Found problems: 85335
2002 Romania National Olympiad, 4
Let $I\subseteq \mathbb{R}$ be an interval and $f:I\rightarrow\mathbb{R}$ a function such that:
\[|f(x)-f(y)|\le |x-y|,\quad\text{for all}\ x,y\in I. \]
Show that $f$ is monotonic on $I$ if and only if, for any $x,y\in I$, either $f(x)\le f\left(\frac{x+y}{2}\right)\le f(y)$ or $f(y)\le f\left(\frac{x+y}{2}\right)\le f(x)$.
2017 Azerbaijan Team Selection Test, 3
Consider fractions $\frac{a}{b}$ where $a$ and $b$ are positive integers.
(a) Prove that for every positive integer $n$, there exists such a fraction $\frac{a}{b}$ such that $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}+1$.
(b) Show that there are infinitely many positive integers $n$ such that no such fraction $\frac{a}{b}$ satisfies $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}$.
2011 Nordic, 3
Find all functions $f$ such that
\[f(f(x) + y) = f(x^2-y) + 4yf(x)\]
for all real numbers $x$ and $y$.
2014 Indonesia MO Shortlist, N5
Prove that we can give a color to each of the numbers $1,2,3,...,2013$ with seven distinct colors (all colors are necessarily used) such that for any distinct numbers $a,b,c$ of the same color, then $2014\nmid abc$ and the remainder when $abc$ is divided by $2014$ is of the same color as $a,b,c$.
2001 USA Team Selection Test, 4
There are 51 senators in a senate. The senate needs to be divided into $n$ committees so that each senator is on one committee. Each senator hates exactly three other senators. (If senator A hates senator B, then senator B does [i]not[/i] necessarily hate senator A.) Find the smallest $n$ such that it is always possible to arrange the committees so that no senator hates another senator on his or her committee.
2022 Kazakhstan National Olympiad, 4
$P$ and $Q$ are points on angle bisectors of two adjacent angles. Let $PA$, $PB$, $QC$ and $QD$ be altitudes on the sides of these adjacent angles. Prove that lines $AB$, $CD$ and $PQ$ are concurrent.
2017 Thailand Mathematical Olympiad, 10
A lattice point is defined as a point on the plane with integer coordinates. Show that for all positive integers $n$, there is a circle on the plane with exactly n lattice points in its interior (not including its boundary).
2019 European Mathematical Cup, 1
Every positive integer is marked with a number from the set $\{ 0,1,2\}$, according to the following rule:
$$\text{if a positive integer }k\text{ is marked with }j,\text{ then the integer }k+j\text{ is marked with }0.$$
Let $S$ denote the sum of marks of the first $2019$ positive integers. Determine the maximum possible value of $S$.
[i]Proposed by Ivan Novak[/i]
PEN A Problems, 3
Let $a$ and $b$ be positive integers such that $ab+1$ divides $a^{2}+b^{2}$. Show that \[\frac{a^{2}+b^{2}}{ab+1}\] is the square of an integer.
1953 Miklós Schweitzer, 10
[b]10.[/b] Consider a point performing a random walk on a planar triangular lattice and suppose that it moves away with equal probability from any lattice point along any one of the six lattice lines issuing from it. Prove that if the walk is continued indefinitely, then the point will return to its starting point with probability 1. [b](P. 5)[/b]
PEN G Problems, 7
Show that $ \pi$ is irrational.
2015 Chile TST Ibero, 4
Let $x, y \in \mathbb{R}^+$. Prove that:
\[
\left( 1 + \frac{1}{x} \right) \left( 1 + \frac{1}{y} \right) \geq \left( 1 + \frac{2}{x + y} \right)^2.
\]
1998 Balkan MO, 4
Prove that the following equation has no solution in integer numbers: \[ x^2 + 4 = y^5. \] [i]Bulgaria[/i]
2023 Tuymaada Olympiad, 6
In the plane $n$ segments with lengths $a_1, a_2, \dots , a_n$ are drawn. Every ray beginning at the point $O$ meets at least one of the segments. Let $h_i$ be the distance from $O$ to the $i$-th segment (not the line!) Prove the inequality
\[\frac{a_1}{h_1}+\frac{a_2}{h_2} + \ldots + \frac{a_i}{h_i} \geqslant 2 \pi.\]
2001 All-Russian Olympiad Regional Round, 8.5
Let $a, b, c, d, e$ and $f$ be some numbers, and $ a \cdot c \cdot e \ne 0$.It is known that the values of the expressions $|ax+b|+|cx+d| $and $|ex+f|$ equal at all values of $x$. Prove that $ad = bc$.
2022 Greece Junior Math Olympiad, 2
Let $ABC$ be an isosceles triangle, and point $D$ in its interior such that $$D \hat{B} C=30^\circ, D \hat{B}A=50^\circ, D \hat{C}B=55^\circ$$
(a) Prove that $\hat B=\hat C=80^\circ$.
(b) Find the measure of the angle $D \hat{A} C$.
