Found problems: 85335
2023 Indonesia MO, 2
Determine all functions $f : \mathbb{R} \to \mathbb{R}$ such that the following equation holds for every real $x,y$:
\[ f(f(x) + y) = \lfloor x + f(f(y)) \rfloor. \]
[b]Note:[/b] $\lfloor x \rfloor$ denotes the greatest integer not greater than $x$.
STEMS 2021 Math Cat B, Q4
Let $n$ be a fixed positive integer.
- Show that there exist real polynomials $p_1, p_2, p_3, \cdots, p_k \in \mathbb{R}[x_1, \cdots, x_n]$ such that
\[(x_1 + x_2 + \cdots + x_n)^2 + p_1(x_1, \cdots, x_n)^2 + p_2(x_1, \cdots, x_n)^2 + \cdots + p_k(x_1, \cdots, x_n)^2 = n(x_1^2 + x_2^2 + \cdots + x_n^2)\]
- Find the least natural number $k$, depending on $n$, such that the above polynomials $p_1, p_2, \cdots, p_k$ exist.
1992 AMC 12/AHSME, 29
An "unfair" coin has a $2/3$ probability of turning up heads. If this coin is tossed $50$ times, what is the probability that the total number of heads is even?
$ \textbf{(A)}\ 25\left(\frac{2}{3}\right)^{50}\qquad\textbf{(B)}\ \frac{1}{2}\left(1 - \frac{1}{3^{50}}\right)\qquad\textbf{(C)}\ \frac{1}{2}\qquad\textbf{(D)}\ \frac{1}{2}\left(1 + \frac{1}{3^{50}}\right)\qquad\textbf{(E)}\ \frac{2}{3} $
2008 IMO Shortlist, 7
Let $ ABCD$ be a convex quadrilateral with $ BA\neq BC$. Denote the incircles of triangles $ ABC$ and $ ADC$ by $ \omega_{1}$ and $ \omega_{2}$ respectively. Suppose that there exists a circle $ \omega$ tangent to ray $ BA$ beyond $ A$ and to the ray $ BC$ beyond $ C$, which is also tangent to the lines $ AD$ and $ CD$. Prove that the common external tangents to $ \omega_{1}$ and $\omega_{2}$ intersect on $ \omega$.
[i]Author: Vladimir Shmarov, Russia[/i]
1999 Romania National Olympiad, 4
a) Let $a,b\in R$, $a <b$. Prove that $x \in (a,b)$ if and only if there exists $\lambda \in (0,1)$ such that $x=\lambda a +(1-\lambda)b$.
b) If the function $f: R \to R$ has the property:
$$f (\lambda x+(1-\lambda) y) < \lambda f(x) + (1-\lambda)f(y), \forall x,y \in R, x\ne y, \forall \lambda \in (0,1), $$ prove that one cannot find four points on the function’s graph that are the vertices of a parallelogram
III Soros Olympiad 1996 - 97 (Russia), 10.8
The distance between cities $A$ and $B$ is $30$ km. Three tourists went from $A$ to $B$. The three of them have two bicycles: a racing bike, on which each of them rides at a speed of $30$ km/h, and a tourist bike, on which they can travel at a speed of $20$ km/h. Each of them can walk at a speed of $6$ km/h. Any bicycle can be left on the road, where it will lie until another tourist can use it. Tourists want to get to $B$ in the shortest time possible, with the end time of the trip corresponding to the moment the last of them arrives at $B$. What is this shortest time?
2022 Harvard-MIT Mathematics Tournament, 3
Let $ABCD$ and $AEF G$ be unit squares such that the area of their intersection is $\frac{20}{21}$ . Given that $\angle BAE < 45^o$, $\tan \angle BAE$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100a + b$.
2013 National Olympiad First Round, 36
A chess club consists of at least $10$ and at most $50$ members, where $G$ of them are female, and $B$ of them are male with $G>B$. In a chess tournament, each member plays with any other member exactly one time. At each game, the winner gains $1$, the loser gains $0$ and both player gains $1/2$ point when a tie occurs. At the tournament, it is observed that each member gained exactly half of his/her points from the games played against male members. How many different values can $B$ take?
$
\textbf{(A)}\ 5
\qquad\textbf{(B)}\ 4
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 2
\qquad\textbf{(E)}\ 1
$
1966 IMO Shortlist, 17
Let $ABCD$ and $A^{\prime }B^{\prime}C^{\prime }D^{\prime }$ be two arbitrary parallelograms in the space, and let $M,$ $N,$ $P,$ $Q$ be points dividing the segments $AA^{\prime },$ $BB^{\prime },$ $CC^{\prime },$ $DD^{\prime }$ in equal ratios.
[b]a.)[/b] Prove that the quadrilateral $MNPQ$ is a parallelogram.
[b]b.)[/b] What is the locus of the center of the parallelogram $MNPQ,$ when the point $M$ moves on the segment $AA^{\prime }$ ?
(Consecutive vertices of the parallelograms are labelled in alphabetical order.
2016 JBMO Shortlist, 2
Let ${ABC}$ be a triangle with $\angle BAC={{60}^{{}^\circ }}$. Let $D$ and $E$ be the feet of the perpendiculars from ${A}$ to the external angle bisectors of $\angle ABC$ and $\angle ACB$, respectively. Let ${O}$ be the circumcenter of the triangle ${ABC}$. Prove that the circumcircles of the triangles ${ADE}$and ${BOC}$ are tangent to each other.
2024 Nepal TST, P4
Vlad draws 100 rays in the Euclidean plane. David then draws a line $\ell$ and pays Vlad one pound for each ray that $\ell$ intersects. Naturally, David wants to pay as little as possible. What is the largest amount of money that Vlad can get from David?
