Olympiad problems

2009 Kosovo National Mathematical Olympiad, 4

Tags: number theory

Prove that $n^{11}-n$ is divisible by $11$.

2023 Bulgarian Autumn Math Competition, 12.1

Tags: algebra

Let $x_0, x_1, \ldots$ be a sequence of real numbers such that $x_0=1$ and $x_{n+1}=\sin(x_n)+\frac{\pi} {2}-1$ for all $n \geq 0$. Show that the sequence converges and find its limit.

2014 Purple Comet Problems, 17

Tags: geometry

Right triangle $ABC$ has a right angle at $C$. Point $D$ on side $\overline{AB}$ is the base of the altitude of $\triangle ABC$ from $C$. Point $E$ on side $\overline{BC}$ is the base of the altitude of $\triangle CBD$ from $D$. Given that $\triangle ACD$ has area $48$ and $\triangle CDE$ has area $40$, find the area of $\triangle DBE$.

2001 National Olympiad First Round, 28

Tags:

The towns $A,B,C,D,E$ are located clockwise on a circular road such that the distance between $A$ and $B$, $B$ and $C$, $C$ and $D$, $E$ and $A$ are $5$, $5$, $2$, $1$ and $4$ km respectively. A health center will be located on that road such that the maximum of the shortest distance to each town will be minimum. How many alternative locations are there for the health center? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{None of the preceding} $

2019 Romania National Olympiad, 1

Tags: inequalities , algebra

a) Prove that for $x,y \ge 1$, holds $$x+y - \frac{1}{x}- \frac{1}{y} \ge 2\sqrt{xy} -\frac{2}{\sqrt{xy}}$$ b) Prove that for $a,b,c,d \ge 1$ with $abcd=16$ , holds $$a+b+c+d-\frac{1}{a}-\frac{1}{b}-\frac{1}{c}-\frac{1}{d}\ge 6$$

2005 Baltic Way, 18

Tags: modular arithmetic , number theory

Let $x$ and $y$ be positive integers and assume that $z=\frac{4xy}{x+y}$ is an odd integer. Prove that at least one divisor of $z$ can be expressed in the form $4n-1$ where $n$ is a positive integer.

1991 AMC 8, 3

Tags:

Two hundred thousand times two hundred thousand equals $\text{(A)}\ \text{four hundred thousand} \qquad \text{(B)}\ \text{four million} \qquad \text{(C)}\ \text{forty thousand} \\ \text{(D)}\ \text{four hundred million} \qquad \text{(E)}\ \text{forty billion}$

2024 USAMO, 3

Tags: geometry

Let $m$ be a positive integer. A triangulation of a polygon is [i]$m$-balanced[/i] if its triangles can be colored with $m$ colors in such a way that the sum of the areas of all triangles of the same color is the same for each of the $m$ colors. Find all positive integers $n$ for which there exists an $m$-balanced triangulation of a regular $n$-gon. [i]Note[/i]: A triangulation of a convex polygon $\mathcal{P}$ with $n \ge 3$ sides is any partitioning of $\mathcal{P}$ into $n-2$ triangles by $n-3$ diagonals of $\mathcal{P}$ that do not intersect in the polygon's interior. [i]Proposed by Krit Boonsiriseth[/i]

2013 Romania National Olympiad, 3

Tags: floor function , algebra , fractional part , integer part

Find all real $x > 0$ and integer $n > 0$ so that $$ \lfloor x \rfloor+\left\{ \frac{1}{x}\right\}= 1.005 \cdot n.$$

Denmark (Mohr) - geometry, 2021.4

Tags: angle bisector , geometry , area

Given triangle $ABC$ with $|AC| > |BC|$. The point $M$ lies on the angle bisector of angle $C$, and $BM$ is perpendicular to the angle bisector. Prove that the area of triangle AMC is half of the area of triangle $ABC$. [img]https://cdn.artofproblemsolving.com/attachments/4/2/1b541b76ec4a9c052b8866acbfea9a0ce04b56.png[/img]

1971 Spain Mathematical Olympiad, 8

Tags: number theory , divide

Among the $2n$ numbers $1, 2, 3, . . . , 2n$ are chosen in any way $n + 1$ different numbers. Prove that among the chosen numbers there are at least two, such that one divides the other.

2021 Regional Olympiad of Mexico West, 4

Tags: number theory , combinatorics , coloring

Some numbers from $1$ to $100$ are painted red so that the following two conditions are met: $\bullet$ The number $1 $ is painted red. $\bullet$ If the numbers other than $a$ and $b$ are painted red then no number between $a$ and $b$ divides the number $ab$. What is the maximum number of numbers that can be painted red?

2012 BMT Spring, 7

Tags: algebra

Suppose Bob begins walking at a constant speed from point $N$ to point $S$ along the path indicated by the following figure. [img]https://cdn.artofproblemsolving.com/attachments/6/2/f5819267020f2bd38e52c6e873a2cf91ce8c49.png[/img] After Bob has walked a distance of $x$, Alice begins walking at point $N$, heading towards point $S$ along the same path. Alice walks $1.28$ times as fast as Bob when they are on the same line segment and $1.06$ times as fast as Bob otherwise. For what value of $x$ do Alice and Bob meet at point $S$?

