Olympiad problems

1986 IMO Longlists, 49

Let $C_1, C_2$ be circles of radius $1/2$ tangent to each other and both tangent internally to a circle $C$ of radius $1$. The circles $C_1$ and $C_2$ are the first two terms of an infinite sequence of distinct circles $C_n$ defined as follows: $C_{n+2}$ is tangent externally to $C_n$ and $C_{n+1}$ and internally to $C$. Show that the radius of each $C_n$ is the reciprocal of an integer.

2006 MOP Homework, 6

Tags: symmetry , geometry

Let $P$ be a convex polygon in the plane. A real number is assigned to each point in the plane so that the sum of the numbers assigned to the vertices of any polygon similar to $P$ is equal to $0$. Prove that all the assigned numbers are equal to $0$.

1994 Austrian-Polish Competition, 4

Tags: combinatorics , polygon

The vertices of a regular $n + 1$-gon are denoted by $P_0,P_1,...,P_n$ in some order ($n \ge 2$). Each side of the polygon is assigned a natural number as follows: if the endpoints of the side are $P_i$ and $P_j$, then the assigned number equals $|i - j |$. Let S be the sum of all $n + 1$ assigned numbers. (a) Given $n$, what is the smallest possible value of $S$? (b) If $P_0$ is fixed, how many different assignments are there for which $S$ attains the smallest value?

1998 AIME Problems, 8

Tags: limit

Except for the first two terms, each term of the sequence $1000, x, 1000-x,\ldots$ is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first negative term encounted. What positive integer $x$ produces a sequence of maximum length?

1999 Greece JBMO TST, 5

Tags: geometry , locus , locus problems , area of triangle , area , symmetry

$\Phi$ is the union of all triangles that are symmetric of the triangle $ABC$ wrt a point $O$, as point $O$ moves along the triangle's sides. If the area of the triangle is $E$, find the area of $\Phi$.

KoMaL A Problems 2024/2025, A. 892

Tags: number theory

Given two integers, $k$ and $d$ such that $d$ divides $k^3 - 2$. Show that there exists integers $a$, $b$, $c$ satisfying $d = a^3 + 2b^3 + 4c^3 - 6abc$. [i]Proposed by Csongor Beke and László Bence Simon, Cambridge[/i]

2014 Turkey MO (2nd round), 2

Tags: modular arithmetic , number theory

Find all all positive integers x,y,and z satisfying the equation $x^3=3^y7^z+8$

2002 All-Russian Olympiad Regional Round, 11.1

Tags: number theory , algebra , rational , irrational

The real numbers $x$ and $y$ are such that for any distinct odd primes $p$ and $q$ the number $x^p + y^q$ is rational. Prove that $x$ and $y$ are rational numbers.

2022 Durer Math Competition Finals, 13

Tags: geometry , tangent circle

Circle $k_1$ has radius $10$, externally touching circle $k_2$ with radius $18$. Circle $k_3$ touches both circles, as well as the line $e$ determined by their centres. Let $k_4$ be the circle touching $k_2$ and $k_3$ externally (other than $k_1$) whose center lies on line $e$. What is the radius of $k_4$?

2022 HMNT, 3

Tags: geometry

Let $ABCD$ be a rectangle with $AB=8$ and $AD=20$. Two circles of radius $5$ are drawn with centers in the interior of the rectangle - one tangent to $AB$ and $AD$, and the other passing through both $C$ and $D$. What is the area inside the rectangle and outside of both circles?

2011 Akdeniz University MO, 4

Tags: geometry , circumcircle

Let an acute-angled triangle $ABC$'s circumcircle is $S$. $S$'s tangent from $B$ and $C$ intersects at point $M$. A line, lies $M$ and parallel to $[AB]$ intersects with $S$ at points $D$ and $E$, intersect with $[AC]$ at point $F$. Prove that $$[DF]=[FE]$$

1990 National High School Mathematics League, 2

Tags:

$E=\{1,2,\cdots,200\},G=\{a_1,a_2,\cdots,a_{100}\}\subset E$. $G$ satisfies the following: (1)For any $1\geq i<j\geq100$, a_i+a_j\neq201. (2)$\sum_{i=1}^{100}a_i=10080$. Prove that the number of odd numbers in $G$ is a multiple of $4$, and the sum the square of all numbers in $G$ is fixed.

PEN P Problems, 3

Tags: function , algebra , floor function , additive number theory

Prove that infinitely many positive integers cannot be written in the form \[{x_{1}}^{3}+{x_{2}}^{5}+{x_{3}}^{7}+{x_{4}}^{9}+{x_{5}}^{11},\] where $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\in \mathbb{N}$.

