Olympiad problems

2018 BMT Spring, 8

Tags: geometry

What is the largest possible area of a triangle with largest side length $39$ and inradius $10$?

2024 Korea National Olympiad, 2

Tags: algebra

For a sequence of positive integers $\{x_n\}$ where $x_1 = 2$ and $x_{n + 1} - x_n \in \{0, 3\}$ for all positve integers $n$, then $\{x_n\}$ is called a "frog sequence". Find all real numbers $d$ that satisfy the following condition. [b](Condition)[/b] For two frog sequence $\{a_n\}, \{b_n\}$, if there exists a positive integer $n$ such that $a_n = 1000b_n$, then there exists a positive integer $m$ such that $a_m = d\cdot b_m$.

2022 CCA Math Bonanza, L2.1

Tags:

Given that a duck found that $5-2\sqrt{3}i$ is one of the roots of $-259 + 107x - 17x^2 + x^3$, what is the sum of the real parts of the other two roots? [i]2022 CCA Math Bonanza Lightning Round 2.1[/i]

2018 Cyprus IMO TST, 3

Tags: inequalities

Find all triples $(\alpha, \beta, \gamma)$ of positive real numbers for which the expression $$K = \frac{\alpha+3 \gamma}{\alpha + 2\beta + \gamma} + \frac{4\beta}{\alpha+\beta+2\gamma} - \frac{8 \gamma}{\alpha+ \beta + 3\gamma}$$ obtains its minimum value.

1993 Greece National Olympiad, 5

Tags: algebra , polynomial

Let $P_0(x) = x^3 + 313x^2 - 77x - 8$. For integers $n \ge 1$, define $P_n(x) = P_{n - 1}(x - n)$. What is the coefficient of $x$ in $P_{20}(x)$?

1991 Arnold's Trivium, 20

Tags: calculus , derivative , parameterization , function , inequalities

Find the derivative of the solution of the equation $\ddot{x} =x + A\dot{x}^2$, with initial conditions $x(0) = 1$, $\dot{x}(0) = 0$, with respect to the parameter $A$ for $A = 0$.

2023 India EGMO TST, P1

Tags: geometry

Let $r > 0$ be a real number. All the interior points of the disc $D(r)$ of radius $r$ are colored with one of two colors, red or blue. [list][*]If $r > \frac{\pi}{\sqrt{3}}$, show that we can find two points $A$ and $B$ in the interior of the disc such that $AB = \pi$ and $A,B$ have the same color [*]Does the conclusion in (a) hold if $r > \frac{\pi}{2}$?[/list] [i]Proposed by S Muralidharan[/i]

1973 AMC 12/AHSME, 3

Tags:

The stronger Goldbach conjecture states that any even integer greater than 7 can be written as the sum of two different prime numbers. For such representations of the even number 126, the largest possible difference between the two primes is $ \textbf{(A)}\ 112 \qquad \textbf{(B)}\ 100 \qquad \textbf{(C)}\ 92 \qquad \textbf{(D)}\ 88 \qquad \textbf{(E)}\ 80$

2003 National High School Mathematics League, 2

Tags:

If $a,b\in\mathbb{R},ab\neq 0$, the the possible figure of $ax-y+b=0$ and $bx^2+ay^2=ab$ is [center][img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvYy80Lzc3NGNjZWNiN2ZjYzIxMTJlYWE5NDlmZmQ0ZjE1NzgwNmNhM2JiLnBuZw==&rn=MTI0MjQ1ODUyMTI0MjQyNTI0MjUyNS5wbmc=[/img][/center]

2005 India IMO Training Camp, 1

Tags: geometry

Consider a $n$-sided polygon inscribed in a circle ($n \geq 4$). Partition the polygon into $n-2$ triangles using [b]non-intersecting[/b] diagnols. Prove that, irrespective of the triangulation, the sum of the in-radii of the triangles is a constant.

1966 IMO Shortlist, 18

Tags: trigonometric equation , parameterization , trigonometry , parabola

Solve the equation $\frac{1}{\sin x}+\frac{1}{\cos x}=\frac 1p$ where $p$ is a real parameter. Discuss for which values of $p$ the equation has at least one real solution and determine the number of solutions in $[0, 2\pi)$ for a given $p.$

1992 Vietnam National Olympiad, 1

Tags: 3d geometry , tetrahedron , geometry

Let $ABCD$ be a tetrahedron satisfying i)$\widehat{ACD}+\widehat{BCD}=180^{0}$, and ii)$\widehat{BAC}+\widehat{CAD}+\widehat{DAB}=\widehat{ABC}+\widehat{CBD}+\widehat{DBA}=180^{0}$. Find value of $[ABC]+[BCD]+[CDA]+[DAB]$ if we know $AC+CB=k$ and $\widehat{ACB}=\alpha$.

