(E)}\ 5$
2014 PUMaC Geometry B, 1
Triangle $ABC$ has lengths $AB=20$, $AC=14$, $BC=22$. The median from $B$ intersects $AC$ at $M$ and the angle bisector from $C$ intersects $AB$ at $N$ and the median from $B$ at $P$. Let $\dfrac pq=\dfrac{[AMPN]}{[ABC]}$ for positive integers $p$, $q$ coprime. Note that $[ABC]$ denotes the area of triangle $ABC$. Find $p+q$.
2021 Romania Team Selection Test, 1
Consider a fixed triangle $ABC$ such that $AB=AC.$ Let $M$ be the midpoint of $BC.$ Let $P$ be a variable point inside $\triangle ABC,$ such that $\angle PBC=\angle PCA.$ Prove that the sum of the measures of $\angle BPM$ and $\angle APC$ is constant.
2014 Contests, Problem 1
Consider a square of side length $12$ centimeters. Irina draws another square that has $8$ centimeters more of perimeter than the original square. What is the area of the square drawn by Irina?
2011 AIME Problems, 3
Let $L$ be the line with slope $\tfrac{5}{12}$ that contains the point $A=(24,-1)$, and let $M$ be the line perpendicular to line $L$ that contains the point $B=(5,6)$. The original coordinate axes are erased, and line $L$ is made the $x$-axis, and line $M$ the $y$-axis. In the new coordinate system, point $A$ is on the positive $x$-axis, and point $B$ is on the positive $y$-axis. The point $P$ with coordinates $(-14,27)$ in the original system has coordinates $(\alpha,\beta)$ in the new coordinate system. Find $\alpha+\beta$.
2009 Turkey Team Selection Test, 1
Find all $ f: Q^ \plus{} \to\ Z$ functions that satisfy $ f \left(\frac {1}{x} \right) \equal{} f(x)$ and $ (x \plus{} 1)f(x \minus{} 1) \equal{} xf(x)$ for all rational numbers that are bigger than 1.
2010 IMO Shortlist, 8
Given six positive numbers $a,b,c,d,e,f$ such that $a < b < c < d < e < f.$ Let $a+c+e=S$ and $b+d+f=T.$ Prove that
\[2ST > \sqrt{3(S+T)\left(S(bd + bf + df) + T(ac + ae + ce) \right)}.\]
[i]Proposed by Sung Yun Kim, South Korea[/i]
2022 Baltic Way, 11
Let $ABC$ be a triangle with circumcircle $\Gamma$ and circumcentre $O$. The circle with centre on the line $AB$ and passing through the points $A$ and $O$ intersects $\Gamma$ again in $D$. Similarly, the circle with centre on the line $AC$ and passing through the points $A$ and $O$ intersects $\Gamma$ again in $E$. Prove that $BD$ is parallel with $CE$.
2005 Brazil National Olympiad, 5
Let $ABC$ be a triangle with all angles $\leq 120^{\circ}$. Let $F$ be the Fermat point of triangle $ABC$, that is, the interior point of $ABC$ such that $\angle AFB = \angle BFC = \angle CFA = 120^\circ$. For each one of the three triangles $BFC$, $CFA$ and $AFB$, draw its Euler line - that is, the line connecting its circumcenter and its centroid.
Prove that these three Euler lines pass through one common point.
[i]Remark.[/i] The Fermat point $F$ is also known as the [b]first Fermat point[/b] or the [b]first Toricelli point[/b] of triangle $ABC$.
[i]Floor van Lamoen[/i]
2017 Iran Team Selection Test, 6
In the unit squares of a transparent $1 \times 100$ tape, numbers $1,2,\cdots,100$ are written in the ascending order.We fold this tape on it's lines with arbitrary order and arbitrary directions until we reach a $1 \times1$ tape with $100$ layers.A permutation of the numbers $1,2,\cdots,100$ can be seen on the tape, from the top to the bottom.
Prove that the number of possible permutations is between $2^{100}$ and $4^{100}$.
([i]e.g.[/i] We can produce all permutations of numbers $1,2,3$ with a $1\times3$ tape)
[i]Proposed by Morteza Saghafian[/i]
2012 AMC 10, 10
How many ordered pairs of positive integers $(M,N)$ staisfy the equation $\frac{M}{6}=\frac{6}{N}$?
$ \textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10 $
2015 BMT Spring, 5
Let $x$ and $y$ be real numbers satisfying the equation $x^2-4x+y^2+3=0$. If the maximum and minimum values of $x^2+y^2$ are $M$ and $m$ respectively, compute the numerical value of $M-m$.
1994 Romania TST for IMO, 2:
Let $n$ be a positive integer. Find the number of polynomials $P(x)$ with coefficients in $\{0, 1, 2, 3\}$ for which $P(2) = n$.
2012 NIMO Summer Contest, 6
When Eva counts, she skips all numbers containing a digit divisible by 3. For example, the first ten numbers she counts are 1, 2, 4, 5, 7, 8, 11, 12, 14, 15. What is the $100^{\text{th}}$ number she counts?
[i]Proposed by Eugene Chen[/i]
2012 Albania National Olympiad, 5
Let $ABC$ be a triangle where $AC\neq BC$. Let $P$ be the foot of the altitude taken from $C$ to $AB$; and let $V$ be the orthocentre, $O$ the circumcentre of $ABC$, and $D$ the point of intersection between the radius $OC$ and the side $AB$. The midpoint of $CD$ is $E$.
a) Prove that the reflection $V'$ of $V$ in $AB$ is on the circumcircle of the triangle $ABC$.
b) In what ratio does the segment $EP$ divide the segment $OV$?
2005 MOP Homework, 7
Let $x_{1,1}$, $x_{2,1}$, ..., $x_{n,1}$, $n \ge 2$, be a sequence of integers and assume that not all $x_{i,1}$ are equal. For $k \ge 2$, if sequence $\{x_{i,k}\}^n_{i=1}$ is defined, we define sequence $\{x_{i,k+1}\}^n_{i=1}$ as \[x_{i,k+1}=\frac{1}{2}(x_{i,k}+x_{i+1,k}),\] for $i=1, 2, ..., n$, (where $x_{n+1,k}=x_{1,k}$). Show that if $n$ is odd then there exist indices $j$ and $k$ such that $x_{j,k}$ is not an integer.