Olympiad problems

2001 CentroAmerican, 2

Let $ a,b$ and $ c$ real numbers such that the equation $ ax^2\plus{}bx\plus{}c\equal{}0$ has two distinct real solutions $ p_1,p_2$ and the equation $ cx^2\plus{}bx\plus{}a\equal{}0$ has two distinct real solutions $ q_1,q_2$. We know that the numbers $ p_1,q_1,p_2,q_2$ in that order, form an arithmetic progression. Show that $ a\plus{}c\equal{}0$.

2007 Princeton University Math Competition, 1

Tags: number theory , prime factorization

If you multiply all positive integer factors of $24$, you get $24^x$. Find $x$.

1965 AMC 12/AHSME, 17

Tags:

Given the true statement: The picnic on Sunday will not be held only if the weather is not fair. We can then conclude that: $ \textbf{(A)}\ \text{If the picnic is held, Sunday's weather is undoubtedly fair.}$ $ \textbf{(B)}\ \text{If the picnic is not held, Sunday's weather is possibly unfair.}$ $ \textbf{(C)}\ \text{If it is not fair Sunday, the picnic will not be held.}$ $ \textbf{(D)}\ \text{If it is fair Sunday, the picnic may be held.}$ $ \textbf{(E)}\ \text{If it is fair Sunday, the picnic must be held.}$

2010 Miklós Schweitzer, 4

Tags: abstract algebra

Prove that if $ n \geq 2 $ and $ I_ {1}, I_ {2}, \ldots, I_ {n} $ are idealized in a unit-element commutative ring such that any nonempty $ H \subseteq \{ 1,2, \dots, n \} $ then if $ \sum_ {h \in H} I_ {h} $ Is ideal $$ I_ {2} I_ {3} I_ {4} \dots I_ {n} + I_ {1} I_ {3} I_ {4} \dots I_ {n} + \dots + I_ {1} I_ {2} \dots I_ {n-1} $$also Is ideal.

1998 Croatia National Olympiad, Problem 3

Tags: geometry , square , triangle , right triangle

Points $E$ and $F$ are chosen on the sides $AB$ and $BC$ respectively of a square $ABCD$ such that $BE=BF$. Let $BN$ be an altitude of the triangle $BCE$. Prove that the triangle $DNF$ is right-angled.

2020 Junior Balkan Team Selection Tests - Moldova, 11

Tags: circumcircle , geometry

Let $\triangle ABC$ be an acute triangle. The bisector of $\angle ACB$ intersects side $AB$ in $D$. The circumcircle of triangle $ADC$ intersects side $BC$ in $C$ and $E$ with $C \neq E$. The line parallel to $AE$ which passes through $B$ intersects line $CD$ in $F$. Prove that the triangle $\triangle AFB$ is isosceles.

2023 HMNT, 8

Tags:

Six standard fair six-sided dice are rolled and arranged in a row at random. Compute the expected number of dice showing the same number as the sixth die in the row.

1953 AMC 12/AHSME, 47

Tags: inequalities , calculus , logarithm

If $ x$ is greater than zero, then the correct relationship is: $ \textbf{(A)}\ \log (1\plus{}x) \equal{} \frac{x}{1\plus{}x} \qquad\textbf{(B)}\ \log (1\plus{}x) < \frac{x}{1\plus{}x} \\ \textbf{(C)}\ \log(1\plus{}x) > x \qquad\textbf{(D)}\ \log (1\plus{}x) < x \qquad\textbf{(E)}\ \text{none of these}$

2018 Moscow Mathematical Olympiad, 1

Tags: algebra , inequalities

$a_1,a_2,...,a_{81}$ are nonzero, $a_i+a_{i+1}>0$ for $i=1,...,80$ and $a_1+a_2+...+a_{81}<0$. What is sign of $a_1*a_2*...*a_{81}$?

2011 China Second Round Olympiad, 3

Tags: inequalities , algebra , logarithm

Let $a,b$ be positive reals such that $\frac{1}{a}+\frac{1}{b}\leq2\sqrt2$ and $(a-b)^2=4(ab)^3$. Find $\log_a b$.

2009 Romania Team Selection Test, 1

Tags: inequalities , geometry

Let $ABCD$ be a circumscribed quadrilateral such that $AD>\max\{AB,BC,CD\}$, $M$ be the common point of $AB$ and $CD$ and $N$ be the common point of $AC$ and $BD$. Show that \[90^{\circ}<m(\angle AND)<90^{\circ}+\frac{1}{2}m(\angle AMD).\] Fixed, thank you Luis.

