Olympiad problems

2023 Canada National Olympiad, 1

Tags: combinatorics

William is thinking of an integer between 1 and 50, inclusive. Victor can choose a positive integer $m$ and ask William: "does $m$ divide your number?", to which William must answer truthfully. Victor continues asking these questions until he determines William's number. What is the minimum number of questions that Victor needs to guarantee this?

2008 District Olympiad, 1

Tags: function , real analysis , ftc , integral , calculus , integration

Let $ f:[0,1]\longrightarrow\mathbb{R} $ be a countinuous function such that $$ \int_0^1 f(x)dx=\int_0^1 xf(x)dx. $$ Show that there is a $ c\in (0,1) $ such that $ f(c)=\int_0^c f(x)dx. $

1998 Chile National Olympiad, 2

Tags: locus , semicircle , geometry

Given a semicircle of diameter $ AB $, with $ AB = 2r $, be $ CD $ a variable string, but of fixed length $ c $. Let $ E $ be the intersection point of lines $ AC $ and $ BD $, and let $ F $ be the intersection point of lines $ AD $ and $ BC $. a) Prove that the lines $ EF $ and $ AB $ are perpendicular. b) Determine the locus of the point $ E $. c) Prove that $ EF $ has a constant measure, and determine it based on $ c $ and $ r $.

2011 Today's Calculation Of Integral, 681

Tags: calculus , integration , trigonometry , calculus computations

Evaluate $\int_0^{\frac{\pi}{2}} \sqrt{1-2\sin 2x+3\cos ^ 2 x}\ dx.$ [i]2011 University of Occupational and Environmental Health/Medicine　entrance exam[/i]

2006 Turkey Team Selection Test, 2

Tags: geometry , rectangle , calculus , integration , function , combinatorics

How many ways are there to divide a $2\times n$ rectangle into rectangles having integral sides, where $n$ is a positive integer?

1965 AMC 12/AHSME, 27

Tags: algebra , polynomial

When $ y^2 \plus{} my \plus{} 2$ is divided by $ y \minus{} 1$ the quotient is $ f(y)$ and the remainder is $ R_1$. When $ y^2 \plus{} my \plus{} 2$ is divided by $ y \plus{} 1$ the quotient is $ g(y)$ and the remainder is $ R_2$. If $ R_1 \equal{} R_2$ then $ m$ is: $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ \minus{} 1 \qquad \textbf{(E)}\ \text{an undetermined constant}$

1988 Polish MO Finals, 3

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

Find the largest possible volume for a tetrahedron which lies inside a hemisphere of radius $1$.

2004 USA Team Selection Test, 2

Tags: modular arithmetic , invariant , induction , algebra , polynomial , vector

Assume $n$ is a positive integer. Considers sequences $a_0, a_1, \ldots, a_n$ for which $a_i \in \{1, 2, \ldots , n\}$ for all $i$ and $a_n = a_0$. (a) Suppose $n$ is odd. Find the number of such sequences if $a_i - a_{i-1} \not \equiv i \pmod{n}$ for all $i = 1, 2, \ldots, n$. (b) Suppose $n$ is an odd prime. Find the number of such sequences if $a_i - a_{i-1} \not \equiv i, 2i \pmod{n}$ for all $i = 1, 2, \ldots, n$.

1982 IMO, 3

Tags: limit , geometric sequence , geometric series , inequality , minimization

Consider infinite sequences $\{x_n\}$ of positive reals such that $x_0=1$ and $x_0\ge x_1\ge x_2\ge\ldots$. [b]a)[/b] Prove that for every such sequence there is an $n\ge1$ such that: \[ {x_0^2\over x_1}+{x_1^2\over x_2}+\ldots+{x_{n-1}^2\over x_n}\ge3.999. \] [b]b)[/b] Find such a sequence such that for all $n$: \[ {x_0^2\over x_1}+{x_1^2\over x_2}+\ldots+{x_{n-1}^2\over x_n}<4. \]

2021 SYMO, Q6

Tags: algebra , polynomial

Let $P(x)$ and $Q(x)$ be non-constant integer-coefficient polynomials such that for any integer $x\in \mathbb Z$, there exists integer $y\in \mathbb Z$ such that $P(x)=Q(y)$. Prove that the degree of $Q$ divides the degree of $P$.

2021 Purple Comet Problems, 12

Tags:

Let $L_1$ and $L_2$ be perpendicular lines, and let $F$ be a point at a distance $18$ from line $L_1$ and a distance $25$ from line $L_2$. There are two distinct points, $P$ and $Q$, that are each equidistant from $F$, from line $L_1$, and from line $L_2$. Find the area of $\triangle{FPQ}$.

2006 AMC 12/AHSME, 9

Tags:

Oscar buys 13 pencils and 3 erasers for $ \$1.00$. A pencil costs more than an eraser, and both items cost a whole number of cents. What is the total cost, in cents, of one pencil and one eraser? $ \textbf{(A) } 10\qquad \textbf{(B) } 12\qquad \textbf{(C) } 15\qquad \textbf{(D) } 18\qquad \textbf{(E) } 20$

Kyiv City MO 1984-93 - geometry, 1984.8.3

Tags: geometry , right triangle , construction

Construct a right triangle given the lengths of segments of the medians $m_a,m_b$ corresponding on its legs.

