Olympiad problems

1999 Harvard-MIT Mathematics Tournament, 1

Find all twice differentiable functions $f(x)$ such that $f^{\prime \prime}(x)=0$, $f(0)=19$, and $f(1)=99$.

1997 All-Russian Olympiad Regional Round, 8.4

Tags: combinatorics

The company employs 50,000 people. For each of them, the sum of the number of his immediate superiors and his immediate subordinates is equal to 7. On Monday, each employee of the enterprise issues an order and gives a copy of this order to each of his direct subordinates (if there are any). Further, every day an employee takes all the basics he received on the previous day and either distributes them copies to all your direct subordinates, or, if any, he is not there, he carries out orders himself. It turned out that on Friday no papers were transferred to the institution. Prove that the enterprise has at least 97 bosses over whom there are no bosses.

2013 Harvard-MIT Mathematics Tournament, 3

Tags: hmmt , modular arithmetic

Find the rightmost non-zero digit of the expansion of $(20)(13!)$.

2015 BMT Spring, 7

Tags: geometry

Define $A = (1, 0, 0)$, $B = (0, 1, 0)$, and $P$ as the set of all points $(x, y, z)$ such that $x+y+z = 0$. Let $P$ be the point on $P$ such that $d = AP + P B$ is minimized. Find $d^2$.

2020 Germany Team Selection Test, 1

Tags: combinatorics , induction , local argument

You are given a set of $n$ blocks, each weighing at least $1$; their total weight is $2n$. Prove that for every real number $r$ with $0 \leq r \leq 2n-2$ you can choose a subset of the blocks whose total weight is at least $r$ but at most $r + 2$.

1998 Baltic Way, 6

Tags: polynomial , algebra

Let $P$ be a polynomial of degree $6$ and let $a,b$ be real numbers such that $0<a<b$. Suppose that $P(a)=P(-a),P(b)=P(-b),P'(0)=0$. Prove that $P(x)=P(-x)$ for all real $x$.

1999 India Regional Mathematical Olympiad, 7

Tags: quadratic , algebra , polynomial

Find the number of quadratic polynomials $ax^2 + bx +c$ which satisfy the following: (a) $a,b,c$ are distinct; (b) $a,b,c \in \{ 1,2,3,\cdots 1999 \}$; (c) $x+1$ divides $ax^2 + bx+c$.

2013-2014 SDML (High School), 6

Tags:

The total number of edges in two regular polygons is $2014$, and the total number of diagonals is $1,014,053$. How many edges does the polygon with the smaller number [of] edges have?

2016 Macedonia JBMO TST, 4

Tags: algebra , inequality

Let $x$, $y$, and $z$ be positive real numbers. Prove that $\sqrt {\frac {xy}{x^2 + y^2 + 2z^2}} + \sqrt {\frac {yz}{y^2 + z^2 + 2x^2}}+\sqrt {\frac {zx}{z^2 + x^2 + 2y^2}} \le \frac{3}{2}$. When does equality hold?

1961 AMC 12/AHSME, 21

Tags: geometry

Medians $AD$ and and $CE$ of triangle $ABC$ intersect in $M$. The midpoint of $AE$ is $N$. Let the area of triangle $MNE$ be $k$ times the area of triangle $ABC$. Then $k$ equals: ${{ \textbf{(A)}\ \frac{1}{6} \qquad\textbf{(B)}\ \frac{1}{8} \qquad\textbf{(C)}\ \frac{1}{9} \qquad\textbf{(D)}\ \frac{1}{12} }\qquad\textbf{(E)}\ \frac{1}{16} } $

2012 Princeton University Math Competition, A6

Tags: combinatorics

Two white pentagonal pyramids, with side lengths all the same, are glued to each other at their regular pentagon bases. Some of the resulting $10$ faces are colored black. How many rotationally distinguishable colorings may result?

2022-2023 OMMC, 22

Tags: abstract algebra

Find the number of ordered pairs of integers $(x, y)$ with $0 \le x, y \le 40$ where $$\frac{x^2-xy^2+1}{41}$$ is an integer.