Revenge EL(S)MO 2024, 3

Tags: square , diophantine equation , algebra

Find all solutions to \[ (abcde)^2 = a^2+b^2+c^2+d^2+e^2+f^2. \] in integers. Proposed by [i]Seongjin Shim[/i]

2013 Today's Calculation Of Integral, 870

Tags: calculus , integration , ellipse , hyperbola , geometry , inequalities , conic

Consider the ellipse $E: 3x^2+y^2=3$ and the hyperbola $H: xy=\frac 34.$ (1) Find all points of intersection of $E$ and $H$. (2) Find the area of the region expressed by the system of inequality \[\left\{ \begin{array}{ll} 3x^2+y^2\leq 3 &\quad \\ xy\geq \frac 34 , &\quad \end{array} \right.\]

2016 Iran Team Selection Test, 4

Tags: algebra

Suppose that a sequence $a_1,a_2,\ldots$ of positive real numbers satisfies \[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\] for every positive integer $k$. Prove that $a_1+a_2+\ldots+a_n\geq n$ for every $n\geq2$.

2017 Adygea Teachers' Geometry Olympiad, 1

Tags: geometry , trapezoid , area

Find the area of the $MNRK$ trapezoid with the lateral side $RK = 3$ if the distances from the vertices $M$ and $N$ to the line $RK$ are $5$ and $7$, respectively.

1993 AMC 8, 25

Tags: geometry

A checkerboard consists of one-inch squares. A square card, $1.5$ inches on a side, is placed on the board so that it covers part or all of the area of each of $n$ squares. The maximum possible value of $n$ is $\text{(A)}\ 4\text{ or }5 \qquad \text{(B)}\ 6\text{ or }7\qquad \text{(C)}\ 8\text{ or }9 \qquad \text{(D)}\ 10\text{ or }11 \qquad \text{(E)}\ 12\text{ or more}$

2018 USA TSTST, 3

Tags: geometry , circumcircle

Let $ABC$ be an acute triangle with incenter $I$, circumcenter $O$, and circumcircle $\Gamma$. Let $M$ be the midpoint of $\overline{AB}$. Ray $AI$ meets $\overline{BC}$ at $D$. Denote by $\omega$ and $\gamma$ the circumcircles of $\triangle BIC$ and $\triangle BAD$, respectively. Line $MO$ meets $\omega$ at $X$ and $Y$, while line $CO$ meets $\omega$ at $C$ and $Q$. Assume that $Q$ lies inside $\triangle ABC$ and $\angle AQM = \angle ACB$. Consider the tangents to $\omega$ at $X$ and $Y$ and the tangents to $\gamma$ at $A$ and $D$. Given that $\angle BAC \neq 60^{\circ}$, prove that these four lines are concurrent on $\Gamma$. [i]Evan Chen and Yannick Yao[/i]

Kharkiv City MO Seniors - geometry, 2012.11.4

Tags: geometry , parallel , incircle

The incircle $\omega$ of triangle $ABC$ touches its sides $BC, CA$ and $AB$ at points $D, E$ and $E$, respectively. Point $G$ lies on circle $\omega$ in such a way that $FG$ is a diameter. Lines $EG$ and $FD$ intersect at point $H$. Prove that $AB \parallel CH$.

Russian TST 2020, P2

Tags: number theory , functional equation

Find all functions $f:\mathbb Z_{>0}\to \mathbb Z_{>0}$ such that $a+f(b)$ divides $a^2+bf(a)$ for all positive integers $a$ and $b$ with $a+b>2019$.

2019 HMNT, 9

Tags: geometry , trapezoid

Let $ABCD$ be an isosceles trapezoid with $AD = BC = 255$ and $AB = 128$. Let $M$ be the midpoint of $CD$ and let $N$ be the foot of the perpendicular from $A$ to $CD$. If $\angle MBC = 90^o$, compute $\tan\angle NBM$.

2022 Adygea Teachers' Geometry Olympiad, 4

Tags: geometry , 3d geometry , pyramid

In a regular hexagonal pyramid $SABCDEF$, a plane is drawn through vertex $A$ and the midpoints of edges $SC$ and $CE$. Find the ratio in which this plane divides the volume of the pyramid.

2014 Contests, 2

Tags: algebra , sum of cubes , sequence , digit , number theory

The first term of a sequence is $2014$. Each succeeding term is the sum of the cubes of the digits of the previous term. What is the $2014$ th term of the sequence?

2004 National Chemistry Olympiad, 56

Tags:

How many structural isomers have the formula $\ce{C3H6Cl2}$? $ \textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4$

2007 Junior Balkan MO, 1

Tags: algebra , quadratic , function , inequalities

Let $a$ be positive real number such that $a^{3}=6(a+1)$. Prove that the equation $x^{2}+ax+a^{2}-6=0$ has no real solution.

2019 Tournament Of Towns, 1

Tags: inequalities , number theory , prime , number of divisors , prime factorization , factor , divisor

Let us call the number of factors in the prime decomposition of an integer $n > 1$ the complexity of $n$. For example, [i]complexity [/i] of numbers $4$ and $6$ is equal to $2$. Find all $n$ such that all integers between $n$ and $2n$ have complexity a) not greater than the complexity of $n$. b) less than the complexity of $n$. (Boris Frenkin)