Olympiad problems

1962 AMC 12/AHSME, 5

Tags: ratio

If the radius of a circle is increased by $ 1$ unit, the ratio of the new circumference to the new diameter is: $ \textbf{(A)}\ \pi \plus{} 2 \qquad \textbf{(B)}\ \frac{2 \pi \plus{} 1}{2} \qquad \textbf{(C)}\ \pi \qquad \textbf{(D)}\ \frac{2 \pi \minus{} 1}{2} \qquad \textbf{(E)}\ \pi \minus{} 2$

1986 IMO Longlists, 71

Tags: modular arithmetic , geometry

Two straight lines perpendicular to each other meet each side of a triangle in points symmetric with respect to the midpoint of that side. Prove that these two lines intersect in a point on the nine-point circle.

2002 Bulgaria National Olympiad, 2

Tags: circumcircle , inradius , incenter , pythagorean theorem , geometry

Consider the orthogonal projections of the vertices $A$, $B$ and $C$ of triangle $ABC$ on external bisectors of $ \angle ACB$, $ \angle BAC$ and $ \angle ABC$, respectively. Prove that if $d$ is the diameter of the circumcircle of the triangle, which is formed by the feet of projections, while $r$ and $p$ are the inradius and the semiperimeter of triangle $ABC$, prove that $r^2+p^2=d^2$ [i]Proposed by Alexander Ivanov[/i]

1974 AMC 12/AHSME, 26

Tags: calculus , integration

The number of distinct positive integral divisors of $(30)^4$ excluding $1$ and $(30)^4$ is $ \textbf{(A)}\ 100 \qquad\textbf{(B)}\ 125 \qquad\textbf{(C)}\ 123 \qquad\textbf{(D)}\ 30 \qquad\textbf{(E)}\ \text{none of these} $

2019 USMCA, 30

Tags:

Let $ABC$ be a triangle with $BC = a$, $CA = b$, and $AB = c$. The $A$-excircle is tangent to $\overline{BC}$ at $A_1$; points $B_1$ and $C_1$ are similarly defined. Determine the number of ways to select positive integers $a$, $b$, $c$ such that [list] [*] the numbers $-a+b+c$, $a-b+c$, and $a+b-c$ are even integers at most 100, and [*] the circle through the midpoints of $\overline{AA_1}$, $\overline{BB_1}$, and $\overline{CC_1}$ is tangent to the incircle of $\triangle ABC$. [/list]

2022 Belarusian National Olympiad, 9.4

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Numbers $1,2,\ldots,50$ are written on the board. Anya does the following operation: removes the numbers $a$ and $b$ from the board and writes their sum - $a+b$, after which also notes down the number $ab(a+b)$. After $49$ of this operations only one number was left on the board. Anya summed up all the $49$ numbers in her notes and got $S$. a) Prove that $S$ does not depend on the order of Anya's actions. b) Calculate $S$.

2016 Tournament Of Towns, 3

Tags: combinatorics , combinatorial geometry

Given a square with side $10$. Cut it into $100$ congruent quadrilaterals such that each of them is inscribed into a circle with diameter $\sqrt{3}$. [i](5 points)[/i] [i]Ilya Bogdanov[/i]

2013 Mediterranean Mathematics Olympiad, 3

Tags: inequalities

Let $x,y,z$ be positive reals for which: $\sum (xy)^{2}=6xyz$ Prove that: $\sum \sqrt{\frac{x}{x+yz}}\geq \sqrt{3}$.

2009 Balkan MO Shortlist, C1

Tags: symmetry , geometry , rectangle , combinatorics

A $ 9 \times 12$ rectangle is partitioned into unit squares. The centers of all the unit squares, except for the four corner squares and eight squares sharing a common side with one of them, are coloured red. Is it possible to label these red centres $ C_1,C_2,\ldots ,C_{96}$ in such way that the following to conditions are both fulfilled i) the distances $C_1C_2,\ldots ,C_{95}C_{96}, C_{96}C_{1}$ are all equal to $ \sqrt {13}$, ii) the closed broken line $ C_1C_2\ldots C_{96}C_1$ has a centre of symmetry? [i]Bulgaria[/i]

1966 IMO Longlists, 56

Tags: geometry , 3d geometry , tetrahedron , circumcircle , sphere

In a tetrahedron, all three pairs of opposite (skew) edges are mutually perpendicular. Prove that the midpoints of the six edges of the tetrahedron lie on one sphere.

