Olympiad problems

2008 Sharygin Geometry Olympiad, 7

Tags: ratio , geometry

(F.Nilov) Given isosceles triangle $ ABC$ with base $ AC$ and $ \angle B \equal{} \alpha$. The arc $ AC$ constructed outside the triangle has angular measure equal to $ \beta$. Two lines passing through $ B$ divide the segment and the arc $ AC$ into three equal parts. Find the ratio $ \alpha / \beta$.

2022 CMIMC, 2.7 1.3

Tags: geometry

Let $\Gamma_1, \Gamma_2, \Gamma_3$ be three pairwise externally tangent circles with radii $1,2,3,$ respectively. A circle passes through the centers of $\Gamma_2$ and $\Gamma_3$ and is externally tangent to $\Gamma_1$ at a point $P.$ Suppose $A$ and $B$ are the centers of $\Gamma_2$ and $\Gamma_3,$ respectively. What is the value of $\frac{{PA}^2}{{PB}^2}?$ [i]Proposed by Kyle Lee[/i]

2014 AMC 10, 15

Tags:

David drives from his home to the airport to catch a flight. He drives $35$ miles in the first hour, but realizes that he will be $1$ hour late if he continues at this speed. He increases his speed by $15$ miles per hour for the rest of the way to the airport and arrives $30$ minutes early. How many miles is the airport from his home? $\textbf{(A) }140\qquad \textbf{(B) }175\qquad \textbf{(C) }210\qquad \textbf{(D) }245\qquad \textbf{(E) }280\qquad$

2023 ELMO Shortlist, C8

Tags: combinatorics

Let $n\ge3$ be a fixed integer, and let $\alpha$ be a fixed positive real number. There are $n$ numbers written around a circle such that there is exactly one $1$ and the rest are $0$'s. An [i]operation[/i] consists of picking a number $a$ in the circle, subtracting some positive real $x\le a$ from it, and adding $\alpha x$ to each of its neighbors. Find all pairs $(n,\alpha)$ such that all the numbers in the circle can be made equal after a finite number of operations. [i]Proposed by Anthony Wang[/i]

2005 iTest, 28

Tags: probability

Yoknapatawpha County has $500,000$ families. Each family is expected to continue to have children until it has a girl, at which point each family stops having children. If the probability of having a boy is $50\%$, and no families have either fertility problems or multiple children per birthing, how many families are expected to have at least $5$ children?

2000 JBMO ShortLists, 16

Tags: inequalities

Find all the triples $(x,y,z)$ of real numbers such that \[2x\sqrt{y-1}+2y\sqrt{z-1}+2z\sqrt{x-1} \ge xy+yz+zx \]

2003 Germany Team Selection Test, 1

Tags: function , combinatorics

At a chess tournament the winner gets 1 point and the defeated one 0 points. A tie makes both obtaining $\frac{1}{2}$ points. 14 players, none of them equally aged, participated in a competition where everybody played against all the other players. After the competition a ranking was carried out. Of the two players with the same number of points the younger received the better ranking. After the competition Jan realizes that the best three players together got as many points as the last 9 players obtained points together. And Joerg noted that the number of ties was maximal. Determine the number of ties.

1994 Spain Mathematical Olympiad, 1

Tags: arithmetic progression , number theory , perfect square

Prove that if an arithmetic progression contains a perfect square, then it contains infinitely many perfect squares.

2010 IFYM, Sozopol, 4

Tags: natural number , algebra

Let $x,y\in \mathbb{N}$ and $k=\frac{x^2+y^2}{2xy+1}$. Determine all natural values of $k$.

1975 Spain Mathematical Olympiad, 8

Tags: probability , combinatorics

Two real numbers between $0$ and $1$ are randomly chosen. Calculate the probability that any one of them is less than the square of the other.

2010 Today's Calculation Of Integral, 586

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

Evaluate $ \int_0^1 \frac{x^{14}}{x^2\plus{}1}\ dx$.

2020 Dutch BxMO TST, 5

Tags: number theory , product , prime

A set S consisting of $2019$ (different) positive integers has the following property: [i]the product of every 100 elements of $S$ is a divisor of the product of the remaining $1919$ elements[/i]. What is the maximum number of prime numbers that $S$ can contain?

1987 Dutch Mathematical Olympiad, 4

Tags: geometry , 3d geometry , dodecahedron , geometric inequality , tetrahedron

On each side of a regular tetrahedron with edges of length $1$ one constructs exactly such a tetrahedron. This creates a dodecahedron with $8$ vertices and $18$ edges. We imagine that the dodecahedron is hollow. Calculate the length of the largest line segment that fits entirely within this dodecahedron.

