Olympiad problems

2013 ELMO Shortlist, 3

Find all $f:\mathbb{R}\to\mathbb{R}$ such that for all $x,y\in\mathbb{R}$, $f(x)+f(y) = f(x+y)$ and $f(x^{2013}) = f(x)^{2013}$. [i]Proposed by Calvin Deng[/i]

2010 India IMO Training Camp, 6

Tags: floor function , induction , combinatorics unsolved , combinatorics

Let $n\ge 2$ be a given integer. Show that the number of strings of length $n$ consisting of $0'$s and $1'$s such that there are equal number of $00$ and $11$ blocks in each string is equal to \[2\binom{n-2}{\left \lfloor \frac{n-2}{2}\right \rfloor}\]

2017 AMC 12/AHSME, 8

Tags: AMC , AMC 12 , 2017 AMC 10A , 2017 AMC 12A , 3D geometry , AMC 10

The region consisting of all points in three-dimensional space within $3$ units of line segment $\overline{AB}$ has volume $216\pi$. What is the length $AB$? $\textbf{(A)}\ 6\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 20\qquad\textbf{(E)}\ 24$

1980 Miklós Schweitzer, 5

Tags: group theory , abstract algebra , superior algebra , superior algebra unsolved

Let $ G$ be a transitive subgroup of the symmetric group $ S_{25}$ different from $ S_{25}$ and $ A_{25}$. Prove that the order of $ G$ is not divisible by $ 23$. [i]J. Pelikan[/i]

2018 Federal Competition For Advanced Students, P2, 5

Tags: combinatorics , number theory

On a circle $2018$ points are marked. Each of these points is labeled with an integer. Let each number be larger than the sum of the preceding two numbers in clockwise order. Determine the maximal number of positive integers that can occur in such a configuration of $2018$ integers. [i](Proposed by Walther Janous)[/i]

2010 Bulgaria National Olympiad, 2

Tags: function , floor function , induction , algebra proposed , algebra

Let $f: \mathbb N \to \mathbb N$ be a function such that $f(1)=1$ and \[f(n)=n - f(f(n-1)), \quad \forall n \geq 2.\] Prove that $f(n+f(n))=n $ for each positive integer $n.$

2001 Greece Junior Math Olympiad, 1

Tags: inequalities

Let $a, b, x, y$ be positive real numbers such that $a+b=1$. Prove that $\frac{1}{\frac{a}{x}+\frac{b}{y}}\leq ax+by$ and find when equality holds.

2012 Today's Calculation Of Integral, 822

Tags: calculus , integration , limit , calculus computations

For $n=0,\ 1,\ 2,\ \cdots$, let $a_n=\int_{n}^{n+1} \{xe^{-x}-(n+1)e^{-n-1}(x-n)\}\ dx,$ $b_n=\int_{n}^{n+1} \{xe^{-x}-(n+1)e^{-n-1}\}\ dx.$ Find $\lim_{n\to\infty} \sum_{k=0}^n (a_k-b_k).$

2021 Latvia Baltic Way TST, P15

Tags: number theory , trivial

Denote by $s(n)$ the sum of all natural divisors of $n$ which are smaller than $n$. Does there exist a positive integer $a$ such that the equation $$s(n)=a+n$$ has infinitely many solutions in positive integers?

2008 Croatia Team Selection Test, 3

Tags: geometry , circumcircle , geometry proposed

Point $ M$ is taken on side $ BC$ of a triangle $ ABC$ such that the centroid $ T_c$ of triangle $ ABM$ lies on the circumcircle of $ \triangle ACM$ and the centroid $ T_b$ of $ \triangle ACM$ lies on the circumcircle of $ \triangle ABM$. Prove that the medians of the triangles $ ABM$ and $ ACM$ from $ M$ are of the same length.

1965 All Russian Mathematical Olympiad, 069

Tags: algebra

A spy airplane flies on the circle with the centre $A$ and radius $10$ km. Its speed is $1000$ km/h. At a certain moment, a rocket , that has same speed with the airplane, is launched from point $A$ and moves along on the straight line connecting the airplane and point $A$.How long after launch will the rocket hit the plane?

2021 Auckland Mathematical Olympiad, 3

Tags: number theory , Digits

Alice and Bob are independently trying to figure out a secret password to Cathy’s bitcoin wallet. Both of them have already figured out that: $\bullet$ it is a $4$-digit number whose first digit is $5$. $\bullet$ it is a multiple of $9$; $\bullet$ The larger number is more likely to be a password than a smaller number. Moreover, Alice figured out the second and the third digits of the password and Bob figured out the third and the fourth digits. They told this information to each other but not actual digits. After that the conversation followed: Alice: ”I have no idea what the number is.” Bob: ”I have no idea too.” After that both of them knew which number they should try first. Identify this number

1980 Poland - Second Round, 1

Tags: combinatorics , vector

Students $ A $ and $ B $ play according to the following rules: student $ A $ selects a vector $ \overrightarrow{a_1} $ of length 1 in the plane, then student $ B $ gives the number $ s_1 $, equal to $ 1 $ or $ - $1; then the student $ A $ chooses a vector $ \overrightarrow{a_1} $ of length $ 1 $, and in turn the student $ B $ gives a number $ s_2 $ equal to $ 1 $ or $ -1 $ etc. $ B $ wins if for a certain $ n $ vector $ \sum_{j=1}^n \varepsilon_j \overrightarrow{a_j} $ has a length greater than the number $ R $ determined before the start of the game. Prove that student $B$ can achieve a win in no more than $R^2 + 1$ steps regardless of partner $A$'s actions.