2015 CCA Math Bonanza, I7

Tags:

Harry Potter would like to purchase a new owl which cost him 2 Galleons, a Sickle, and 5 Knuts. There are 23 Knuts in a Sickle and 17 Sickles in a Galleon. He currently has no money, but has many potions, each of which are worth 9 Knuts. How many potions does he have to exhange to buy this new owl? [i]2015 CCA Math Bonanza Individual Round #7[/i]

2013 Online Math Open Problems, 2

Tags:

The number $123454321$ is written on a blackboard. Evan walks by and erases some (but not all) of the digits, and notices that the resulting number (when spaces are removed) is divisible by $9$. What is the fewest number of digits he could have erased? [i]Ray Li[/i]

1983 USAMO, 2

Tags: calculus , derivative , quadratic , algebra , marvio

Prove that the roots of\[x^5 + ax^4 + bx^3 + cx^2 + dx + e = 0\] cannot all be real if $2a^2 < 5b$.

2009 Mediterranean Mathematics Olympiad, 1

Tags: system of equations , algebra , inequalities

Determine all integers $n\ge1$ for which there exists $n$ real numbers $x_1,\ldots,x_n$ in the closed interval $[-4,2]$ such that the following three conditions are fulfilled: - the sum of these real numbers is at least $n$. - the sum of their squares is at most $4n$. - the sum of their fourth powers is at least $34n$. [i](Proposed by Gerhard Woeginger, Austria)[/i]

2002 Tuymaada Olympiad, 3

Tags: algebra , polynomial , integer polynomial , trinomial , quadratic trinomial , power of 2

Is there a quadratic trinomial with integer coefficients, such that all values which are natural to be natural powers of two?

2010 AMC 12/AHSME, 19

Tags: probability , ellipse , inequalities , conic

Each of 2010 boxes in a line contains a single red marble, and for $ 1 \le k \le 2010$, the box in the $ kth$ position also contains $ k$ white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops when she first draws a red marble. Let $ P(n)$ be the probability that Isabella stops after drawing exactly $ n$ marbles. What is the smallest value of $ n$ for which $ P(n) < \frac {1}{2010}$? $ \textbf{(A)}\ 45 \qquad \textbf{(B)}\ 63 \qquad \textbf{(C)}\ 64 \qquad \textbf{(D)}\ 201 \qquad \textbf{(E)}\ 1005$

1989 AMC 8, 12

Tags:

$\frac{1-\frac{1}{3}}{1-\frac{1}{2}} =$ $\text{(A)}\ \frac{1}{3} \qquad \text{(B)}\ \frac{2}{3} \qquad \text{(C)}\ \frac{3}{4} \qquad \text{(D)}\ \frac{3}{2} \qquad \text{(E)}\ \frac{4}{3}$

2021 MIG, 19

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Aprameya graphs the equation $2x = y + 4$ on the coordinate plane. It turns out that there is a unique point with a positive integer coordinate and a negative integer coordinate lying on Aprameya's graph. What is the sum of the coordinates of this point? $\textbf{(A) }{-}3\qquad\textbf{(B) }{-}1\qquad\textbf{(C) }0\qquad\textbf{(D) }1\qquad\textbf{(E) }2$

2022 CCA Math Bonanza, I14

Tags:

Let $ABC$ be a triangle with side lengths $AB=6, AC=7,$ and $BC=8.$ Let $H$ be the orthocenter of $\triangle ABC$ and $H'$ be the reflection of $H$ across the midpoint $M$ of $BC.$ $\tfrac{[ABH']}{[ACH']}$ can be expressed as $\frac{p}{q}$. Find $p+q$. [i]2022 CCA Math Bonanza Individual Round #14[/i]

1966 Miklós Schweitzer, 2

Tags: advanced fields

Characterize those configurations of $ n$ coplanar straight lines for which the sum of angles between all pairs of lines is maximum. [i]L.Fejes-Toth, A. Heppes[/i]

2021 Iran MO (2nd Round), 4

Tags: geometry , combinatorics , combinatorial geometry

$n$ points are given on a circle $\omega$. There is a circle with radius smaller than $\omega$ such that all these points lie inside or on the boundary of this circle. Prove that we can draw a diameter of $\omega$ with endpoints not belonging to the given points such that all the $n$ given points remain in one side of the diameter.

2005 Today's Calculation Of Integral, 91

Tags: calculus , integration , calculus computations

Prove the following inequality. \[ \sum_{n=0}^\infty \int_0^1 x^{4011} (1-x^{2006})^\frac{n-1}{2006}\ dx<\frac{2006}{2005} \]