2024 HMNT, 16

Tags: guts

Compute $$\frac{2+3+\cdots+100}{1}+\frac{3+4+\cdots+100}{1+2}+\cdots+\frac{100}{1+2+\cdots+99}.$$

1987 Austrian-Polish Competition, 8

Tags: arc , curve , integer sidelengths , geometry , combinatorial geometry , circles

A circle of perimeter $1$ has been dissected into four equal arcs $B_1, B_2, B_3, B_4$. A closed smooth non-selfintersecting curve $C$ has been composed of translates of these arcs (each $B_j$ possibly occurring several times). Prove that the length of $C$ is an integer.

2012 USAJMO, 6

Tags: geometry , geometric transformation

Let $P$ be a point in the plane of $\triangle ABC$, and $\gamma$ a line passing through $P$. Let $A', B', C'$ be the points where the reflections of lines $PA, PB, PC$ with respect to $\gamma$ intersect lines $BC, AC, AB$ respectively. Prove that $A', B', C'$ are collinear.

2016 Brazil Team Selection Test, 1

Tags: functional equation , algebra , functional equation in n

Determine all functions $f$ from the set of non-negative integers to itself such that $f(a + b) = f(a) + f(b) + f(c) + f(d)$, whenever $a, b, c, d$, are non-negative integers satisfying $2ab = c^2 + d^2$.

1997 Moldova Team Selection Test, 7

Tags: collinear , geometry

Let $ABC$ be a triangle with orthocenter $H$. Let the circle $\omega$ have $BC$ as the diameter. Draw tangents $AP$, $AQ$ to the circle $\omega $ at the point $P, Q$ respectively. Prove that $ P,H,Q$ lie on the same line .

2010 District Olympiad, 3

Tags: floor function , algebra

For any real number $ x$ prove that: \[ x\in \mathbb{Z}\Leftrightarrow \lfloor x\rfloor \plus{}\lfloor 2x\rfloor\plus{}\lfloor 3x\rfloor\plus{}...\plus{}\lfloor nx\rfloor\equal{}\frac{n(\lfloor x\rfloor\plus{}\lfloor nx\rfloor)}{2}\ ,\ (\forall)n\in \mathbb{N}^*\]

2012 China Second Round Olympiad, 3

Tags: inequalities

Suppose that $x,y,z\in [0,1]$. Find the maximal value of the expression \[\sqrt{|x-y|}+\sqrt{|y-z|}+\sqrt{|z-x|}.\]

2003 China Team Selection Test, 2

Tags: number theory

Let $x<y$ be positive integers and $P=\frac{x^3-y}{1+xy}$. Find all integer values that $P$ can take.

Kyiv City MO Seniors 2003+ geometry, 2013.11.3

Tags: geometry , bisects segment , diameter , perpendicular

The segment $AB$ is the diameter of the circle. The points $M$ and $C$ belong to this circle and are located in different half-planes relative to the line $AB$. From the point $M$ the perpendiculars $MN$ and $MK$ are drawn on the lines $AB$ and $AC$, respectively. Prove that the line $KN$ intersects the segment $CM$ in its midpoint. (Igor Nagel)

Kvant 2023, M2752

Tags: combinatorics

A square grid $100 \times 100$ is tiled in two ways - only with dominoes and only with squares $2 \times 2$. What is the least number of dominoes that are entirely inside some square $2 \times 2$?

2000 Tournament Of Towns, 1

Tags: number theory , greatest common divisor

Positive integers $m$ and $n$ have no common divisor greater than one. What is the largest possible value of the greatest common divisor of $m + 2000n$ and $n + 2000m$ ? (S Zlobin)

2000 Portugal MO, 6

Tags: combinatorics

In a tournament, $n$ players participate. Each player plays each other exactly once, with no ties. A player $A$ is said to be [i]champion [/i] if, for every other player $B$, one of the following two situations occurs: (a) $A$ beat $B$; (b) $A$ beat a player $C$ who in turn beat $B$. Prove that in such a tournament there cannot be exactly two champions.

Today's calculation of integrals, 900

Tags: calculus , integration , trigonometry , calculus computations

Find $\sum_{k=0}^n \frac{(-1)^k}{2k+1}\ _n C_k.$