Olympiad problems

2005 National Olympiad First Round, 31

Tags:

If the equation system \[\begin{array}{rcl} f(x) + g(x) &=& 0 \\ f(x)-(g(x))^3 &=& 0 \end{array}\] has more than one real roots, where $a,b,c,d$ are reals and $f(x)=x^2 + ax+b$, $g(x)=x^2 + cx + d$, at most how many distinct real roots can the equation $f(x)g(x) = 0$ have? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4 $

2005 iTest, 7

Find the coefficient of the fourth term of the expansion of $(x+y)^{15}$.

1985 AMC 12/AHSME, 15

Tags: logarithm

If $ a$ and $ b$ are positive numbers such that $ a^b \equal{} b^a$ and $ b \equal{} 9a$, then the value of $ a$ is: $ \textbf{(A)}\ 9\qquad \textbf{(B)}\ \frac {1}{9}\qquad \textbf{(C)}\ \sqrt [9] {9}\qquad \textbf{(D)}\ \sqrt [3] {9}\qquad \textbf{(E)}\ \sqrt [4] {3}$

2014 BMT Spring, 3

Tags: number theory

Together, Abe and Bob have less than or equal to \$ $100$. When Corey asks them how much money they have, Abe says that the reciprocal of his money added to Bob’s money is thirteen times as much as the sum of Abe’s money and the reciprocal of Bob’s money. If Abe and Bob both have integer amounts of money, how many possible values are there for Abe’s money?

1992 China National Olympiad, 3

Tags: combinatorics

Given a $9\times 9$ grid, we assign either $+1$ or $-1$ to each square on the grid. We define an [i]adjustment[/i] as follow: for each square on the grid, we make a product of all numbers of those squares which share a common side with the square (excluding itself).Then we have $81$ products. Next we replace all the squares’ values with their corresponding products. Determine if we can make all values in the grid equal to $1$ through finite [i]adjustments[/i].

1975 Polish MO Finals, 2

Tags: geometry , 3d geometry , tetrahedron , combinatorial geometry

On the surface of a regular tetrahedron of edge length $1$ are given finitely many segments such that every two vertices of the tetrahedron can be joined by a polygonal line consisting of given segments. Can the sum of the lengths of the given segments be less than $1+\sqrt3 $?

2016 Oral Moscow Geometry Olympiad, 5

Tags: geometry , excenter , concurrency

Points $I_A, I_B, I_C$ are the centers of the excircles of $ABC$ related to sides $BC, AC$ and $AB$ respectively. Perpendicular from $I_A$ to $AC$ intersects the perpendicular from $I_B$ to $B_C$ at point $X_C$. The points $X_A$ and $X_B$. Prove that the lines $I_AX_A, I_BX_B$ and $I_CX_C$ intersect at the same point.

2022 Dutch BxMO TST, 1

Tags: number theory , functional

Find all functions $f : Z_{>0} \to Z_{>0}$ for which $f(n) | f(m) - n$ if and only if $n | m$ for all natural numbers $m$ and $n$.

2025 239 Open Mathematical Olympiad, 2

Tags: geometry

$AD$, $BE$, $CF$ are the heights of the acute—angled triangle $ABC$. A perpendicular is drawn to the segment $DE$ at point $E$. It intersects the height of $AD$ at point $G$. The point $J$ is chosen on the segment $BD$ in such a way that $BJ = CD$. The circumscribed circle of a triangle $BD$ intersects the segment $BE$ at point $Q$. Prove that the points $J$, $Q$, and $G$ are collinear.

2023 USA IMO Team Selection Test, 4

Tags: floor function , function , algebra

Let $\lfloor \bullet \rfloor$ denote the floor function. For nonnegative integers $a$ and $b$, their [i]bitwise xor[/i], denoted $a \oplus b$, is the unique nonnegative integer such that $$ \left \lfloor \frac{a}{2^k} \right \rfloor+ \left\lfloor\frac{b}{2^k} \right\rfloor - \left\lfloor \frac{a\oplus b}{2^k}\right\rfloor$$ is even for every $k \ge 0$. Find all positive integers $a$ such that for any integers $x>y\ge 0$, we have \[ x\oplus ax \neq y \oplus ay. \] [i]Carl Schildkraut[/i]

1990 Tournament Of Towns, (278) 3

Tags: combinatorics , geometry , combinatorial geometry

A finite set $M$ of unit squares on the plane is considered. The sides of the squares are parallel to the coordinate axes and the squares are allowed to intersect. It is known that the distance between the centres of any pair of squares is no greater than $2$. Prove that there exists a unit square (not necessarily belonging to $M$) with sides parallel to the coordinate axes and which has at least one common point with each of the squares in $M$. (A Andjans, Riga)

2015 Germany Team Selection Test, 2

Tags: integer , positive , consecutive , form , number theory

A positive integer $n$ is called [i]naughty[/i] if it can be written in the form $n=a^b+b$ with integers $a,b \geq 2$. Is there a sequence of $102$ consecutive positive integers such that exactly $100$ of those numbers are naughty?

1969 IMO Longlists, 8

Tags: function , algebra , functional equation , continuous function

Find all functions $f$ defined for all $x$ that satisfy the condition $xf(y) + yf(x) = (x + y)f(x)f(y),$ for all $x$ and $y.$ Prove that exactly two of them are continuous.