2015 Saint Petersburg Mathematical Olympiad, 1

Tags: algebra

$x,y$ are real numbers such that $$x^2+y^2=1 , 20x^3-15x=3$$Find the value of $|20y^3-15y|$.(K. Tyshchuk)

2021 CMIMC Integration Bee, 2

Tags:

$$\int\frac{\ln^2(x)}{x}\,dx$$ [i]Proposed by Connor Gordon[/i]

2001 China Team Selection Test, 3

Tags: function , polynomial , functional equation , algebra

For a given natural number $k > 1$, find all functions $f:\mathbb{R} \to \mathbb{R}$ such that for all $x, y \in \mathbb{R}$, $f[x^k + f(y)] = y +[f(x)]^k$.

2021 CMIMC, 13

Tags: number theory

Let $p=3\cdot 10^{10}+1$ be a prime and let $p_n$ denote the probability that $p\mid (k^k-1)$ for a random $k$ chosen uniformly from $\{1,2,\cdots,n\}$. Given that $p_n\cdot p$ converges to a value $L$ as $n$ goes to infinity, what is $L$? [i]Proposed by Vijay Srinivasan[/i]

2010 Contests, 2

Tags: quadratic , algebra , polynomial

Let $P_1(x) = ax^2 - bx - c$, $P_2(x) = bx^2 - cx - a$, $P_3(x) = cx^2 - ax - b$ be three quadratic polynomials. Suppose there exists a real number $\alpha$ such that $P_1(\alpha) = P_2(\alpha) = P_3(\alpha)$. Prove that $a = b = c$.

2014 NZMOC Camp Selection Problems, 5

Tags: geometry , concyclic

Let $ABC$ be an acute angled triangle. Let the altitude from $C$ to $AB$ meet $AB$ at $C'$ and have midpoint $M$, and let the altitude from $B$ to $AC$ meet $AC$ at $B'$ and have midpoint $N$. Let $P$ be the point of intersection of $AM$ and $BB'$ and $Q$ the point of intersection of $AN$ and $CC'$. Prove that the point $M, N, P$ and $Q$ lie on a circle.

PEN A Problems, 108

Tags: arithmetic sequence , divisibility theory

For each integer $n>1$, let $p(n)$ denote the largest prime factor of $n$. Determine all triples $(x, y, z)$ of distinct positive integers satisfying [list] [*] $x, y, z$ are in arithmetic progression, [*] $p(xyz) \le 3$. [/list]

2000 Iran MO (3rd Round), 1

Tags: number theory

Does there exist a natural number $N$ which is a power of$2$, such that one can permute its decimal digits to obtain a different power of $2$?

2005 AMC 10, 15

Tags: geometry , 3d geometry , factorial , modular arithmetic , number theory

How many positive integer cubes divide $ 3!\cdot 5!\cdot 7!$? $ \textbf{(A)}\ 2\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 4\qquad \textbf{(D)}\ 5\qquad \textbf{(E)}\ 6$

2023 IMC, 6

Tags: combinatorics , matrix , linear algebra

Ivan writes the matrix $\begin{pmatrix} 2 & 3\\ 2 & 4\end{pmatrix}$ on the board. Then he performs the following operation on the matrix several times: [b]1.[/b] he chooses a row or column of the matrix, and [b]2.[/b] he multiplies or divides the chosen row or column entry-wise by the other row or column, respectively. Can Ivan end up with the matrix $\begin{pmatrix} 2 & 4\\ 2 & 3\end{pmatrix}$ after finitely many steps?

2021 Bosnia and Herzegovina Junior BMO TST, 4

Tags: combinatorics

Let $n$ be a nonzero natural number and let $S = \{1, 2, . . . , n\}$. A $3 \times n$ board is called [i]beautiful [/i] if it can be completed with numbers from the set $S$ like this as long as the following conditions are met: $\bullet$ on each line, each number from the set S appears exactly once, $\bullet$ on each column the sum of the products of two numbers on that column is divisible by $n$ (that is, if the numbers $a, b, c$ are written on a column, it must be $ab + bc + ca$ be divisible by $n$). For which values of the natural number $n$ are there beautiful tables ¸and for which values do not exist? Justify your answer.

1950 AMC 12/AHSME, 42

Tags:

The equation $ x^{x^{x}}...\equal{}2$ is satisfied when $x$ is equal to: $\textbf{(A)}\ \infty \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ \sqrt[4]{2} \qquad \textbf{(D)}\ \sqrt{2} \qquad \textbf{(E)}\ \text{None of these}$