Olympiad problems

2009 Today's Calculation Of Integral, 439

Tags: calculus , integration , inequalities , analytic geometry , calculus computations , logarithm

Find the volume of the solid defined by the inequality $ x^2 \plus{} y^2 \plus{} \ln (1 \plus{} z^2)\leq \ln 2$. Note that you may not directively use double integral here for Japanese high school students who don't study it.

1935 Moscow Mathematical Olympiad, 002

Tags: geometric construction , geometry

Given the lengths of two sides of a triangle and that of the bisector of the angle between these sides, construct the triangle.

2013 CHMMC (Fall), 3

Tags: combinatorics

Bill plays a game in which he rolls two fair standard six-sided dice with sides labeled one through six. He wins if the number on one of the dice is three times the number on the other die. If Bill plays this game three times, compute the probability that he wins at least once.

2009 Croatia Team Selection Test, 2

Tags: geometry , rectangle , analytic geometry , combinatorics

Every natural number is coloured in one of the $ k$ colors. Prove that there exist four distinct natural numbers $ a, b, c, d$, all coloured in the same colour, such that $ ad \equal{} bc$, $ \displaystyle \frac b a$ is power of 2 and $ \displaystyle \frac c a$ is power of 3.

2014 Iran Team Selection Test, 2

Tags: geometry , incenter , circumcircle , geometric transformation

Point $D$ is an arbitary point on side $BC$ of triangle $ABC$. $I$,$I_1$ and$I_2$ are the incenters of triangles $ABC$,$ABD$ and $ACD$ respectively. $M\not=A$ and $N\not=A$ are the intersections of circumcircle of triangle $ABC$ and circumcircles of triangles $IAI_1$ and $IAI_2$ respectively. Prove that regardless of point $D$, line $MN$ goes through a fixed point.

1958 February Putnam, A1

Tags: root , polynomial

If $a_0 , a_1 ,\ldots, a_n$ are real number satisfying $$ \frac{a_0 }{1} + \frac{a_1 }{2} + \ldots + \frac{a_n }{n+1}=0,$$ show that the equation $a_n x^n + \ldots +a_1 x+a_0 =0$ has at least one real root.

2004 Postal Coaching, 3

Tags: number theory , inequalities

Let $a,b,c,d,$ be real and $ad-bc = 1$. Show that $Q = a^2 + b^2 + c^2 + d^2 + ac +bd$ $\not= 0, 1, -1$

1998 Finnish National High School Mathematics Competition, 3

Tags: ratio , geometric sequence , algebra

Consider the geometric sequence $1/2, \ 1 / 4, \ 1 / 8,...$ Can one choose a subsequence, finite or infinite, for which the ratio of consecutive terms is not $1$ and whose sum is $1/5?$

1968 Polish MO Finals, 6

Tags: combinatorial geometry , combinatorics , geometry

Consider a set of $n > 3$ points in the plane, no three of which are collinear, and a natural number $k < n$. Prove the following statements: (a) If $k \le \frac{n}{2}$, then each point can be connected with at least k other points by segments so that no three segments form a triangle. (b) If $k \ge \frac{n}{2}$, and each point is connected with at least k other points by segments, then some three segments form a triangle.

2019 PUMaC Team Round, 9

Tags: number theory , algebra

Find the integer $\sqrt[5]{55^5 + 3183^5 + 28969^5 + 85282^5}$.

II Soros Olympiad 1995 - 96 (Russia), 11.7

Tags: geometry , 3d geometry , prism , volume

Three edges of a parallelepiped lie on three intersecting diagonals of the lateral faces of a triangular prism. Find the ratio of the volumes of the parallelepiped and the prism.

2001 Junior Balkan MO, 3

Tags: trigonometry , geometry

Let $ABC$ be an equilateral triangle and $D$, $E$ points on the sides $[AB]$ and $[AC]$ respectively. If $DF$, $EF$ (with $F\in AE$, $G\in AD$) are the interior angle bisectors of the angles of the triangle $ADE$, prove that the sum of the areas of the triangles $DEF$ and $DEG$ is at most equal with the area of the triangle $ABC$. When does the equality hold? [i]Greece[/i]

2024 Dutch IMO TST, 1

Tags: circumcircle , tangent , geometry

Let $ABC$ be a triangle with orthocenter $H$ and circumcircle $\Gamma$. Let $D$ be the reflection of $A$ in $B$ and let $E$ the reflection of $A$ in $C$. Let $M$ be the midpoint of segment $DE$. Show that the tangent to $\Gamma$ in $A$ is perpendicular to $HM$.