2013 NIMO Problems, 1

Tags:

Let $a$, $b$, $c$, $d$, $e$ be positive reals satisfying \begin{align*} a + b &= c \\ a + b + c &= d \\ a + b + c + d &= e.\end{align*} If $c=5$, compute $a+b+c+d+e$. [i]Proposed by Evan Chen[/i]

1999 National Olympiad First Round, 19

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$ k$ black pieces are placed on $ k$ consecutive squares of top row starting from upper left of a $ 2\times 5$ board. We are placing white pieces on empty squares one by one in arbitrary order. Two squares is said to adjacent if they have common vertex. When a white piece is placed on a square, the pieces on adjacent squares change their color. For which $ k$, when all the squares are filled, it is possible that color of every piece is white? $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{None}$

2021 Romania National Olympiad, 2

Tags: vector , geometry , trigonometry

Let $P_0, P_1,\ldots, P_{2021}$ points on the unit circle of centre $O$ such that for each $n\in \{1,2,\ldots, 2021\}$ the length of the arc from $P_{n-1}$ to $P_n$ (in anti-clockwise direction) is in the interval $\left[\frac{\pi}2,\pi\right]$. Determine the maximum possible length of the vector: \[\overrightarrow{OP_0}+\overrightarrow{OP_1}+\ldots+\overrightarrow{OP_{2021}}.\] [i]Mihai Iancu[/i]

2019 CCA Math Bonanza, I13

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Convex quadrilateral $ABCD$ has $AB=20$, $BC=CD=26$, and $\angle{ABC}=90^\circ$. Point $P$ is on $DA$ such that $\angle{PBA}=\angle{ADB}$. If $PB=20$, compute the area of $ABCD$. [i]2019 CCA Math Bonanza Individual Round #13[/i]

2023 Iranian Geometry Olympiad, 2

Tags: geometry

A convex hexagon $ABCDEF$ with an interior point $P$ is given. Assume that $BCEF$ is a square and both $ABP$ and $PCD$ are right isosceles triangles with right angles at $B$ and $C$, respectively. Lines $AF$ and $DE$ intersect at $G$. Prove that $GP$ is perpendicular to $BC$. [i]Proposed by Patrik Bak - Slovakia[/i]

2011 Saudi Arabia IMO TST, 1

Tags: algebra , inequalities

Let $a, b, c$ be real numbers such that $ab + bc + ca = 1$. Prove that $$\frac{(a + b)^2 + 1}{c^2+2}+\frac{(b + c)^2 + 1}{a^2+2}+ \frac{(c + a)^2 + 1}{b^2+2} \ge 3$$

2025 CMIMC Algebra/NT, 10

Tags: algebra , number theory

Let $a_n$ be a recursively defined sequence with $a_0=2024$ and $a_{n+1}=a_n^3+5a_n^2+10a_n+6$ for $n\ge 0.$ Determine the value of $$\sum_{n=0}^{\infty} \frac{2^n(a_n+1)}{a_n^2+3a_n+4}.$$

2021 Science ON Seniors, 2

Tags: number theory , divisibility , order

Find all pairs $(p,q)$ of prime numbers such that $$p^q-4~|~q^p-1.$$ [i](Vlad Robu)[/i]

2023 Brazil National Olympiad, 4

Tags: number theory , diophantine equation , minimum value , algebra

Determine the smallest integer $k$ for which there are three distinct positive integers $a$, $b$ and $c$, such that $$a^2 =bc \text{ and } k = 2b+3c-a.$$

2014 Moldova Team Selection Test, 4

Tags: combinatorics

On a circle $n \geq 1$ real numbers are written, their sum is $n-1$. Prove that one can denote these numbers as $x_1, x_2, ..., x_n$ consecutively, starting from a number and moving clockwise, such that for any $k$ ($1\leq k \leq n$) $ x_1 + x_2+...+x_k \geq k-1$.

1952 Moscow Mathematical Olympiad, 213

Tags: geometric sequence , integer , sum , algebra

Given a geometric progression whose denominator $q$ is an integer not equal to $0$ or $-1$, prove that the sum of two or more terms in this progression cannot equal any other term in it.

2023 HMNT, 5

Tags: number theory

Compute the unique positive integer $n$ such that $\frac{n^3-1989}{n}$ is a perfect square.

2010 Contests, 2

Tags: number theory

Determine the number of positive integers $n$ for which $(n+15)(n+2010)$ is a perfect square.

2024 Australian Mathematical Olympiad, P2

Tags: geometry

Let $ABCD$ be a cyclic quadrilateral. Point $P$ is on line $CB$ such that $CP=CA$and $B$ lies between $C$ and $P$. Point $Q$ is on line $CD$ such that $CQ=CA$ and $D$ lies between $C$ and $Q$. Prove that the incentre of triangle $ABD$ lies on line $PQ.$

1968 Miklós Schweitzer, 11

Tags: probability , probability and stats

Let $ A_1,...,A_n$ be arbitrary events in a probability field. Denote by $ C_k$ the event that at least $ k$ of $ A_1,...,A_n$ occur. Prove that \[ \prod_{k=1}^n P(C_k) \leq \prod_{k=1}^n P(A_k).\] [i]A. Renyi[/i]