Found problems: 85335
2017 IMO Shortlist, N2
Let $ p \geq 2$ be a prime number. Eduardo and Fernando play the following game making moves alternately: in each move, the current player chooses an index $i$ in the set $\{0,1,2,\ldots, p-1 \}$ that was not chosen before by either of the two players and then chooses an element $a_i$ from the set $\{0,1,2,3,4,5,6,7,8,9\}$. Eduardo has the first move. The game ends after all the indices have been chosen .Then the following number is computed:
$$M=a_0+a_110+a_210^2+\cdots+a_{p-1}10^{p-1}= \sum_{i=0}^{p-1}a_i.10^i$$.
The goal of Eduardo is to make $M$ divisible by $p$, and the goal of Fernando is to prevent this.
Prove that Eduardo has a winning strategy.
[i]Proposed by Amine Natik, Morocco[/i]
2012 Peru IMO TST, 1
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that $$\displaystyle{f(f(x)) = \frac{x^2 - x}{2}\cdot f(x) + 2-x,}$$ for all $x \in \mathbb{R}.$ Find all possible values of $f(2).$
2021 Kyiv City MO Round 1, 9.3
Let $a_n = 1 + \frac{2}{n} - \frac{2}{n^3} - \frac{1}{n^4}$. For which smallest positive integer $n$ does the value of $P_n = a_2a_3a_4 \ldots a_n$ exceed $100$?
2013 Kazakhstan National Olympiad, 2
Given triangle ABC with incenter I. Let P,Q be point on circumcircle such that $\angle API=\angle CPI$ and $\angle BQI=\angle CQI$.Prove that $BP,AQ$ and $OI$ are concurrent.
1998 Romania National Olympiad, 1
We consider the nonzero matrices $A_0, A_1, \ldots, A_n \in \mathcal{M}_2(\mathbb{R}),$ $n \ge 2,$ with the properties: $A_0 \neq aI_2$ for any $a \in \mathbb{R}$ and $A_0A_k=A_kA_0$ for $k= \overline{1,n}.$ Prove that
a) $\det \left(\sum\limits_{k=1}^n A_k^2 \right) \ge 0$;
b) If $\det \left(\sum\limits_{k=1}^n A_k^2 \right) = 0$ and $A_2 \ne aA_1$ for any $a \in \mathbb{R},$ then $\sum\limits_{k=1}^n A_k^2=O_2.$
2019 Thailand TST, 2
In a classroom of at least four students, when any four of them take seats around a round table, there is always someone who either knows both of his neighbors, or does not know either of his neighbors. Prove that it is possible to divide the students into two groups so that in one of them, all students knows one another, and in the other, none of the students know each other.
[i]Note: If $A$ knows $B$, then $B$ knows $A$ as well.[/i]
2017 Regional Olympiad of Mexico West, 2
From a point $P$, two tangent lines are drawn to a circle $\Gamma$, which touch it at points $A$ and $B$. A circle $\Phi$ is drawn with center at $P$ and passes through $A$ and $B$ and is taken a point $R$ that is on the circumference $\Phi$ and in the interior of $\Gamma$. The straight line $PR$ intersects $\Gamma$ at the points $S$ and $Q$. The straight lines $AR$ and $BR$ meet $\Gamma$ again at points $C$ and $D$, respectively. Prove that $CD$ passes through the midpoint of $SQ$.
1977 Polish MO Finals, 2
Show that for every convex polygon there is a circle passing through three consecutive vertices of the polygon and containing the entire polygon
1993 Poland - First Round, 11
A triangle with perimeter $2p$ is inscribed in a circle of radius $R$ and also circumscribed on a circle of radius $r$. Prove that $p < 2(R+r)$.
2000 China Team Selection Test, 3
For positive integer $a \geq 2$, denote $N_a$ as the number of positive integer $k$ with the following property: the sum of squares of digits of $k$ in base a representation equals $k$. Prove that:
a.) $N_a$ is odd;
b.) For every positive integer $M$, there exist a positive integer $a \geq 2$ such that $N_a \geq M$.
2010 Contests, 2
For a positive integer $n$, we define the function $f_n(x)=\sum_{k=1}^n |x-k|$ for all real numbers $x$. For any two-digit number $n$ (in decimal representation), determine the set of solutions $\mathbb{L}_n$ of the inequality $f_n(x)<41$.
[i](41st Austrian Mathematical Olympiad, National Competition, part 1, Problem 2)[/i]
ICMC 6, 1
The city of Atlantis is built on an island represented by $[ -1, 1]$, with skyline initially given by $f(x) = 1 - |x| $. The sea level is currently $y=0$, but due to global warming, it is rising at a rate of $0.01$ a year. For any position $-1 < x < 1$, while the building at $x$ is not completely submerged, then it is instantaneously being built upward at a rate of $r$ per year, where $r$ is the distance (along the $x$-axis) from this building to the nearest completely submerged building.
How long will it be until Atlantis becomes completely submerged?
[i]Proposed by Ethan Tan[/i]
2021 MIG, 25
Thelma writes a list of four digits consisting of $1$, $3$, $5$, and $7$, and each digit can appear one time, multiples time, or not at all. The list has a unique [i]mode[/i], or the number that appears the most. Thelma removes two numbers of that mode from the list; her list now has no unique mode! How many lists are possible? Suppose that all possible lists are unordered.
