Olympiad problems

2007 AMC 8, 1

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Theresa's parents have agreed to buy her tickets to see her favorite band if she spends an average of $10$ hours per week helping around the house for $6$ weeks. For the first $5$ weeks, she helps around the house for $8$, $11$, $7$, $12$ and $10$ hours. How many hours must she work during the final week to earn the tickets? $\textbf{(A)}\ 9 \qquad \textbf{(B)}\ 10 \qquad \textbf{(C)}\ 11 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 13$

2001 Moldova National Olympiad, Problem 4

Tags: geometry

In a triangle $ABC$ the altitude $AD$ is drawn. Points $M$ on side $AC$ and $N$ on side $AB$ are taken so that $\angle MDA=\angle NDA$. Prove that the lines $AD,BM$ and $CN$ are concurrent.

2016 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: limit , sequence , algebra

Let $a_1=1$ and $a_{n+1}=a_{n}+\frac{1}{2a_n}$ for $n \geq 1$. Prove that $a)$ $n \leq a_n^2 < n + \sqrt[3]{n}$ $b)$ $\lim_{n\to\infty} (a_n-\sqrt{n})=0$

2023 MOAA, 12

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Andy is planning to flip a fair coin 10 times. Among the 10 flips, Valencia randomly chooses one flip to exchange Andy's fair coin with her special coin which lands on heads with a probability of $\frac{1}{4}$. If the coin is exchanged in a certain flip, then that flip, along with all following flips will be performed with the special coin. The expected number of heads Andy flips can be expressed as $\frac{m}{n}$ where $m$ and $n$ are positive integers. Find $m+n$. [i]Proposed by Andy Xu[/i]

2023 Durer Math Competition Finals, 5

Tags: geometry

King Minos divided his rectangular island of Crete between his 3 sons as follows: he built a wall along one diagonal of the island and gave one half of the island to his eldest son. Then, in the remaining triangular area, from the right-angled vertex he built a wall perpendicular to the other wall. Of the two areas thus obtained, the larger was given to the middle son and the smaller to the youngest. Each of the three sons had the largest possible square palace built on his own land. How many times is the area of the eldest son’s palace larger than the area of the youngest son’s palace if the side lengths of the island are $30$ m and $210$ m?

KoMaL A Problems 2018/2019, A. 728

Tags: number theory

Floyd the flea makes jumps on the positive integers. On the first day he can jump to any positive integer. From then on, every day he jumps to another number that is not more than twice his previous day's place. [list=a] [*]Show that Floyd can make infinitely many jumps in such a way that he never arrives at any number with the same sum of decimal digits as at a previous place.[/*] [*]Can the flea jump this way if we consider the sum of binary digits instead of decimal digits?[/*] [/list]

2014 Canadian Mathematical Olympiad Qualification, 6

Tags: midpoint , triangle , geometry

Given a triangle $A, B, C, X$ is on side $AB$, $Y$ is on side $AC$, and $P$ and $Q$ are on side $BC$ such that $AX = AY , BX = BP$ and $CY = CQ$. Let $XP$ and $YQ$ intersect at $T$. Prove that $AT$ passes through the midpoint of $PQ$.

2005 Mediterranean Mathematics Olympiad, 1

Tags: combinatorics

The professor tells Peter the product of two positive integers and Sam their sum. At first, nobody of them knows the number of the other. One of them says: [i]You can't possibly guess my number[/i]. Then the other says: [i]You are wrong, the number is 136[/i]. Which number did the professor tell them respectively? Give reasons for your claim.

2013 NIMO Summer Contest, 6

Tags: geometry

Let $ABC$ and $DEF$ be two triangles, such that $AB=DE=20$, $BC=EF=13$, and $\angle A = \angle D$. If $AC-DF=10$, determine the area of $\triangle ABC$. [i]Proposed by Lewis Chen[/i]

1986 IMO Longlists, 44

Tags: incenter , circumcircle , collinearity , geometry

The circle inscribed in a triangle $ABC$ touches the sides $BC,CA,AB$ in $D,E, F$, respectively, and $X, Y,Z$ are the midpoints of $EF, FD,DE$, respectively. Prove that the centers of the inscribed circle and of the circles around $XYZ$ and $ABC$ are collinear.

2012 JBMO ShortLists, 4

Tags: geometry , circumcircle , rhombus , angle bisector

Let $ABC$ be an acute-angled triangle with circumcircle $\omega$ , and let $O$ , $H$ be the triangle's circumcenter and orthocenter respectively . Let also $A^{'}$ be the point where the angle bisector of the angle $BAC$ meets $\omega$ . If $A^{'}H=AH$ , then find the measure of the angle $BAC$.

1998 May Olympiad, 2

Tags: rectangle , combinatorics , combinatorial geometry , geometry

There are $1998$ rectangular pieces $2$ cm wide and $3$ cm long and with them squares are assembled (without overlapping or gaps). What is the greatest number of different squares that can be had at the same time?

2014 NIMO Problems, 2

Tags: induction

I'm thinking of a five-letter word that rhymes with ``angry'' and ``hungry''. What is it?

2000 Junior Balkan Team Selection Tests - Romania, 4

Tags: geometry

On the hypotenuse $ BC $ of an isosceles right triangle $ ABC $ let $ M,N $ such that $ BM^2-MN^2+NC^2=0. $ Show that $ \angle MAN= 45^{\circ } . $ [i]Cristinel Mortici[/i]

2001 CentroAmerican, 2

Tags: quadratic , function , arithmetic sequence , absolute value

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

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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

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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.