2017 Brazil Team Selection Test, 1

Tags: combinatorics , binary , string

The leader of an IMO team chooses positive integers $n$ and $k$ with $n > k$, and announces them to the deputy leader and a contestant. The leader then secretly tells the deputy leader an $n$-digit binary string, and the deputy leader writes down all $n$-digit binary strings which differ from the leader’s in exactly $k$ positions. (For example, if $n = 3$ and $k = 1$, and if the leader chooses $101$, the deputy leader would write down $001, 111$ and $100$.) The contestant is allowed to look at the strings written by the deputy leader and guess the leader’s string. What is the minimum number of guesses (in terms of $n$ and $k$) needed to guarantee the correct answer?

1994 All-Russian Olympiad, 8

Tags: combinatorics , combinatorial geometry , grid

A plane is divided into unit squares by two collections of parallel lines. For any $n\times n$ square with sides on the division lines, we define its frame as the set of those unit squares which internally touch the boundary of the $n\times n$ square. Prove that there exists only one way of covering a given $100\times 100$ square whose sides are on the division lines with frames of $50$ squares (not necessarily contained in the $100\times 100$ square). (A. Perlin)

2002 District Olympiad, 4

Tags: perimeter , geometry , 3d geometry , cube , geometric inequality

The cube $ABCDA' B' C' D' $has of length a. Consider the points $K \in [AB], L \in [CC' ], M \in [D'A']$. a) Show that $\sqrt3 KL \ge KB + BC + CL$ b) Show that the perimeter of triangle $KLM$ is strictly greater than $2a\sqrt3$.

2005 Tournament of Towns, 2

Tags: number theory

Prove that one of the digits 1, 2 and 9 must appear in the base-ten expression of $n$ or $3n$ for any positive integer $n$. [i](4 points)[/i]

1996 Singapore Team Selection Test, 3

Tags: sequence , number theory , divide , divisible

Let $S = \{0, 1, 2, .., 1994\}$. Let $a$ and $b$ be two positive numbers in $S$ which are relatively prime. Prove that the elements of $S$ can be arranged into a sequence $s_1, s_2, s_3,... , s_{1995}$ such that $s_{i+1} - s_i \equiv \pm a$ or $\pm b$ (mod $1995$) for $i = 1, 2, ... , 1994$

2004 IMO, 5

Tags: isogonal conjugate , angle chasing , easier p5 , geometry

In a convex quadrilateral $ABCD$, the diagonal $BD$ bisects neither the angle $ABC$ nor the angle $CDA$. The point $P$ lies inside $ABCD$ and satisfies \[\angle PBC=\angle DBA\quad\text{and}\quad \angle PDC=\angle BDA.\] Prove that $ABCD$ is a cyclic quadrilateral if and only if $AP=CP$.

2009 Stanford Mathematics Tournament, 8

Tags: geometry , probability

Three points are randomly placed on a circle. What is the probability that they lie on the same semicircle

2005 Bundeswettbewerb Mathematik, 3

Tags: trigonometry , geometry

Let $ABC$ be a triangle with sides $a$, $b$, $c$ and (corresponding) angles $A$, $B$, $C$. Prove that if $3A + 2B = 180^{\circ}$, then $a^2+bc=c^2$. [b]Additional problem:[/b] Prove that the converse also holds, i. e. prove the following: Let $ABC$ be an arbitrary triangle. Then, $3A + 2B = 180^{\circ}$ if and only if $a^2+bc=c^2$. [b]Similar problem:[/b] Let $ABC$ be an arbitrary triangle. Then, $3A + 2B = 360^{\circ}$ if and only if $a^2-bc=c^2$.

2020 Saint Petersburg Mathematical Olympiad, 5.

Tags: geometry , circumcircle

Rays $\ell, \ell_1, \ell_2$ have the same starting point $O$, such that the angle between $\ell$ and $\ell_2$ is acute and the ray $\ell_1$ lies inside this angle. The ray $\ell$ contains a fixed point of $F$ and an arbitrary point $L$. Circles passing through $F$ and $L$ and tangent to $\ell_1$ at $L_1$, and passing through $F$ and $L$ and tangent to $\ell_2$ at $L_2$. Prove that the circumcircle of $\triangle FL_1L_2$ passes through a fixed point other than $F$ independent on $L$.

2017 OMMock - Mexico National Olympiad Mock Exam, 3

Tags: square , number theory , system

Let $x, y, z$ be positive integers such that $xy=z^2+2$. Prove that there exist integers $a, b, c, d$ such that the following equalities are satisfied: \begin{eqnarray*} x=a^2+2b^2\\ y=c^2+d^2\\ z=ac+2bd\\ \end{eqnarray*} [i]Proposed by Isaac Jiménez[/i]

2025 Belarusian National Olympiad, 9.1

Tags: geometry

Altitudes $BE$ and $CF$ of triangle $ABC$ intersect in $H$. A perpendicular $HT$ from $H$ to $EF$ is drawn. Circumcircles $ABC$ and $BHT$ intersect at $B$ and $X$. Prove that $\angle TXA= \angle BAC$. [i]Vadzim Kamianetski[/i]

May Olympiad L1 - geometry, 2010.1

Tags: 3d geometry , geometry

A closed container in the shape of a rectangular parallelepiped contains $1$ liter of water. If the container rests horizontally on three different sides, the water level is $2$ cm, $4$ cm and $5$ cm. Calculate the volume of the parallelepiped.