Olympiad problems

2021 European Mathematical Cup, 3

Let $\mathbb{N}$ denote the set of all positive integers. Find all functions $f:\mathbb{N}\to\mathbb{N}$ such that $$x^2-y^2+2y(f(x)+f(y))$$ is a square of an integer for all positive integers $x$ and $y$.

2018 Iran Team Selection Test, 6

Tags: combinatorics , sequence

$a_1,a_2,\ldots,a_n$ is a sequence of positive integers that has at least $\frac {2n}{3}+1$ distinct numbers and each positive integer has occurred at most three times in it. Prove that there exists a permutation $b_1,b_2,\ldots,b_n$ of $a_i $'s such that all the $n$ sums $b_i+b_{i+1}$ are distinct ($1\le i\le n $ , $b_{n+1}\equiv b_1 $) [i]Proposed by Mohsen Jamali[/i]

1989 Putnam, B6

Tags: probability , expected value , calculus , integration

Let $(x_1,x_2,\ldots,x_n)$ be a point chosen at random in the $n$-dimensional region defined by $0<x_1<x_2<\ldots<x_n<1$, denoting $x_0=0$ and $x_{n+1}=1$. Let $f$ be a continuous function on $[0,1]$ with $f(1)=0$. Show that the expected value of the sum $$\sum_{i=0}^n(x_{i+1}-x_i)f(x_{i+1})$$is $\int^1_0f(t)P(t)dt$., where $P$ is a polynomial of degree $n$, independent of $f$, with $0\le P(t)\le1$ for $0\le t\le1$.

2011 Dutch IMO TST, 5

Tags: combinatorics , coloring

Find all triples $(a, b, c)$ of positive integers with $a+b+c = 10$ such that there are $a$ red, $b$ blue and $c$ green points (all different) in the plane satisfying the following properties: $\bullet$ for each red point and each blue point we consider the distance between these two points, the sum of these distances is $37$, $\bullet$ for each green point and each red point we consider the distance between these two points, the sum of these distances is $30$, $\bullet$ for each blue point and each green point we consider the distance between these two points, the sum of these distances is $1$.

2010 AMC 8, 16

Tags: ratio

A square and a circle have the same area. What is the ratio of the side length of the square to the radius of the circle? $ \textbf{(A)}\ \frac{\sqrt{\pi}}{2} \qquad\textbf{(B)}\ \sqrt{\pi} \qquad\textbf{(C)}\ \pi \qquad\textbf{(D)}\ 2\pi \qquad\textbf{(E)}\ \pi^{2}$

2005 Slovenia Team Selection Test, 6

Tags: inequalities , algebra

Let $a,b,c > 0$ and $ab+bc+ca = 1$. Prove the inequality $3\sqrt[3]{\frac{1}{abc} +6(a+b+c) }\le \frac{\sqrt[3]3}{abc}$

V Soros Olympiad 1998 - 99 (Russia), 9.5

Tags: trapezoid , geometry

In the trapezoid $ABCD$ with bases $BC = a$, $AD = b$, the equality holds: $\angle BAC + \angle ACD = 180^o$. The straight line $AC$ intersects the common tangents to the circumcircles of the triangles $ABC$ and $ACD$ at the points Find $PQ$.

II Soros Olympiad 1995 - 96 (Russia), 11.2

Tags: geometry , 3d geometry

A cylindrical glass filled to the brim with water stands on a horizontal plane. The height of the glass is $2$ times the diameter of the base. At what angle must the glass be tilted from the vertical so that exactly $1/3$ of the water it contains pours out?

2014 Taiwan TST Round 3, 2

Tags: geometry , rhombus , trigonometry

In a triangle $ABC$, let $D$ and $E$ be the feet of the angle bisectors of angles $A$ and $B$, respectively. A rhombus is inscribed into the quadrilateral $AEDB$ (all vertices of the rhombus lie on different sides of $AEDB$). Let $\varphi$ be the non-obtuse angle of the rhombus. Prove that $\varphi \le \max \{ \angle BAC, \angle ABC \}$.

2021 Caucasus Mathematical Olympiad, 5

Tags: geometry , geometry inequality

A triangle $\Delta$ with sidelengths $a\leq b\leq c$ is given. It appears that it is impossible to construct a triangle from three segments whose lengths are equal to the altitudes of $\Delta$. Prove that $b^2>ac$.

2021 Iran Team Selection Test, 6

Tags: set theory , combinatorics , graph theory , coloring

May Olympiad L1 - geometry, 2011.3

Tags: geometry , rectangle , area of a triangle , area

In the rectangle $ABCD, BC = 5, EC = 1/3 CD$ and $F$ is the point where $AE$ and $BD$ are cut. The triangle $DFE$ has area $12$ and the triangle $ABF$ has area $27$. Find the area of the quadrilateral $BCEF$ . [img]https://1.bp.blogspot.com/-4w6e729AF9o/XNY9hqHaBaI/AAAAAAAAKL0/eCaNnWmgc7Yj9uV4z29JAvTcWCe21NIMgCK4BGAYYCw/s400/may%2B2011%2Bl1.png[/img]