2005 Today's Calculation Of Integral, 64

Tags: trigonometry , integration , algebra , calculus , polynomial , calculus computations

Let $f(t)$ be the cubic polynomial for $t$ such that $\cos 3x=f(\cos x)$ holds for all real number $x$. Evaluate \[\int_0^1 \{f(t)\}^2 \sqrt{1-t^2}dt\]

2016 NIMO Problems, 2

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Michael, David, Evan, Isabella, and Justin compete in the NIMO Super Bowl, a round-robin cereal-eating tournament. Each pair of competitors plays exactly one game, in which each competitor has an equal chance of winning (and there are no ties). The probability that none of the five players wins all of his/her games is $\tfrac{m}{n}$ for relatively prime positive integers $m$, $n$. Compute $100m + n$. [i]Proposed by Evan Chen[/i]

1992 IMO Longlists, 34

Tags: number theory , linear algebra , system of equations , additive number theory

Let $a, b, c$ be integers. Prove that there are integers $p_1, q_1, r_1, p_2, q_2, r_2$ such that \[a = q_1r_2 - q_2r_1, b = r_1p_2 - r_2p_1, c = p_1q_2 - p_2q_1.\]

2016 Purple Comet Problems, 25

Tags: trigonometry

For $n$ measured in degrees, let $T(n) = \cos^2(30^\circ -n) - \cos(30^\circ -n)\cos(30^\circ +n) +\cos^2(30^\circ +n)$. Evaluate $$ 4\sum^{30}_{n=1} n \cdot T(n).$$

2012 Centers of Excellency of Suceava, 1

Tags: algebra , formula

Let be three nonzero rational numbers $ a,b,c $ under the relation $ (a+b)(b+c)(c+a)=a^2b^2c^2. $ Show that the expression $ \sqrt[3]{3+1/a^3+1/b^3+1/c^3} $ is rational. [i]Ion Bursuc[/i]

2008 JBMO Shortlist, 11

Tags: number theory

Determine the greatest number with $n$ digits in the decimal representation which is divisible by $429$ and has the sum of all digits less than or equal to $11$.

2013 NZMOC Camp Selection Problems, 7

Tags: number theory , digit

In a sequence of positive integers an inversion is a pair of positions such that the element in the position to the left is greater than the element in the position to the right. For instance the sequence $2,5,3,1,3$ has five inversions - between the first and fourth positions, the second and all later positions, and between the third and fourth positions. What is the largest possible number of inversions in a sequence of positive integers whose sum is $2014$?

Kvant 2024, M2797

Tags: algebra , inequalities , inequality

For real numbers $0 \leq a_1 \leq a_2 \leq ... \leq a_n$ and $0 \leq b_1 \leq b_2 \leq ... \leq b_n$ prove that \[ \left( \frac{a_1}{1 \cdot 2}+\frac{a_2}{2 \cdot 3}+...+\frac{a_n}{n(n+1)} \right) \times \left( \frac{b_1}{1 \cdot 2}+\frac{b_2}{2 \cdot 3}+...+\frac{b_n}{n(n+1)} \right) \leq \frac{a_1b_1}{1 \cdot 2}+\frac{a_2b_2}{2 \cdot 3}+...+\frac{a_nb_n}{n(n+1)}.\] [i]Proposed by A. Antropov[/i]

2020 Harvard-MIT Mathematics Tournament, 10

Tags: hmmt

Let $n$ be a fixed positive integer, and choose $n$ positive integers $a_1, \ldots , a_n$. Given a permutation $\pi$ on the first $n$ positive integers, let $S_{\pi}=\{i\mid \frac{a_i}{\pi(i)} \text{ is an integer}\}$. Let $N$ denote the number of distinct sets $S_{\pi}$ as $\pi$ ranges over all such permutations. Determine, in terms of $n$, the maximum value of $N$ over all possible values of $a_1, \ldots , a_n$. [i]Proposed by James Lin.[/i]

2011 Saudi Arabia BMO TST, 3

Tags: circumcircle , area , equal area , geometry

In an acute triangle $ABC$ the angle bisector $AL$, $L \in BC$, intersects its circumcircle at $N$. Let $K$ and $M$ be the projections of $L$ onto sides $AB$ and $AC$. Prove that triangle $ABC$ and quadrilateral $A K N M$ have equal areas.

2008 Hanoi Open Mathematics Competitions, 2

Tags: number theory , diophantine equation , diophantine

Find all pairs $(m, n)$ of positive integers such that $m^2 + 2n^2 = 3(m + 2n)$

2010 Contests, 2

Tags: combinatorics , inequalities

There are $100$ random, distinct real numbers corresponding to $100$ points on a circle. Prove that you can always choose $4$ consecutive points in such a way that the sum of the two numbers corresponding to the points on the outside is always greater than the sum of the two numbers corresponding to the two points on the inside.