[i]Proposed by Vlad Spătaru[/i]
2021 Korea Winter Program Practice Test, 6
Is there exist a sequence $a_0,a_1,a_2,\cdots $ consisting of non-zero integers that satisfies the following condition?
[b]Condition[/b]: For all integers $n$ ($\ge 2020$), equation
$$a_n x^n+a_{n-1}x^{n-1}+\cdots +a_0=0$$
has a real root with its absolute value larger than $2.001$.
2023 Federal Competition For Advanced Students, P1, 3
Given a positive integer $n$, find the proportion of the subsets of $\{1,2, \ldots, 2n\}$ such that their smallest element is odd.
2021 Peru IMO TST, P2
For any positive integers $a,b,c,n$, we define
$$D_n(a,b,c)=\mathrm{gcd}(a+b+c,a^2+b^2+c^2,a^n+b^n+c^n).$$
1. Prove that if $n$ is a positive integer not divisible by $3$, then for any positive integer $k$, there exist three integers $a,b,c$ such that $\mathrm{gcd}(a,b,c)=1$, and $D_n(a,b,c)>k$.
2. For any positive integer $n$ divisible by $3$, find all values of $D_n(a,b,c)$, where $a,b,c$ are three positive integers such that $\mathrm{gcd}(a,b,c)=1$.
2020 Saint Petersburg Mathematical Olympiad, 3.
On the side $AD$ of the convex quadrilateral $ABCD$ with an acute angle at $B$, a point $E$ is marked.
It is known that $\angle CAD = \angle ADC=\angle ABE =\angle DBE$.
(Grade 9 version) Prove that $BE+CE<AD$.
(Grade 10 version) Prove that $\triangle BCE$ is isosceles.(Here the condition that $\angle B$ is acute is not necessary.)
2009 VJIMC, Problem 1
Let $ABC$ be a non-degenerate triangle in the euclidean plane. Define a sequence $(C_n)_{n=0}^\infty$ of points as follows: $C_0:=C$, and $C_{n+1}$ is the incenter of the triangle $ABC_n$. Find $\lim_{n\to\infty}C_n$.
2013 Miklós Schweitzer, 7
Suppose that ${f: \Bbb{R} \rightarrow \Bbb{R}}$ is an additive function $($that is ${f(x+y) = f(x)+f(y)}$ for all ${x, y \in \Bbb{R}})$ for which ${x \mapsto f(x)f(\sqrt{1-x^2})}$ is bounded of some nonempty subinterval of ${(0,1)}$. Prove that ${f}$ is continuous.
[i]Proposed by Zoltán Boros[/i]
2006 India IMO Training Camp, 1
Find all triples $(a,b,c)$ such that $a,b,c$ are integers in the set $\{2000,2001,\ldots,3000\}$ satisfying $a^2+b^2=c^2$ and $\text{gcd}(a,b,c)=1$.
2013 NIMO Problems, 8
Find the number of positive integers $n$ for which there exists a sequence $x_1, x_2, \cdots, x_n$ of integers with the following property: if indices $1 \le i \le j \le n$ satisfy $i+j \le n$ and $x_i - x_j$ is divisible by $3$, then $x_{i+j} + x_i + x_j + 1$ is divisible by $3$.
[i]Based on a proposal by Ivan Koswara[/i]
2006 MOP Homework, 7
Let $n$ be a given integer greater than two, and let $S = \{1, 2,...,n\}$.
Suppose the function $f : S^k \to S$ has the property that $f(a) \ne f(b)$ for every pair $a$ and $b$ of elements in $S^k$ with $a$ and $b$ differ in all components. Prove that $f$ is a function of one of its elements.
1969 IMO Shortlist, 52
Prove that a regular polygon with an odd number of edges cannot be partitioned into four pieces with equal areas by two lines that pass through the center of polygon.
1992 IMO, 2
Let $\,{\mathbb{R}}\,$ denote the set of all real numbers. Find all functions $\,f: {\mathbb{R}}\rightarrow {\mathbb{R}}\,$ such that \[ f\left( x^{2}+f(y)\right) =y+\left( f(x)\right) ^{2}\hspace{0.2in}\text{for all}\,x,y\in \mathbb{R}. \]
2022 May Olympiad, 1
In a $7\times7$ board, some squares are painted red. Let $a$ be the number of rows that have an odd number of red squares and let $b$ be the number of columns that have an odd number of red squares. Find all possible values of $a+b$. For each value found, give a example of how the board can be painted.
2021 JHMT HS, 8
Triangle $ABC,$ with $BC = 48,$ is inscribed in a circle $\Omega$ of radius $49\sqrt{3}.$ There is a unique circle $\omega$ that is tangent to $\overline{AB}$ and $\overline{AC}$ and internally tangent to $\Omega.$ Let $D,$ $E,$ and $F$ be the points at which $\omega$ is tangent to $\Omega,$ $\overline{AB},$ and $\overline{AC},$ respectively. The rays $\overrightarrow{DE}$ and $\overrightarrow{DF}$ intersect $\Omega$ at points $X$ and $Y,$ respectively, such that $X \neq D$ and $Y \neq D.$ Compute $XY.$
1969 IMO Shortlist, 47
$C$ is a point on the semicircle diameter $AB$, between $A$ and $B$. $D$ is the foot of the perpendicular from $C$ to $AB$. The circle $K_1$ is the incircle of $ABC$, the circle $K_2$ touches $CD,DA$ and the semicircle, the circle $K_3$ touches $CD,DB$ and the semicircle. Prove that $K_1,K_2$ and $K_3$ have another common tangent apart from $AB$.