2021 The Chinese Mathematics Competition, Problem 7

Tags: differential equation

Let $f(x)$ be a bounded continuous function on $[0,+\infty)$. Prove that every solutions of the equation $y''+14y'+13y=f(x)$ are bounded continuous functions on $[0,+\infty)$

1961 Poland - Second Round, 3

Tags: trigonometry

Prove that for any angles $x,y,z$ holds the equality $$1-\cos^2x-\cos^2y- y-\cos^2z +2 \cos x \cos y \cos z= 4 \sin \frac{x+y+z}{2} \sin \frac{x+y-z}{2} \sin \frac{x-y+z}{2} \sin\frac{-x-y+z}{2}. $$

2017 ASDAN Math Tournament, 19

Tags:

How many ways can you tile a $2\times5$ rectangle with $2\times1$ dominoes of $4$ different colors if no two dominoes of the same color may be adjacent?

2012 AMC 8, 22

Tags:

Let $R$ be a set of nine distinct integers. Six of the elements are 2, 3, 4, 6, 9, and 14. What is the number of possible values of the median of $R$ ? $\textbf{(A)}\hspace{.05in}4 \qquad \textbf{(B)}\hspace{.05in}5 \qquad \textbf{(C)}\hspace{.05in}6 \qquad \textbf{(D)}\hspace{.05in}7 \qquad \textbf{(E)}\hspace{.05in}8 $

1965 All Russian Mathematical Olympiad, 071

Tags: algebra

On the surface of the planet lives one inhabitant, that can move with the speed not greater than $u$. A spaceship approaches to the planet with its speed $v$. Prove that if $v/u > 10$ , the spaceship can find the inhabitant, even it is trying to hide.

2008 Stanford Mathematics Tournament, 13

Tags:

Let N be the number of distinct rearrangements of the 34 letters in SUPERCALIFRAGILISTICEXPIALIDOCIOUS. How many positive factors does N have?

2020 Turkey Team Selection Test, 4

Tags: function , inequality , inequalities

Let $Z^+$ be positive integers set. $f:\mathbb{Z^+}\to\mathbb{Z^+}$ is a function and we show $ f \circ f \circ ...\circ f $ with $f_l$ for all $l\in \mathbb{Z^+}$ where $f$ is repeated $l$ times. Find all $f:\mathbb{Z^+}\to\mathbb{Z^+}$ functions such that $$ (n-1)^{2020}< \prod _{l=1}^{2020} {f_l}(n)< n^{2020}+n^{2019} $$ for all $n\in \mathbb{Z^+}$

1999 Rioplatense Mathematical Olympiad, Level 3, 6

Tags: combinatorics

At a big New Year's Eve party, each guest receives two hats: one red and one blue. At the beginning of the party, all the guests put on the red hat. Several times throughout the evening, the announcer announces the name of one of the guests and, at that moment, the named and each of his friends change the hat they are wearing for the other color. Show that the announcer can make it so that all the guests are wearing the blue hat when the party is over. Note: All guests remain at the party from start to finish.

2005 Today's Calculation Of Integral, 8

Tags: calculus , integration , trigonometry , function , calculus computations , logarithm

Calculate the following indefinite integrals. [1] $\int x(x^2+3)^2 dx$ [2] $\int \ln (x+2) dx$ [3] $\int x\cos x dx$ [4] $\int \frac{dx}{(x+2)^2}dx$ [5] $\int \frac{x-1}{x^2-2x+3}dx$

2009 All-Russian Olympiad, 3

Tags: geometry , 3d geometry , sphere , pyramid , circumcircle , symmetry , perpendicular bisector

Let $ ABCD$ be a triangular pyramid such that no face of the pyramid is a right triangle and the orthocenters of triangles $ ABC$, $ ABD$, and $ ACD$ are collinear. Prove that the center of the sphere circumscribed to the pyramid lies on the plane passing through the midpoints of $ AB$, $ AC$ and $ AD$.

1962 Vietnam National Olympiad, 1

Tags: inequalities

Prove that for positive real numbers $ a$, $ b$, $ c$, $ d$, we have \[ \frac{1}{\frac{1}{a}\plus{}\frac{1}{b}}\plus{}\frac{1}{\frac{1}{c}\plus{}\frac{1}{d}}\le\frac{1}{\frac{1}{a\plus{}c}\plus{}\frac{1}{b\plus{}d}}\]

2014 Stars Of Mathematics, 3

Tags: algebra , combinatorics

Let positive integers $M$, $m$, $n$ be such that $1\leq m \leq n$, $1\leq M \leq \dfrac {m(m+1)} {2}$, and let $A \subseteq \{1,2,\ldots,n\}$ with $|A|=m$. Prove there exists a subset $B\subseteq A$ with $$0 \leq \sum_{b\in B} b - M \leq n-m.$$ ([i]Dan Schwarz[/i])