2012 AMC 12/AHSME, 12

Tags: geometry

A square region $ABCD$ is externally tangent to the circle with equation $x^2+y^2=1$ at the point $(0,1)$ on the side $CD$. Vertices $A$ and $B$ are on the circle with equation $x^2+y^2=4$. What is the side length of this square? $ \textbf{(A)}\ \frac{\sqrt{10}+5}{10}\qquad\textbf{(B)}\ \frac{2\sqrt{5}}{5}\qquad\textbf{(C)}\ \frac{2\sqrt{2}}{3}\qquad\textbf{(D)}\ \frac{2\sqrt{19}-4}{5}\qquad\textbf{(E)}\ \frac{9-\sqrt{17}}{5} $

2004 National High School Mathematics League, 4

Tags: vector , ratio , geometry

$O$ is a point inside $\triangle ABC$, and $\overrightarrow{OA}+2\overrightarrow{OB}+3\overrightarrow{OC}=\overrightarrow{0}$, then the ratio of the area of $\triangle ABC$ to $\triangle AOC$ is $\text{(A)}2\qquad\text{(B)}\frac{3}{2}\qquad\text{(C)}3\qquad\text{(D)}\frac{5}{3}$

1986 IMO Longlists, 35

Tags: triangle inequality , inequalities

Establish the maximum and minimum values that the sum $|a| + |b| + |c|$ can have if $a, b, c$ are real numbers such that the maximum value of $|ax^2 + bx + c|$ is $1$ for $-1 \leq x \leq 1.$

2011 AMC 12/AHSME, 2

Tags: inequalities

Josanna's test scores to date are 90, 80, 70, 60, and 85. Her goal is to raise her test average at least 3 points with her next test. What is the minimum test score she would need to accomplish this goal? $ \textbf{(A)}\ 80 \qquad \textbf{(B)}\ 82 \qquad \textbf{(C)}\ 85 \qquad \textbf{(D)}\ 90 \qquad \textbf{(E)}\ 95 $

2005 AMC 8, 12

Tags:

Big Al, the ape, ate 100 bananas from May 1 through May 5. Each day he ate six more bananas than on the previous day. How many bananas did Big Al eat on May 5? $ \textbf{(A)}\ 20\qquad\textbf{(B)}\ 22\qquad\textbf{(C)}\ 30\qquad\textbf{(D)}\ 32\qquad\textbf{(E)}\ 34 $

2007 Harvard-MIT Mathematics Tournament, 30

Tags: geometry , circumcircle , cyclic quadrilateral

$ABCD$ is a cyclic quadrilateral in which $AB=3$, $BC=5$, $CD=6$, and $AD=10$. $M$, $I$, and $T$ are the feet of the perpendiculars from $D$ to lines $AB$, $AC$, and $BC$ respectively. Determine the value of $MI/IT$.

2021 Iran MO (3rd Round), 2

Tags: function , number theory

Find all functions $f:\mathbb{N} \to \mathbb{N}$ such that for any two positive integers $a$ and $b$ we have $$ f^a(b) + f^b(a) \mid 2(f(ab) +b^2 -1)$$ Where $f^n(m)$ is defined in the standard iterative manner.

2017 Poland - Second Round, 5

Tags: combinatorics , graph theory

Gourmet Jan compared $n$ restaurants ($n$ is a positive integer). Each pair of restaurants was compared in two categories: tastiness of food and quality of service. For some pairs Jan couldn't tell which restaurant was better in one category, but never in two categories. Moreover, if Jan thought restaurant $A$ was better than restaurant $B$ in one category and restaurant $B$ was better than restaurant $C$ in the same category, then $A$ is also better than $C$ in that category. Prove there exists a restaurant $R$ such that every other restaurant is worse than $R$ in at least one category.

1975 All Soviet Union Mathematical Olympiad, 212

Tags: inequalities , algebra

Prove that for all the positive numbers $a,b,c$ the following inequality is valid: $$a^3+b^3+c^3+3abc>ab(a+b)+bc(b+c)+ac(a+c)$$

1996 AMC 12/AHSME, 1

Tags:

The addition below is incorrect. What is the largest digit that can be changed to make the addition correct? \[ \begin{array}{cccc} & 6 & 4 & 1 \\ & 8 & 5 & 2 \\ + & 9 & 7 & 3 \\ \hline 2 & 4 & 5 & 6 \end{array} \] $\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 8$

2024 Israel National Olympiad (Gillis), P7

Tags: path , grid , rook , combinatorics

A rook stands in one cell of an infinite square grid. A different cell was colored blue and mines were placed in $n$ additional cells: the rook cannot stand on or pass through them. It is known that the rook can reach the blue cell in finitely many moves. Can it do so (for every $n$ and such a choice of mines, starting point, and blue cell) in at most [b](a)[/b] $1.99n+100$ moves? [b](b)[/b] $2n-2\sqrt{3n}+100$ moves? [b]Remark.[/b] In each move, the rook goes in a vertical or horizontal line.

2012 Estonia Team Selection Test, 5

Tags: inequalities , max , algebra

Let $x, y, z$ be positive real numbers whose sum is $2012$. Find the maximum value of $$ \frac{(x^2 + y^2 + z^2)(x^3 + y^3 + z^3)}{(x^4 + y^4 + z^4)}$$

2016 AMC 10, 22

Tags: number theory , divisor

For some positive integer $n$, the number $110n^3$ has $110$ positive integer divisors, including $1$ and the number $110n^3$. How many positive integer divisors does the number $81n^4$ have? $\textbf{(A) }110 \qquad \textbf{(B) } 191 \qquad \textbf{(C) } 261 \qquad \textbf{(D) } 325 \qquad \textbf{(E) } 425$