Found problems: 85335
1996 Putnam, 6
Let $c\ge 0$ be a real number. Give a complete description with proof of the set of all continuous functions $f: \mathbb{R}\to \mathbb{R}$ such that $f(x)=f(x^2+c)$ for all $x\in \mathbb{R}$.
2018 Estonia Team Selection Test, 5
Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.
2016 Harvard-MIT Mathematics Tournament, 2
Let $ABC$ be a triangle with $AB = 13$, $BC = 14$, $CA = 15$. Let $H$ be the orthocenter of $ABC$. Find the distance between the circumcenters of triangles $AHB$ and $AHC$.
2018 Junior Balkan Team Selection Tests - Moldova, 2
Let $ABC$ be an acute triangle.Let $OF \| BC$ where $O$ is the circumcenter and $F$ is between $A$ and $B$.Let $H$ be the orthocenter.Let $M$ be the midpoint of $AH$.Prove that $\angle FMC=90$.
1981 Romania Team Selection Tests, 1.
Show that for every real number $x$ we have
\[\max(|\sin x|,|\sin (x+1)|)>\frac13.\]
1970 Polish MO Finals, 2
Consider three sequences $(a_n)_{n=1}^{^\infty}$, $(b_n)_{n=1}^{^\infty}$ , $(c_n)_{n=1}^{^\infty}$, each of which has pairwisedistinct terms. Prove that there exist two indices $k$ and $l$ for which $k < l$, $$a_k < a_l
, b_k < b_l , \,\,\, and \,\,\, c_k < c_l.$$
2022 VIASM Summer Challenge, Problem 2
Let $S$ be the set of real numbers $k$ with the following property: for all set of real numbers $(a,b,c)$ satisfying $ab+bc+ca=1$, we always have the inequality:$$\frac{a}{{\sqrt {{a^2} + ab + {b^2} + k} }} + \frac{b}{{\sqrt {{b^2} + bc + {c^2} + k} }} + \frac{c}{{\sqrt {{c^2} + ca + {a^2} + k} }} \ge \sqrt {\frac{3}{{k + 1}}} .$$
a) Assume that $k\in S$. Prove that: $k\ge 2$.
b) Prove that: $2\in S$.
1998 National Olympiad First Round, 21
In an acute triangle $ ABC$, let $ D$ be a point on $ \left[AC\right]$ and $ E$ be a point on $ \left[AB\right]$ such that $ \angle ADB\equal{}\angle AEC\equal{}90{}^\circ$. If perimeter of triangle $ AED$ is 9, circumradius of $ AED$ is $ \frac{9}{5}$ and perimeter of triangle $ ABC$ is 15, then $ \left|BC\right|$ is
$\textbf{(A)}\ 5 \qquad\textbf{(B)}\ \frac{24}{5} \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ \frac{48}{5}$
2019 Oral Moscow Geometry Olympiad, 5
On sides $AB$ and $BC$ of a non-isosceles triangle $ABC$ are selected points $C_1$ and $A_1$ such that the quadrilateral $AC_1A_1C$ is cyclic. Lines $CC_1$ and $AA_1$ intersect at point $P$. Line $BP$ intersects the circumscribed circle of triangle $ABC$ at the point $Q$. Prove that the lines $QC_1$ and $CM$, where $M$ is the midpoint of $A_1C_1$, intersect at the circumscribed circles of triangle $ABC$.
2008 Harvard-MIT Mathematics Tournament, 22
For a positive integer $ n$, let $ \theta(n)$ denote the number of integers $ 0 \leq x < 2010$ such that $ x^2 \minus{} n$ is divisible by $ 2010$. Determine the remainder when $ \displaystyle \sum_{n \equal{} 0}^{2009} n \cdot \theta(n)$ is divided by $ 2010$.
2019 Auckland Mathematical Olympiad, 1
Function $f$ satisfies the equation $f(\cos x) = \cos (17x)$. Prove that it also satisfies the equation $f(\sin x) = \sin (17x)$.
2019 BMT Spring, 15
A group of aliens from Gliese $667$ Cc come to Earth to test the hypothesis that mathematics is indeed a universal language. To do this, they give you the following information about their mathematical system:
$\bullet$ For the purposes of this experiment, the Gliesians have decided to write their equations in the same syntactic format as in Western math. For example, in Western math, the expression “$5+4$” is interpreted as running the “$+$” operation on numbers $5$ and $4$. Similarly, in Gliesian math, the expression $\alpha \gamma \beta$ is interpreted as running the “$\gamma $” operation on numbers $\alpha$ and $ \beta$.
$\bullet$ You know that $\gamma $ and $\eta$ are the symbols for addition and multiplication (which works the same in Gliesian math as in Western math), but you don’t know which is which. By some bizarre coincidence, the symbol for equality is the same in Gliesian math as it is in Western math; equality is denoted with an “$=$” symbol between the two equal values.
$\bullet$ Two symbols that look exactly the same have the same meaning. Two symbols that are different have different meanings and, therefore, are not equal.
They then provide you with the following equations, written in Gliesian, which are known to be true:
[img]https://cdn.artofproblemsolving.com/attachments/b/e/e2e44c257830ce8eee7c05535046c17ae3b7e6.png[/img]
2012 India PRMO, 2
A triangle with perimeter $7$ has integer sidelengths. What is the maximum possible area of such a triangle?
