Olympiad problems

2002 Tournament Of Towns, 5

Does there exist a regular triangular prism that can be covered (without overlapping) by different equilateral triangles? (One is allowed to bend the triangles around the edges of the prism.)

2021 Saudi Arabia Training Tests, 13

Tags: geometry , equal angles , angle

Let $ABCD$ be a quadrilateral with $\angle A = \angle B = 90^o$, $AB = AD$. Denote $E$ as the midpoint of $AD$, suppose that $CD = BC + AD$, $AD > BC$. Prove that $\angle ADC = 2\angle ABE$.

2015 Romania Team Selection Tests, 2

Tags: graph theory , combinatorics

Let $n$ be an integer greater than $1$, and let $p$ be a prime divisor of $n$. A confederation consists of $p$ states, each of which has exactly $n$ airports. There are $p$ air companies operating interstate flights only such that every two airports in different states are joined by a direct (two-way) flight operated by one of these companies. Determine the maximal integer $N$ satisfying the following condition: In every such confederation it is possible to choose one of the $p$ air companies and $N$ of the $np$ airports such that one may travel (not necessarily directly) from any one of the $N$ chosen airports to any other such only by flights operated by the chosen air company.

2024 MMATHS, 11

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Define a sequence $a_{m,n}$ where $a_{m,0}=1,$ and for all other $m,n$ (assuming $m \ge 1$): $$a_{m,n}=\begin{cases} 0 & n<0 \\ 1 & n \equiv 0 \mod{m} \\ a_{m,n-1}+a_{m, n-m} & \text{else} \end{cases}$$ If $\tfrac{a_{2025, (2025^2-1)}}{a_{2025, (2024^2-1)}} = \tfrac{a}{b}$ where $a$ and $b$ are relatively prime positive integers, then what is $a+b$?

1986 IMO Longlists, 49

Tags: rectangle , induction , power of a point , radical axis , geometry

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

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$E=\{1,2,\cdots,200\},G=\{a_1,a_2,\cdots,a_{100}\}\subset E$. $G$ 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

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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

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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

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

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? $\textbf{(A) }{-}3\qquad\textbf{(B) }{-}1\qquad\textbf{(C) }0\qquad\textbf{(D) }1\qquad\textbf{(E) }2$