Olympiad problems

2009 Math Prize For Girls Problems, 15

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 $BAC$

Denmark (Mohr) - geometry, 2012.1

Tags: geometry , circles , area

Inside a circle with radius $6$ lie four smaller circles with centres $A,B,C$ and $D$. The circles touch each other as shown. The point where the circles with centres $A$ and $C$ touch each other is the centre of the big circle. Calculate the area of quadrilateral $ABCD$. [img]https://1.bp.blogspot.com/-FFsiOOdcjao/XzT_oJYuQAI/AAAAAAAAMVk/PpyUNpDBeEIESMsiElbexKOFMoCXRVaZwCLcBGAsYHQ/s0/2012%2BMohr%2Bp1.png[/img]

2004 Germany Team Selection Test, 2

Tags: geometry , locus problems , geometric inequalities

Let $d$ be a diameter of a circle $k$, and let $A$ be an arbitrary point on this diameter $d$ in the interior of $k$. Further, let $P$ be a point in the exterior of $k$. The circle with diameter $PA$ meets the circle $k$ at the points $M$ and $N$. Find all points $B$ on the diameter $d$ in the interior of $k$ such that \[\measuredangle MPA = \measuredangle BPN \quad \text{and} \quad PA \leq PB.\] (i. e. give an explicit description of these points without using the points $M$ and $N$).

2021 CCA Math Bonanza, I4

Tags:

Given that nonzero real numbers $x$ and $y$ satisfy $x+\frac{1}{y}=3$ and $y+\frac{1}{x}=4$, what is $xy+\frac{1}{xy}$? [i]2021 CCA Math Bonanza Individual Round #4[/i]

Indonesia MO Shortlist - geometry, g10

Tags: geometry , incenter , circles

Given two circles with one of the centers of the circle is on the other circle. The two circles intersect at two points $C$ and $D$. The line through $D$ intersects the two circles again at $A$ and $ B$. Let $H$ be the midpoint of the arc $AC$ that does not contain $D$ and the segment $HD$ intersects circle that does not contain $H$ at point $E$. Show that $E$ is the center of the incircle of the triangle $ACD$.

2016 AMC 12/AHSME, 1

Tags: factorial , fraction

What is the value of $\dfrac{11!-10!}{9!}$? $\textbf{(A)}\ 99\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 121\qquad\textbf{(E)}\ 132$

2010 South africa National Olympiad, 2

Tags: trigonometry , geometry

Consider a triangle $ABC$ with $BC = 3$. Choose a point $D$ on $BC$ such that $BD = 2$. Find the value of \[AB^2 + 2AC^2 - 3AD^2.\]

2004 Cono Sur Olympiad, 5

Tags: combinatorial geometry , combinatorics

Using cardboard equilateral triangles of side length $1$, an equilateral triangle of side length $2^{2004}$ is formed. An equilateral triangle of side $1$ whose center coincides with the center of the large triangle is removed. Determine if it is possible to completely cover the remaining surface, without overlaps or holes, using only pieces in the shape of an isosceles trapezoid, each of which is created by joining three equilateral triangles of side $1$.

1985 IMO Longlists, 71

Tags: inequalities , number theory

For every integer $r > 1$ find the smallest integer $h(r) > 1$ having the following property: For any partition of the set $\{1, 2, . . ., h(r)\}$ into $r$ classes, there exist integers $a \geq 0, 1 \leq x \leq y$ such that the numbers $a + x, a + y, a + x + y$ are contained in the same class of the partition.

2010 Romania Team Selection Test, 2

Tags: geometry , circumcircle , incenter

Let $ABC$ be a scalene triangle, let $I$ be its incentre, and let $A_1$, $B_1$ and $C_1$ be the points of contact of the excircles with the sides $BC$, $CA$ and $AB$, respectively. Prove that the circumcircles of the triangles $AIA_1$, $BIB_1$ and $CIC_1$ have a common point different from $I$. [i]Cezar Lupu & Vlad Matei[/i]