Olympiad problems

2006 Moldova National Olympiad, 10.1

Let $a,b$ be the smaller sides of a right triangle. Let $c$ be the hypothenuse and $h$ be the altitude from the right angle. Fint the maximal value of $\frac{c+h}{a+b}$.

2004 Uzbekistan National Olympiad, 2

Tags: geometry , quadratic , number theory

Lenth of a right angle triangle sides are posive integer. Prove that double area of the triangle divides 12.

1998 Tournament Of Towns, 2

Tags: number theory

The units-digit of the square of an integer is 9 and the tens-digit of this square is 0. Prove that the hundreds-digit is even.

How many ordered pairs of real numbers $(x,y)$ satisfy the following system of equations? \begin{align*}x+3y&=3\\ \big||x|-|y|\big|&=1\end{align*} $\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 3 \qquad \textbf{(D) } 4 \qquad \textbf{(E) } 8 $

PEN J Problems, 7

Tags: divisor function

Show that if the equation $\phi(x)=n$ has one solution, it always has a second solution, $n$ being given and $x$ being the unknown.

2023 Harvard-MIT Mathematics Tournament, 8

Tags: combinatorics

A random permutation $a = (a_1, a_2,...,a_{40})$ of $(1, 2,...,40)$ is chosen, with all permutations being equally likely. William writes down a $20 \times 20$ grid of numbers $b_{ij}$ such that $b_{ij} = \max (a_i, a_{j+20})$ for all $1 \le i, j \le 20$, but then forgets the original permutation $a$. Compute the probability that, given the values of $b_{ij}$ alone, there are exactly $2$ permutations $a$ consistent with the grid.

1991 Greece National Olympiad, 1

Tags: algebra , inequalities

Let $a, b$ be two reals such that $a+b<2ab$. Prove that $a+b>2$

2005 Today's Calculation Of Integral, 57

Tags: calculus , integration , trigonometry , calculus computations

Find the value of $n\in{\mathbb{N}}$ satisfying the following inequality. \[\left|\int_0^{\pi} x^2\sin nx\ dx\right|<\frac{99\pi ^ 2}{100n}\]

Novosibirsk Oral Geo Oly VII, 2019.7

Tags: geometry , acute , square

Cut a square into eight acute-angled triangles.

2009 Math Prize For Girls Problems, 15

Tags: number theory , relatively prime

Let $ x \equal{} \sqrt[3]{\frac{4}{25}}\,$. There is a unique value of $ y$ such that $ 0 < y < x$ and $ x^x \equal{} y^y$. What is the value of $ y$? Express your answer in the form $ \sqrt[c]{\frac{a}{b}}\,$, where $ a$ and $ b$ are relatively prime positive integers and $ c$ is a prime number.

1997 Bulgaria National Olympiad, 1

Tags: polynomial , floor function , algebra

Consider the polynomial $P_n(x) = \binom {n}{2}+\binom {n}{5}x+\binom {n}{8}x^2 + \cdots + \binom {n}{3k+2}x^{3k}$ where $n \ge 2$ is a natural number and $k = \left\lfloor \frac{n-2}{3} \right \rfloor$ [b](a)[/b] Prove that $P_{n+3}(x)=3P_{n+2}(x)-3P_{n+1}(x)+(x+1)P_n(x)$ [b](b)[/b] Find all integer numbers $a$ such that $P_n(a^3)$ is divisible by $3^{ \lfloor \frac{n-1}{2} \rfloor}$ for all $n \ge 2$

2023 Turkey EGMO TST, 3

Tags: inequalities , minimum value

Let $x,y,z$ be positive real numbers that satisfy at least one of the inequalities, $2xy>1$, $yz>1$. Find the least possible value of $$xy^3z^2+\frac{4z}{x}-8yz-\frac{4}{yz}$$ .

2019 Moldova EGMO TST, 5

Tags: number theory

Prove that the number $a=2019^{2020}+4^{2019}$ is a composite number.

MOAA Gunga Bowls, 2023.5

Tags:

Andy creates a 3 sided dice with a side labeled $7$, a side labeled $17$, and a side labeled $27$. He then asks Anthony to roll the dice $3$ times. The probability that the product of Anthony's rolls is greater than $2023$ can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Proposed by Andy Xu[/i]

2016 Balkan MO Shortlist, N1

Tags: coprime , phi function , number theory

Find all natural numbers $n$ for which $1^{\phi (n)} + 2^{\phi (n)} +... + n^{\phi (n)}$ is coprime with $n$.

