Olympiad problems

1953 AMC 12/AHSME, 4

Tags: algebra , polynomial

The roots of $ x(x^2\plus{}8x\plus{}16)(4\minus{}x)\equal{}0$ are: $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 0,4 \qquad\textbf{(C)}\ 0,4,\minus{}4 \qquad\textbf{(D)}\ 0,4,\minus{}4,\minus{}4 \qquad\textbf{(E)}\ \text{none of these}$

2021 AMC 12/AHSME Spring, 6

Tags: AMC , AMC 10 , AMC 10 B

An inverted cone with base radius $12 \text{ cm}$ and height $18 \text{ cm}$ is full of water. The water is poured into a tall cylinder whose horizontal base has a radius of $24 \text{ cm}$. What is the height in centimeters of the water in the cylinder? $\textbf{(A) }1.5 \qquad \textbf{(B) }3 \qquad \textbf{(C) }4 \qquad \textbf{(D) }4.5 \qquad \textbf{(E) }6$

2000 Junior Balkan Team Selection Tests - Romania, 1

Tags: inequalities

Let be a natural number $ n\ge 2, n $ real numbers $ b_1,b_2,\ldots ,b_n , $ and $ n-1 $ positive real numbers $ a_1,a_2,\ldots ,a_{n-1} $ such that $ a_1+a_2+\cdots +a_{n-1} =1. $ Prove the inequality $$ b_1^2+\frac{b_2^2}{a_1} +\frac{b_3^2}{a_2} +\cdots +\frac{b_n^2}{a_{n-1}} \ge 2b_1\left( b_2+b_3+\cdots +b_n \right) , $$ and specify when equality is attained. [i]Dumitru Acu[/i]

2018 AMC 10, 18

Tags:

How many nonnegative integers can be written in the form $$a_7\cdot3^7+a_6\cdot3^6+a_5\cdot3^5+a_4\cdot3^4+a_3\cdot3^3+a_2\cdot3^2+a_1\cdot3^1+a_0\cdot3^0,$$ where $a_i\in \{-1,0,1\}$ for $0\le i \le 7$? $\textbf{(A) } 512 \qquad \textbf{(B) } 729 \qquad \textbf{(C) } 1094 \qquad \textbf{(D) } 3281 \qquad \textbf{(E) } 59,048 $

Kvant 2020, M2606

Tags: geometry , Kvant

Three circles $\omega_1,\omega_2$ and $\omega_3$ pass through one point $D{}$. Let $A{}$ be the intersection of $\omega_1$ and $\omega_3$, and $E{}$ be the intersections of $\omega_3$ and $\omega_2$ and $F{}$ be the intersection of $\omega_2$ and $\omega_1$. It is known that $\omega_3$ passes through the center $B{}$ of the circle $\omega_2$. The line $EF$ intersects $\omega_1$ a second time at the point $G{}$. Prove that $\angle GAB=90^\circ$. [i]Proposed by K. Knop[/i]

2021 Caucasus Mathematical Olympiad, 1

Tags: algebra , CMO

Let $a$, $b$, $c$ be real numbers such that $a^2+b=c^2$, $b^2+c=a^2$, $c^2+a=b^2$. Find all possible values of $abc$.

2013 AMC 10, 1

Tags: MATHCOUNTS , AMC

What is $\frac{2+4+6}{1+3+5}-\frac{1+3+5}{2+4+6}$? $\textbf{(A) }-1\qquad\textbf{(B) }\frac5{36}\qquad\textbf{(C) }\frac7{12}\qquad\textbf{(D) }\frac{49}{20}\qquad\textbf{(E) }\frac{43}3$

1990 China National Olympiad, 4

Tags: algebra , system of equations , number theory unsolved , number theory

Given a positive integer number $a$ and two real numbers $A$ and $B$, find a necessary and sufficient condition on $A$ and $B$ for the following system of equations to have integer solution: \[ \left\{\begin{array}{cc} x^2+y^2+z^2=(Ba)^2\\ x^2(Ax^2+By^2)+y^2(Ay^2+Bz^2)+z^2(Az^2+Bx^2)=\dfrac{1}{4}(2A+B)(Ba)^4\end{array}\right. \]

1992 AMC 12/AHSME, 18

Tags: inequalities , AMC

The increasing sequence of positive integers $a_{1},a_{2},a_{3},\ldots$ has the property that $a_{n+2} = a_{n} + a_{n+1}$ for all $n \ge 1$. If $a_{7} = 120$, then $a_{8}$ is $ \textbf{(A)}\ 128\qquad\textbf{(B)}\ 168\qquad\textbf{(C)}\ 193\qquad\textbf{(D)}\ 194\qquad\textbf{(E)}\ 210 $

1976 Putnam, 4

Tags: college contests

Let $r$ be a root of $P(x)=x^3+ax^2+bx-1=0$ and $r+1$ be a root of $y^3+cy^2+dy+1=0,$ where $a,b,c$ and $d$ are integers. Also let $P(x)$ be irreducible over the rational numbers. Express another root $s$ of $P(x)=0$ as a function of $r$ which does not explicitly involve $a,b,c$ or $d.$

2004 Tournament Of Towns, 5

Tags: combinatorics

All sides of a polygonal billiard table are in one of two perpendicular directions. A tiny billiard ball rolls out of the vertex $A$ of an inner $90^o$ angle and moves inside the billiard table, bouncing off its sides according to the law “angle of reflection equals angle of incidence”. If the ball passes a vertex, it will drop in and srays there. Prove that the ball will never return to $A$.

