Olympiad problems

2003 China National Olympiad, 3

Tags: trigonometry , inequalities

Given a positive integer $n$, find the least $\lambda>0$ such that for any $x_1,\ldots x_n\in \left(0,\frac{\pi}{2}\right)$, the condition $\prod_{i=1}^{n}\tan x_i=2^{\frac{n}{2}}$ implies $\sum_{i=1}^{n}\cos x_i\le\lambda$. [i]Huang Yumin[/i]

1979 Vietnam National Olympiad, 1

Tags: algebra , inequalities

Show that for all $x > 1$ there is a triangle with sides, $x^4 + x^3 + 2x^2 + x + 1, 2x^3 + x^2 + 2x + 1, x^4 - 1.$

2017 IFYM, Sozopol, 4

Tags: combinatorics

$n$ students want to equally partition $m$ identical cakes between themselves. What’s the minimal number of pieces of cake one has to cut, so that the upper condition is satisfied? Each cut increases the number of pieces by 1.

2016 Korea USCM, 8

Tags: linear algebra , nilpotent , matrix , commutator

For a $n\times n$ complex valued matrix $A$, show that the following two conditions are equivalent. (i) There exists a $n\times n$ complex valued matrix $B$ such that $AB-BA=A$. (ii) There exists a positive integer $k$ such that $A^k = O$. ($O$ is the zero matrix.)

2018 PUMaC Team Round, 12

Tags:

In right triangle $\triangle{ABC}$, a square $WXYZ$ is inscribed such that vertices $W$ and $X$ lie on hypotenuse $\overline{AB}$, vertex $Y$ lies on leg $\overline{BC}$, and vertex $Z$ lies on leg $\overline{CA}$. Let $\overline{AY}$ and $\overline{BZ}$ intersect at some point $P$. If the length of each side of square $WXYZ$ is $4$, the length of the hypotenuse $\overline{AB}$ is $60$, and the distance between point $P$ and point $G$, where $G$ denotes the centroid of $\triangle{ABC}$, is $\tfrac{a}{b}$, compute the value of $a+b$.

2024 International Zhautykov Olympiad, 3

Tags: number theory

Positive integer $d$ is not perfect square. For each positive integer $n$, let $s(n)$ denote the number of digits $1$ among the first $n$ digits in the binary representation of $\sqrt{d}$ (including the digits before the point). Prove that there exists an integer $A$ such that $s(n)>\sqrt{2n}-2$ for all integers $n\ge A$

2018 Balkan MO Shortlist, A3

Tags: inequalities

Show that for every positive integer $n$ we have: $$\sum_{k=0}^{n}\left(\frac{2n+1-k}{k+1}\right)^k=\left(\frac{2n+1}{1}\right)^0+\left(\frac{2n}{2}\right)^1+...+\left(\frac{n+1}{n+1}\right)^n\leq 2^n$$ [i]Proposed by Dorlir Ahmeti, Albania[/i]

2016 Ukraine Team Selection Test, 8

Tags: geometry , incircle , circumcircle , parallel , incenter

Let $ABC$ be an acute triangle with $AB<BC$. Let $I$ be the incenter of $ABC$, and let $\omega$ be the circumcircle of $ABC$. The incircle of $ABC$ is tangent to the side $BC$ at $K$. The line $AK$ meets $\omega$ again at $T$. Let $M$ be the midpoint of the side $BC$, and let $N$ be the midpoint of the arc $BAC$ of $\omega$. The segment $NT$ intersects the circumcircle of $BIC$ at $P$. Prove that $PM\parallel AK$.

1995 AMC 12/AHSME, 18

Tags: trigonometry , trig identities , law of cosines , law of sines , geometry

Two rays with common endpoint $O$ forms a $30^\circ$ angle. Point $A$ lies on one ray, point $B$ on the other ray, and $AB = 1$. The maximum possible length of $OB$ is $\textbf{(A)}\ 1 \qquad \textbf{(B)}\ \dfrac{1+\sqrt{3}}{\sqrt{2}} \qquad \textbf{(C)}\ \sqrt{3} \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ \dfrac{4}{\sqrt{3}}$

1990 Canada National Olympiad, 3

Tags: geometry , incenter , cyclic quadrilateral , angle bisector

The feet of the perpendiculars from the intersection point of the diagonals of a convex cyclic quadrilateral to the sides form a quadrilateral $q$. Show that the sum of the lengths of each pair of opposite sides of $q$ is equal.

2010 Contests, 3

Tags: calculus , integration , analytic geometry , geometry , pigeonhole principle , combinatorics

[b](a)[/b]Prove that every pentagon with integral coordinates has at least two vertices , whose respective coordinates have the same parity. [b](b)[/b]What is the smallest area possible of pentagons with integral coordinates. Albanian National Mathematical Olympiad 2010---12 GRADE Question 3.

PEN J Problems, 19

Tags: divisor function

Prove that $\sigma(n)\phi(n) < n^2$, but that there is a positive constant $c$ such that $\sigma(n)\phi(n) \ge c n^2$ holds for all positive integers $n$.