2012 National Olympiad First Round, 5
$\triangle ABC$ is given with $|AB|=7, |BC|=12$, and $|CA|=13$. Let $D$ be a point on $[BC]$ such that $|BD|=5$. Let $r_1$ and $r_2$ be the inradii of $\triangle ABD$ and $\triangle ACD$, respectively. What is $r_1/r_2$?
$ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ \frac{13}{12} \qquad \textbf{(C)}\ \frac{7}{5} \qquad \textbf{(D)}\ \frac{3}{2} \qquad \textbf{(E)}\ \text{None}$
Russian TST 2015, P1
A worm is called an [i]adult[/i] if its length is one meter. In one operation, it is possible to cut an adult worm into two (possibly unequal) parts, each of which immediately becomes a worm and begins to grow at a speed of one meter per hour and stops growing once it reaches one meter in length. What is the smallest amount of time in which it is possible to get $n{}$ adult worms starting with one adult worm? Note that it is possible to cut several adult worms at the same time.
2017 ASDAN Math Tournament, 5
Compute the maximum value attained by $f(x)=x^{1/x^2}$.
2023 USA IMO Team Selection Test, 3
Consider pairs $(f,g)$ of functions from the set of nonnegative integers to itself such that
[list]
[*]$f(0) \geq f(1) \geq f(2) \geq \dots \geq f(300) \geq 0$
[*]$f(0)+f(1)+f(2)+\dots+f(300) \leq 300$
[*]for any 20 nonnegative integers $n_1, n_2, \dots, n_{20}$, not necessarily distinct, we have $$g(n_1+n_2+\dots+n_{20}) \leq f(n_1)+f(n_2)+\dots+f(n_{20}).$$
[/list]
Determine the maximum possible value of $g(0)+g(1)+\dots+g(6000)$ over all such pairs of functions.
[i]Sean Li[/i]
1992 AIME Problems, 12
In a game of [i]Chomp[/i], two players alternately take bites from a 5-by-7 grid of unit squares. To take a bite, a player chooses one of the remaining squares, then removes ("eats'') all squares in the quadrant defined by the left edge (extended upward) and the lower edge (extended rightward) of the chosen square. For example, the bite determined by the shaded square in the diagram would remove the shaded square and the four squares marked by $\times.$ (The squares with two or more dotted edges have been removed form the original board in previous moves.)
[asy]
defaultpen(linewidth(0.7));
fill((2,2)--(2,3)--(3,3)--(3,2)--cycle, mediumgray);
int[] array={5, 5, 5, 4, 2, 2, 2, 0};
pair[] ex = {(2,3), (2,4), (3,2), (3,3)};
draw((3,5)--(7,5)^^(4,4)--(7,4)^^(4,3)--(7,3), linetype("3 3"));
draw((4,4)--(4,5)^^(5,2)--(5,5)^^(6,2)--(6,5)^^(7,2)--(7,5), linetype("3 3"));
int i, j;
for(i=0; i<7; i=i+1) {
for(j=0; j<array[i]; j=j+1) {
draw((i,j+1)--(i,j)--(i+1,j));
}
draw((i,array[i])--(i+1,array[i]));
if(array[i]>array[i+1]) {
draw((i+1,array[i])--(i+1,array[i+1]));
}}
for(i=0; i<4; i=i+1) {
draw(ex[i]--(ex[i].x+1, ex[i].y+1), linewidth(1.2));
draw((ex[i].x+1, ex[i].y)--(ex[i].x, ex[i].y+1), linewidth(1.2));
}[/asy]
The object of the game is to make one's opponent take the last bite. The diagram shows one of the many subsets of the set of 35 unit squares that can occur during the game of Chomp. How many different subsets are there in all? Include the full board and empty board in your count.
2021 Tuymaada Olympiad, 8
In a sequence $P_n$ of quadratic trinomials each trinomial, starting with the third, is the sum of the two preceding trinomials. The first two trinomials do not have common roots. Is it possible that $P_n$ has an integral root for each $n$?
2015 IMAR Test, 4
(a) Show that, if $I \subset R$ is a closed bounded interval, and $f : I \to R$ is a non-constant monic polynomial function such that $max_{x\in I}|f(x)|< 2$, then there exists a non-constant monic polynomial function $g : I \to R$ such that $max_{x\in I} |g(x)| < 1$.
(b) Show that there exists a closed bounded interval $I \subset R$ such that $max_{x\in I}|f(x)| \ge 2$ for every non-constant monic polynomial function $f : I \to R$.
Kvant 2024, M2803
Given is a permutation of $1, 2, \ldots, 2023, 2024$ and two positive integers $a, b$, such that for any two adjacent numbers, at least one of the following conditions hold:
1) their sum is $a$;
2) the absolute value of their difference is $b$.
Find all possible values of $b$.
1989 Bulgaria National Olympiad, Problem 5
Prove that the perpendiculars, drawn from the midpoints of the edges of the base of a given tetrahedron to the opposite lateral edges, have a common point if and only if the circumcenter of the tetrahedron, the centroid of the base, and the top vertex of the tetrahedron are collinear.