2020 Brazil Cono Sur TST, 1

Tags: combinatorics

Maria have $14$ days to train for an olympiad. The only conditions are that she cannot train by $3$ consecutive days and she cannot rest by $3$ consecutive days. Determine how many configurations of days(in training) she can reach her goal.

1986 Swedish Mathematical Competition, 4

Tags: system of equations , System , algebra

Prove that $x = y = z = 1$ is the only positive solution of the system \[\left\{ \begin{array}{l} x+y^2 +z^3 = 3\\ y+z^2 +x^3 = 3\\ z+x^2 +y^3 = 3\\ \end{array} \right. \]

1958 AMC 12/AHSME, 43

Tags: Pythagorean Theorem , geometry

$ \overline{AB}$ is the hypotenuse of a right triangle $ ABC$. Median $ \overline{AD}$ has length $ 7$ and median $ \overline{BE}$ has length $ 4$. The length of $ \overline{AB}$ is: $ \textbf{(A)}\ 10\qquad \textbf{(B)}\ 5\sqrt{3}\qquad \textbf{(C)}\ 5\sqrt{2}\qquad \textbf{(D)}\ 2\sqrt{13}\qquad \textbf{(E)}\ 2\sqrt{15}$

LMT Team Rounds 2021+, A22 B23

Tags: algebra

A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five. Ada has been told To write down five haikus plus Two more every hour. Such that she needs to Write down five in the first hour Seven, nine, so on. Ada has so far Forty haikus and writes down Seven every hour. At which hour after She begins will she not have Enough haikus done? [i]Proposed by Ada Tsui[/i]

BIMO 2020, 3

Tags: IMOR , geometry

Let $G$ be the centroid of a triangle $\triangle ABC$ and let $AG, BG, CG$ meet its circumcircle at $P, Q, R$ respectively. Let $AD, BE, CF$ be the altitudes of the triangle. Prove that the radical center of circles $(DQR),(EPR),(FPQ)$ lies on Euler Line of $\triangle ABC$. [i]Proposed by Ivan Chai, Malaysia.[/i]

2002 AMC 12/AHSME, 3

Tags: quadratics

For how many positive integers $ n$ is $ n^2\minus{}3n\plus{}2$ a prime number? $ \textbf{(A)}\ \text{none} \qquad \textbf{(B)}\ \text{one} \qquad \textbf{(C)}\ \text{two} \qquad \textbf{(D)}\ \text{more than two, but finitely many}\\ \textbf{(E)}\ \text{infinitely many}$

1978 IMO Shortlist, 12

Tags: geometry , inradius , circumcircle , incenter , Triangle , IMO , IMO 1978

In a triangle $ABC$ we have $AB = AC.$ A circle which is internally tangent with the circumscribed circle of the triangle is also tangent to the sides $AB, AC$ in the points $P,$ respectively $Q.$ Prove that the midpoint of $PQ$ is the center of the inscribed circle of the triangle $ABC.$

2003 Gheorghe Vranceanu, 1

Tags: floor function , algebra , equation

Solve in $ \mathbb{R}^2 $ the equation $ \lfloor x/y-y/x \rfloor =x^2/y+y/x^2. $

2012 Saint Petersburg Mathematical Olympiad, 6

Tags: geometry , parallelogram

$ABCD$ is parallelogram. Line $l$ is perpendicular to $BC$ at $B$. Two circles passes through $D,C$, such that $l$ is tangent in points $P$ and $Q$. $M$ - midpoint $AB$. Prove that $\angle DMP=\angle DMQ$

2018 May Olympiad, 1

Tags: number theory , Digits , Perfect Squares , Perfect Square

You have a $4$-digit whole number that is a perfect square. Another number is built adding $ 1$ to the unit's digit, subtracting $ 1$ from the ten's digit, adding $ 1$ to the hundred's digit and subtracting $ 1$ from the ones digit of one thousand. If the number you get is also a perfect square, find the original number. It's unique?

2010 Moldova National Olympiad, 12.5

Tags: algebra unsolved , algebra

Prove that exists a infinity of triplets $a, b, c\in\mathbb{R}$ satisfying simultaneously the relations $a+b+c=0$ and $a^4+b^4+c^4=50$. Moldova National Math Olympiad 2010, 12th grade

2004 Estonia National Olympiad, 2

Tags: combinatorics , game , game strategy

Albert and Brita play a game with a bar of $19$ adjacent squares. Initially, there is a button on the middle square of the bar. At every turn Albert mentions one positive integer less than $5$, and Brita moves button a number of squares in the direction of her choice - while doing so however, Brita must not move the button more than twice in one direction order. Prove that Albert can choose the numbers so that by the $19$th turn, Brita to be forced to move the button out of the bar.

2000 All-Russian Olympiad Regional Round, 10.1

Tags: number theory , Composite

$2000$ numbers are considered: $11, 101, 1001, . . $. Prove that at least $99\%$ of these numbers are composite.

2021 Durer Math Competition Finals, 13

Tags: geometry , trapezoid , equal angles

The trapezoid $ABCD$ satisfies $AB \parallel CD$, $AB = 70$, $AD = 32$ and $BC = 49$. We also know that $\angle ABC = 3 \angle ADC$. How long is the base $CD$?