$\textbf{(A) }18\qquad\textbf{(B) }24\qquad\textbf{(C) }30\qquad\textbf{(D) }36\qquad\textbf{(E) }48$
1997 APMO, 4
Triangle $A_1 A_2 A_3$ has a right angle at $A_3$. A sequence of points is now defined by the following iterative process, where $n$ is a positive integer. From $A_n$ ($n \geq 3$), a perpendicular line is drawn to meet $A_{n-2}A_{n-1}$ at $A_{n+1}$.
(a) Prove that if this process is continued indefinitely, then one and only one point $P$ is interior to every triangle $A_{n-2} A_{n-1} A_{n}$, $n \geq 3$.
(b) Let $A_1$ and $A_3$ be fixed points. By considering all possible locations of $A_2$ on the plane, find the locus of $P$.
2011 IMAC Arhimede, 6
Let $a$, $b$ and $c$ be positive real numbers such that $a+b+c=1$. Prove that
$\frac{a}{a^3+b^2c+c^2b} + \frac{b}{b^3+c^2a+a^2c} + \frac{c}{c^3+a^2b+b^2a} \le 1+\frac{8}{27abc}$
2008 Saint Petersburg Mathematical Olympiad, 2
In a kingdom, there are roads open between some cities with lanes both ways, in such a way, that you can come from one city to another using those roads. The roads are toll, and the price for taking each road is distinct. A minister made a list of all routes that go through each city exactly once. The king marked the most expensive road in each of the routes and said to close all the roads that he marked at least once. After that, it became impossible to go from city $A$ to city $B$, from city $B$ to city $C$, and from city $C$ to city $A$. Prove that the kings order was followed incorrectly.
1963 Miklós Schweitzer, 9
Let $ f(t)$ be a continuous function on the interval $ 0 \leq t \leq 1$, and define the two sets of points \[ A_t\equal{}\{(t,0): t\in[0,1]\} , B_t\equal{}\{(f(t),1): t\in [0,1]\}.\] Show that the union of all segments $ \overline{A_tB_t}$ is Lebesgue-measurable, and find the minimum of its measure with respect to all functions $ f$. [A. Csaszar]
2015 Kosovo Team Selection Test, 1
a)Prove that for every n,natural number exist natural numbers a and b such that
$(1-\sqrt{2})^n=a-b\sqrt{2}$ and $a^2-2b^2=(-1)^n$
b)Using first equation prove that for every n exist m such that
$(\sqrt{2}-1)^n=\sqrt{m}-\sqrt{m-1}$
1949-56 Chisinau City MO, 56
Solve the system of equations
$$\begin{cases} \dfrac{x+y}{xy}+\dfrac{xy}{x+y}= a+ \dfrac{1}{a}\\ \\\dfrac{x-y}{xy}+\dfrac{xy}{x-y}= c+ \dfrac{1}{c}\end{cases}$$
2015 CIIM, Problem 4
Let $f:\mathbb{R} \to \mathbb{R}$ a continuos function and $\alpha$ a real number such that $$\lim_{x\to\infty}f(x) = \lim_{x\to-\infty}f(x) = \alpha.$$
Prove that for any $r > 0,$ there exists $x,y \in \mathbb{R}$ such that $y-x = r$ and $f(x) = f(y).$
2015 Romania Team Selection Tests, 3
A Pythagorean triple is a solution of the equation $x^2 + y^2 = z^2$ in positive integers such that $x < y$. Given any non-negative integer $n$ , show that some positive integer appears in precisely $n$ distinct Pythagorean triples.
2019-2020 Winter SDPC, 2
Let $S = \{ a_0, a_1, a_2, a_3, \dots \}$ be a set of positive integers with $1 = a_0 < a_1 < a_2 < a_3 < \dots$. For a subset $T$ of $S$, let $\sigma(T)$ be the sum of the elements of $T$. For instance, $\sigma(\{1, 2, 3\}) = 6$. By convention, $\sigma(\emptyset) = 0$, where $\emptyset$ denotes an empty set. Call a number $n$ representable if there exists a subset $T$ of $S$ such that $\sigma(T) = n$. We aim to prove for any set $S$ satisfying $a_{k+1} \le 2a_k$ for every $k \ge 0$, that all non-negative integers are representable.
(a) Prove there is a unique value of $a_1$, and find this value. Use this to determine, with proof, all possible sets $\{a_0, a_1, a_2, a_3 \}$. (Hint: there are 7 possible sets.)
[Not for credit] I recommend that you show that for all 7 sets in part (a), every integer between $0$ and $a_3 - 1$ is representable. (Note that this does not depend on the values of $a_4, a_5, a_6, \dots$.)
(b) Show that if $a_k \le n \le a_{k+1} - 1$, then $0 \le n - a_k \le a_k - 1$.
(c) Prove that any non-negative integer is representable.
2024 Ukraine National Mathematical Olympiad, Problem 1
Oleksiy wrote several distinct positive integers on the board and calculated all their pairwise sums. It turned out that all digits from $0$ to $9$ appear among the last digits of these sums. What could be the smallest number of integers that Oleksiy wrote?
[i]Proposed by Oleksiy Masalitin[/i]
2006 AMC 10, 9
Francesca uses 100 grams of lemon juice, 100 grams of sugar, and 400 grams of water to make lemonade. There are 25 calories in 100 grams of lemon juice and 386 calories in 100 grams of sugar. Water contains no calories. How many calories are in 200 grams of her lemonade?
$ \textbf{(A) } 129 \qquad \textbf{(B) } 137 \qquad \textbf{(C) } 174 \qquad \textbf{(D) } 223 \qquad \textbf{(E) } 411$
2014 Hanoi Open Mathematics Competitions, 7
Determine the integral part of $A$, where $A =\frac{1}{672}+\frac{1}{673}+... +\frac{1}{2014}$