2021 USMCA, 11

Tags:

Let $f_1 (x) = x^2 - 3$ and $f_n (x) = f_1(f_{n-1} (x))$ for $n \ge 2$. Let $m_n$ be the smallest positive root of $f_n$, and $M_n$ be the largest positive root of $f_n$. If $x$ is the least number such that $M_n \le m_n \cdot x$ for all $n \ge 1$, compute $x^2$.

2013 India IMO Training Camp, 1

Tags: number theory

For a prime $p$, a natural number $n$ and an integer $a$, we let $S_n(a,p)$ denote the exponent of $p$ in the prime factorisation of $a^{p^n} - 1$. For example, $S_1(4,3) = 2$ and $S_2(6,2) = 0$. Find all pairs $(n,p)$ such that $S_n(2013,p) = 100$.

1962 AMC 12/AHSME, 6

Tags: geometry , perimeter

A square and an equilateral triangle have equal perimeters. The area of the triangle is $ 9 \sqrt{3}$ square inches. Expressed in inches the diagonal of the square is: $ \textbf{(A)}\ \frac{9}{2} \qquad \textbf{(B)}\ 2 \sqrt{5} \qquad \textbf{(C)}\ 4 \sqrt{2} \qquad \textbf{(D)}\ \frac{9 \sqrt{2}}{2} \qquad \textbf{(E)}\ \text{none of these}$

2002 Moldova National Olympiad, 1

Tags: ratio

Volume $ A$ equals one fourth of the sum of the volumes $ B$ and $ C$, while volume $ B$ equals one sixth of the sum of the volumes $ C$ and $ A$. Find the ratio of the volume $ C$ to the sum of the volumes $ A$ and $ B$.

1995 Irish Math Olympiad, 5

Tags: function , vector , algebra

Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all real numbers $ x,y$: $ x f(x)\minus{}yf(y)\equal{}(x\minus{}y)f(x\plus{}y)$.

2017 AMC 12/AHSME, 19

Tags:

A square with side length $x$ is inscribed in a right triangle with sides of length $3$, $4$, and $5$ so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length $y$ is inscribed so that one side of the square lies on the hypotenuse of the triangle. What is $\frac{x}{y}$? $\textbf{(A)}\ \frac{12}{13}\qquad\textbf{(B)}\ \frac{35}{37}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ \frac{37}{35}\qquad\textbf{(E)}\ \frac{13}{12}$

2017 Math Prize for Girls Olympiad, 2

Tags:

Let $n$ be a positive integer. Prove that there exist polynomials $P$ and $Q$ with real coefficients such that for every real number $x$, we have $P(x) \ge 0$, $Q(x) \ge 0$, and \[ 1 - x^n = (1 - x)P(x) + (1 + x)Q(x). \]

1974 IMO Longlists, 19

Tags: geometry

Prove that there exists, for $n \geq 4$, a set $S$ of $3n$ equal circles in space that can be partitioned into three subsets $s_5, s_4$, and $s_3$, each containing $n$ circles, such that each circle in $s_r$ touches exactly $r$ circles in $S.$

1997 Tournament Of Towns, (537) 2

Tags: number theory , divisible , divide

Let $a$ and $b$ be positive integers. If $a^2 + b^2$ is divisible by $ab$, prove that $a = b$. (BR Frenkin)

2015 Korea - Final Round, 5

Tags: number theory , recurrence

For a fixed positive integer $k$, there are two sequences $A_n$ and $B_n$. They are defined inductively, by the following recurrences. $A_1 = k$, $A_2 = k$, $A_{n+2} = A_{n}A_{n+1}$ $B_1 = 1$, $B_2 = k$, $B_{n+2} = \frac{B^3_{n+1}+1}{B_{n}}$ Prove that for all positive integers $n$, $A_{2n}B_{n+3}$ is an integer.

1999 Greece Junior Math Olympiad, 2

Tags: number theory , minimum , maximum

Let $n$ be a fixed positive integer and let $x, y$ be positive integers such that $xy = nx+ny$. Determine the minimum and the maximum of $x$ in terms of $n$.

2016 Bosnia And Herzegovina - Regional Olympiad, 4

Tags: function , functional equation , algebra

Find all functions $f : \mathbb{Q} \rightarrow \mathbb{R}$ such that: $a)$ $f(1)+2>0$ $b)$ $f(x+y)-xf(y)-yf(x)=f(x)f(y)+f(x)+f(y)+xy$, $\forall x,y \in \mathbb{Q}$ $c)$ $f(x)=3f(x+1)+2x+5$, $\forall x \in \mathbb{Q}$