2023 CCA Math Bonanza, L1.4

Tags:

Find the area of the shaded region. [i]Lightning 1.4[/i]

2006 France Team Selection Test, 2

Tags: geometry , circumcircle , homothety , triangle -incenter

Given a triangle $ABC$ satisfying $AC+BC=3\cdot AB$. The incircle of triangle $ABC$ has center $I$ and touches the sides $BC$ and $CA$ at the points $D$ and $E$, respectively. Let $K$ and $L$ be the reflections of the points $D$ and $E$ with respect to $I$. Prove that the points $A$, $B$, $K$, $L$ lie on one circle. [i]Proposed by Dimitris Kontogiannis, Greece[/i]

2012 Pre - Vietnam Mathematical Olympiad, 2

Tags: polynomial , algebra

Let $(a_n)$ defined by: $a_0=1, \; a_1=p, \; a_2=p(p-1)$, $a_{n+3}=pa_{n+2}-pa_{n+1}+a_n, \; \forall n \in \mathbb{N}$. Knowing that (i) $a_n>0, \; \forall n \in \mathbb{N}$. (ii) $a_ma_n>a_{m+1}a_{n-1}, \; \forall m \ge n \ge 0$. Prove that $|p-1| \ge 2$.

2022 IOQM India, 7

Tags:

Find the number of maps $f: \{1,2,3\} \rightarrow \{1,2,3,4,5\}$ such that $f(i) \le f(j)$ whenever $i < j$.

2007 Pan African, 3

Tags: geometry , 3d geometry , combinatorics

In a country, towns are connected by roads. Each town is directly connected to exactly three other towns. Show that there exists a town from which you can make a round-trip, without using the same road more than once, and for which the number of roads used is not divisible by $3$. (Not all towns need to be visited.)

2023 Germany Team Selection Test, 3

Tags: polynomial , number theory , prime number , algebra

Let $f(x)$ be a monic polynomial of degree $2023$ with positive integer coefficients. Show that for any sufficiently large integer $N$ and any prime number $p>2023N$, the product \[f(1)f(2)\dots f(N)\] is at most $\binom{2023}{2}$ times divisible by $p$. [i]Proposed by Ashwin Sah[/i]

2005 National High School Mathematics League, 1

Tags: inequalities

The maximum value of $k$ such that the enequality $\sqrt{x-3}+\sqrt{6-x}\geq k$ has a real solution is $\text{(A)}\sqrt6-\sqrt3\qquad\text{(B)}\sqrt3\qquad\text{(C)}\sqrt3+\sqrt6\qquad\text{(D)}\sqrt6$

2017 AIME Problems, 9

Tags:

A special deck of cards contains $49$ cards, each labeled with a number from $1$ to $7$ and colored with one of seven colors. Each number-color combination appears on exactly one card. Sharon will select a set of eight cards from the deck at random. Given that she gets at least one card of each color and at least one card with each number, the probability that Sharon can discard one of her cards and [i]still[/i] have at least one card of each color and at least one card with each number is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

1972 Canada National Olympiad, 4

Tags: parallelogram , analytic geometry , geometry

Describe a construction of quadrilateral $ABCD$ given: (i) the lengths of all four sides; (ii) that $AB$ and $CD$ are parallel; (iii) that $BC$ and $DA$ do not intersect.

2006 Croatia Team Selection Test, 1

Tags: number theory

Find all natural numbers that can be expressed in a unique way as a sum of ﬁve or less perfect squares.

2017 District Olympiad, 4

Tags: superior algebra , idempotence , ring theory

Let $ A $ be a ring that is not a division ring, and such that any non-unit of it is idempotent. Show that: [b]a)[/b] $ \left( U(A) +A\setminus\left( U(A)\cup \{ 0\} \right) \right)\cap U(A) =\emptyset . $ [b]b)[/b] Every element of $ A $ is idempotent.

2017-2018 SDML (Middle School), 4

Tags:

The diagram below shows an equilateral triangle and a square of side length $2$ joined along an edge. What is the area of the shaded triangle? [asy] fill((2,0) -- (2,2) -- (1, 2 + sqrt(3)) -- cycle, gray); draw((0,0) -- (2,0) -- (2,2) -- (1, 2 + sqrt(3)) -- (0,2) -- (0,0)); draw((0,2) -- (2,2)); [/asy]

2023 Dutch Mathematical Olympiad, 1

Tags: number theory

A number is called [i]nillless [/i] if it is integer and positive and contains no zeros. You can make a positive integer nillless by simply omitting the zeros. We denote this with square brackets, for example $[2050] = 25$ and $[13] = 13$. When we multiply, add, and subtract we indicate with square brackets when we omit the zeros. For example, $[[4 \cdot 5] + 7] = [[20] + 7] = [2 + 7] = [9] = 9$ and $[[5 + 5] + 9] = [[10] + 9] = [1 + 9] = [10] = 1$. The following is known about the two numbers $a$ and $b$: $\bullet$ $a$ and $b$ are nillless, $\bullet$ $1 < a < b < 100$, $\bullet$ $[[a \cdot b] - 1] = 1$. Which pairs $(a, b)$ satisfy these three requirements?