2018 Purple Comet Problems, 21

Tags: trigonometry

Let $x$ be in the interval $\left(0, \frac{\pi}{2}\right)$ such that $\sin x - \cos x = \frac12$ . Then $\sin^3 x + \cos^3 x = \frac{m\sqrt{p}}{n}$ , where $m, n$, and $p$ are relatively prime positive integers, and $p$ is not divisible by the square of any prime. Find $m + n + p$.

MOAA Gunga Bowls, 2023.2

Tags: MOAA 2023

Harry wants to put $5$ identical blue books, $3$ identical red books, and $1$ white book on his bookshelf. If no two adjacent books may be the same color, how many distinct arrangements can Harry make? [i]Proposed by Anthony Yang[/i]

2007 AIME Problems, 12

Tags: calculus , integration , logarithms , ratio , AMC , USA(J)MO , USAMO

The increasing geometric sequence $x_{0},x_{1},x_{2},\ldots$ consists entirely of integral powers of $3.$ Given that \[\sum_{n=0}^{7}\log_{3}(x_{n}) = 308\qquad\text{and}\qquad 56 \leq \log_{3}\left ( \sum_{n=0}^{7}x_{n}\right ) \leq 57,\] find $\log_{3}(x_{14}).$

2022 CCA Math Bonanza, TB1

Tags:

How many positive integer factors does the following expression have? \[ \sum_{n=1}^{999} \log_{10} \left(\frac{n+1}{n} \right) \] [i]2022 CCA Math Bonanza Tiebreaker Round #1[/i]

2023 MOAA, 16

Tags: MOAA 2023

Compute the sum $$\frac{\varphi(50!)}{\varphi(49!)}+ \frac{\varphi(51!)}{\varphi(50!)} + \dots + \frac{\varphi(100!)}{\varphi(99!)}$$ where $\varphi(n)$ returns the number of positive integers less than $n$ that are relatively prime to $n$. [i]Proposed by Andy Xu[/i]

2004 Purple Comet Problems, 25

Tags:

In the addition problem \[ \setlength{\tabcolsep}{1mm}\begin{tabular}{cccccc}& W & H & I & T & E\\ + & W & A & T & E & R \\\hline P & I & C & N & I & C\end{tabular} \] each distinct letter represents a diﬀerent digit. Find the number represented by the answer PICNIC.

1993 Korea - Final Round, 2

Tags: geometry , incenter , geometry proposed

Let be given a triangle $ABC$ with $BC = a, CA = b, AB = c$. Find point $P$ in the plane for which $aAP^{2}+bBP^{2}+cCP^{2}$ is minimum, and compute this minimum.

1941 Moscow Mathematical Olympiad, 073

Tags: congruent triangles , geometry

Given a quadrilateral, the midpoints $A, B, C, D$ of its consecutive sides, and the midpoints of its diagonals, $P$ and $Q$. Prove that $\vartriangle BCP = \vartriangle ADQ$.

2010 ELMO Shortlist, 7

Tags: algebra proposed , algebra

Find the smallest real number $M$ with the following property: Given nine nonnegative real numbers with sum $1$, it is possible to arrange them in the cells of a $3 \times 3$ square so that the product of each row or column is at most $M$. [i]Evan O' Dorney.[/i]

V Soros Olympiad 1998 - 99 (Russia), 11.8

Tags: geometry

Side $BC$ of triangle $ABC$ is equal to $a$ and the opposite angle is equal to $\theta$. A straight line passing through the midpoint of $BC$ and the center of the inscribed circle intersects lines $AB$ and $AC$ at points $M$ and $P$, respectively. Find the area of the quadrilateral (non-convex) $BMPC$.

2011 ELMO Shortlist, 6

Tags: combinatorics proposed , combinatorics

Do there exist positive integers $k$ and $n$ such that for any finite graph $G$ with diameter $k+1$ there exists a set $S$ of at most $n$ vertices such that for any $v\in V(G)\setminus S$, there exists a vertex $u\in S$ of distance at most $k$ from $v$? [i]David Yang.[/i]

2002 USAMTS Problems, 2

Tags: algebra , polynomial

Find four distinct positive integers, $a$, $b$, $c$, and $d$, such that each of the four sums $a+b+c$, $a+b+d$,$a+c+d$, and $b+c+d$ is the square of an integer. Show that infinitely many quadruples $(a,b,c,d)$ with this property can be created.