2009 Singapore Junior Math Olympiad, 2
The set of $2000$-digit integers are divided into two sets: the set $M$ consisting all integers each of which can be represented as the product of two $1000$-digit integers, and the set $N$ which contains the other integers. Which of the sets $M$ and $N$ contains more elements?
2022 AMC 12/AHSME, 5
The point $(-1, -2)$ is rotated $270^{\circ}$ counterclockwise about the point $(3, 1)$. What are the coordinates of its new position?
$\textbf{(A)}\ (-3, -4) \qquad \textbf{(B)}\ (0,5) \qquad \textbf{(C)}\ (2,-1) \qquad \textbf{(D)}\ (4,3) \qquad \textbf{(E)}\ (6,-3)$
2000 Stanford Mathematics Tournament, 24
Peter is randomly filling boxes with candy. If he has 10 pieces of candy and 5 boxes in a row labeled A, B, C, D, and E, how many ways can he distribute the candy so that no two adjacent boxes are empty?
2000 Mongolian Mathematical Olympiad, Problem 5
Given a natural number $n$, find the number of quadruples $(x,y,u,v)$ of integers with $1\le x,y,y,v\le n$ satisfy the following inequalities:
\begin{align*}
&1\le v+x-y\le n,\\
&1\le x+y-u\le n,\\
&1\le u+v-y\le n,\\
&1\le v+x-u\le n.
\end{align*}
2020 Italy National Olympiad, #1
Let $\omega$ be a circle and let $A,B,C,D,E$ be five points on $\omega$ in this order. Define $F=BC\cap DE$, such that the points $F$ and $A$ are on opposite sides, with regard to the line $BE$ and the line $AE$ is tangent to the circumcircle of the triangle $BFE$.
a) Prove that the lines $AC$ and $DE$ are parallel
b) Prove that $AE=CD$
2005 National High School Mathematics League, 12
If the sum of all digits of a number is $7$, then we call it [i]lucky number[/i]. Put all [i]lucky numbers[/i] in order (from small to large): $a_1,a_2,\cdots,a_n,\cdots$. If $a_n=2005$, then $a_{5n}=$________.
1963 Czech and Slovak Olympiad III A, 1
Consider a cuboid$ ABCDA'B'C'D'$ (where $ABCD$ is a rectangle and $AA' \parallel BB' \parallel CC' \parallel DD'$) with $AA' = d$, $\angle ABD' = \alpha, \angle A'D'B = \beta$. Express the lengths x = $AB$, $y = BC$ in terms of $d$ and (acute) angles $\alpha, \beta$. Discuss condition of solvability.
2021 Math Prize for Girls Problems, 12
Let $P_1$, $P_2$, $P_3$, $P_4$, $P_5$, and $P_6$ be six parabolas in the plane, each congruent to the parabola $y = x^2/16$. The vertices of the six parabolas are evenly spaced around a circle. The parabolas open outward with their axes being extensions of six of the circle's radii. Parabola $P_1$ is tangent to $P_2$, which is tangent to $P_3$, which is tangent to $P_4$, which is tangent to $P_5$, which is tangent to $P_6$, which is tangent to $P_1$. What is the diameter of the circle?
2018 Iranian Geometry Olympiad, 2
In acute triangle $ABC, \angle A = 45^o$. Points $O,H$ are the circumcenter and the orthocenter of $ABC$, respectively. $D$ is the foot of altitude from $B$. Point $X$ is the midpoint of arc $AH$ of the circumcircle of triangle $ADH$ that contains $D$. Prove that $DX = DO$.
Proposed by Fatemeh Sajadi
Swiss NMO - geometry, 2006.5
A circle $k_1$ lies within a second circle $k_2$ and touches it at point $A$. A line through $A$ intersects $k_1$ again in $B$ and $k_2$ in $C$. The tangent to $k_1$ through $B$ intersects $k_2$ at points $D$ and $E$. The tangents at $k_1$ passing through $C$ intersects $k_1$ in points $F$ and $G$. Prove that $D, E, F$ and $G$ lie on a circle.
2007 F = Ma, 37
A massless elastic cord (that obeys Hooke's Law) will break if the tension in the cord exceeds $T_{max}$. One end of the cord is attached to a fixed point, the other is attached to an object of mass $3m$. If a second, smaller object of mass m moving at an initial speed $v_0$ strikes the larger mass and the two stick together, the cord will stretch and break, but the final kinetic energy of the two masses will be zero. If instead the two collide with a perfectly elastic one-dimensional collision, the cord will still break, and the larger mass will move off with a final speed of $v_f$. All motion occurs on a horizontal, frictionless surface.
Find $v_f/v_0$.
$ \textbf{(A)}\ 1/\sqrt{12}\qquad\textbf{(B)}\ 1/\sqrt{2}\qquad\textbf{(C)}\ 1/\sqrt{6} \qquad\textbf{(D)}\ 1/\sqrt{3}\qquad\textbf{(E)}\ \text{none of the above} $
2019 India IMO Training Camp, P3
Let $f : \{ 1, 2, 3, \dots \} \to \{ 2, 3, \dots \}$ be a function such that $f(m + n) | f(m) + f(n) $ for all pairs $m,n$ of positive integers. Prove that there exists a positive integer $c > 1$ which divides all values of $f$.