2013 Dutch Mathematical Olympiad, 2

Tags: algebra , system of equations

Find all triples $(x, y, z)$ of real numbers satisfying: $x + y - z = -1$ , $x^2 - y^2 + z^2 = 1$ and $- x^3 + y^3 + z^3 = -1$

2013 Philippine MO, 1

Tags:

1. Determine, with proof, the least positive integer $n$ for which there exist $n$ distinct positive integers, $\left(1-\frac{1}{x_1}\right)\left(1-\frac{1}{x_2}\right)......\left(1-\frac{1}{x_n}\right)=\frac{15}{2013}$

2023 China National Olympiad, 2

Tags: geometry

Let $\triangle ABC$ be an equilateral triangle of side length 1. Let $D,E,F$ be points on $BC,AC,AB$ respectively, such that $\frac{DE}{20} = \frac{EF}{22} = \frac{FD}{38}$. Let $X,Y,Z$ be on lines $BC,CA,AB$ respectively, such that $XY\perp DE, YZ\perp EF, ZX\perp FD$. Find all possible values of $\frac{1}{[DEF]} + \frac{1}{[XYZ]}$.

2002 Canada National Olympiad, 5

Tags: function , modular arithmetic , functional equation , algebra

Let $\mathbb N = \{0,1,2,\ldots\}$. Determine all functions $f: \mathbb N \to \mathbb N$ such that \[ xf(y) + yf(x) = (x+y) f(x^2+y^2) \] for all $x$ and $y$ in $\mathbb N$.

2013 Purple Comet Problems, 11

Tags: percent

After Jennifer walked $r$ percent of the way from her home to the store, she turned around and walked home, got on her bicycle, and bicycled to the store and back home. Jennifer bicycles two and a half times faster than she walks. Find the largest value of $r$ so that returning home for her bicycle was not slower than her walking all the way to and from the store without her bicycle.

1985 IMO Longlists, 31

Tags: ellipse , concurrency , conic , geometry

Let $E_1, E_2$, and $E_3$ be three mutually intersecting ellipses, all in the same plane. Their foci are respectively $F_2, F_3; F_3, F_1$; and $F_1, F_2$. The three foci are not on a straight line. Prove that the common chords of each pair of ellipses are concurrent.

2007 Stanford Mathematics Tournament, 7

Tags:

Find the minimum value of $xy+x+y+\frac{1}{xy}+\frac{1}{x}+\frac{1}{y}$ for $x, y>0$ real.

1992 Tournament Of Towns, (349) 1

Tags: geometry , 3d geometry , combinatorics , combinatorial geometry

We are given a cube with edges of length $n$ cm. At our disposal is a long piece of insulating tape of width $1$ cm. It is required to stick this tape to the cube. The tape may freely cross an edge of the cube on to a different face but it must always be parallel to an edge of the cube. It may not overhang the edge of a face or cross over a vertex. How many pieces of the tape are necessary in order to completely cover the cube? (You may assume that $n$ is an integer.) (A Spivak)

2004 Baltic Way, 17

Tags: rectangle , geometry , inequalities

Consider a rectangle with sidelengths 3 and 4, pick an arbitrary inner point on each side of this rectangle. Let $x, y, z$ and $u$ denote the side lengths of the quadrilateral spanned by these four points. Prove that $25 \leq x^2+y^2+z^2+u^2 \leq 50$.

1998 Bulgaria National Olympiad, 3

Tags: trigonometry , complex number , geometry

On the sides of a non-obtuse triangle $ABC$ a square, a regular $n$-gon and a regular $m$-gon ($m$,$n > 5$) are constructed externally, so that their centers are vertices of a regular triangle. Prove that $m = n = 6$ and find the angles of $\triangle ABC$.

2018 Polish MO Finals, 5

Tags: radical axis , projective geometry , geometry

An acute triangle $ABC$ in which $AB<AC$ is given. Points $E$ and $F$ are feet of its heights from $B$ and $C$, respectively. The line tangent in point $A$ to the circle escribed on $ABC$ crosses $BC$ at $P$. The line parallel to $BC$ that goes through point $A$ crosses $EF$ at $Q$. Prove $PQ$ is perpendicular to the median from $A$ of triangle $ABC$.

2007 AMC 12/AHSME, 25

Tags:

Call a set of integers [i]spacy[/i] if it contains no more than one out of any three consecutive integers. How many subsets of $ \{1,2,3,\ldots,12\},$ including the empty set, are spacy? $ \textbf{(A)}\ 121 \qquad \textbf{(B)}\ 123 \qquad \textbf{(C)}\ 125 \qquad \textbf{(D)}\ 127 \qquad \textbf{(E)}\ 129$

2014 Contests, 2

Tags:

Let $x_1$, $x_2$, …, $x_{10}$ be 10 numbers. Suppose that $x_i + 2 x_{i + 1} = 1$ for each $i$ from 1 through 9. What is the value of $x_1 + 512 x_{10}$?