LMT Team Rounds 2021+, 9

Tags: geometry

Points $X$ and $Y$ on the unit circle centered at $O = (0,0)$ are at $(-1,0)$ and $(0,-1)$ respectively. Points $P$ and $Q$ are on the unit circle such that $\angle P XO = \angle QY O = 30^o$. Let $Z$ be the intersection of line $X P$ and line $Y Q$. The area bounded by segment $Z P$, segment $ZQ$, and arc $PQ$ can be expressed as $a\pi -b$ where $a$ and $b$ are rational numbers. Find $\frac{1}{ab}$ .

PEN G Problems, 27

Tags: irrational number , limit , calculus , function , derivative , integration

Let $1<a_{1}<a_{2}<\cdots$ be a sequence of positive integers. Show that \[\frac{2^{a_{1}}}{{a_{1}}!}+\frac{2^{a_{2}}}{{a_{2}}!}+\frac{2^{a_{3}}}{{a_{3}}!}+\cdots\] is irrational.

2016 AIME Problems, 7

Tags: complex number

For integers $a$ and $b$ consider the complex number \[\dfrac{\sqrt{ab+2016}}{ab+100} - \left(\frac{\sqrt{|a+b|}}{ab+100}\right)i.\] Find the number of ordered pairs of integers $(a, b)$ such that this complex number is a real number.

2024 Portugal MO, 3

Tags: combinatorics

A sequence composed by $0$s and $1$s has at most two consecutive $0$s. How many sequences of length $10$ exist?

1955 Moscow Mathematical Olympiad, 318

Tags: point , maximum , combinatorics , combinatorial geometry

What greatest number of triples of points can be selected from $1955$ given points, so that each two triples have one common point?

2002 Singapore MO Open, 3

Tags: min , algebra , binomial , permutation , inequalities

Let $n$ be a positive integer. Determine the smallest value of the sum $a_1b_1+a_2b_2+...+a_{2n+2}b_{2n+2}$ where $(a_1,a_2,...,a_{2n+2})$ and $(b_1,b_2,...,b_{2n+2})$ are rearrangements of the binomial coefficients $2n+1 \choose 0$, $2n+1 \choose 1$,...,$2n+1 \choose 2n+1$. Justify your answer

2006 India IMO Training Camp, 1

Tags: geometry , inradius , circumcircle , inequalities

Let $ABC$ be a triangle with inradius $r$, circumradius $R$, and with sides $a=BC,b=CA,c=AB$. Prove that \[\frac{R}{2r} \ge \left(\frac{64a^2b^2c^2}{(4a^2-(b-c)^2)(4b^2-(c-a)^2)(4c^2-(a-b)^2)}\right)^2.\]

2005 Federal Competition For Advanced Students, Part 1, 4

Tags: parallelogram , geometric transformation , homothety , reflection , geometry

We're given two congruent, equilateral triangles $ABC$ and $PQR$ with parallel sides, but one has one vertex pointing up and the other one has the vertex pointing down. One is placed above the other so that the area of intersection is a hexagon $A_1A_2A_3A_4A_5A_6$ (labelled counterclockwise). Prove that $A_1A_4$, $A_2A_5$ and $A_3A_6$ are concurrent.

2023 Romania Team Selection Test, P5

Tags: geometry , inequalities

Let $ABCDEF$ be a convex hexagon. The diagonals $AC$ and $BD$ cross at $P,$ the diagonals $AE{}$ and $DF$ cross at $Q,$ and the line $PQ$ crosses the sides $BC$ and $EF$ at $X$ and $Y,{}$ respectively. Prove that the length of the segment $XY$ does not exceed the sum of the lengths of one of the diagonals through $P{}$ and one of the diagonals through $Q{}$. [i]The Problem Selection Committee[/i]

2017 Czech And Slovak Olympiad III A, 5

Tags: arc midpoint , concyclic , circumcircle , geometry

Given is the acute triangle $ABC$ with the intersection of altitudes $H$. The angle bisector of angle $BHC$ intersects side $BC$ at point $D$. Mark $E$ and $F$ the symmetrics of the point $D$ wrt lines $AB$ and $AC$. Prove that the circle circumscribed around the triangle $AEF$ passes through the midpoint of the arc $BAC$