Olympiad problems

2010 Today's Calculation Of Integral, 607

Tags: calculus , integration , analytic geometry , logarithms , geometry , function , parameterization

On the coordinate plane, Let $C$ be the graph of $y=(\ln x)^2\ (x>0)$ and for $\alpha >0$, denote $L(\alpha)$ be the tangent line of $C$ at the point $(\alpha ,\ (\ln \alpha)^2).$ (1) Draw the graph. (2) Let $n(\alpha)$ be the number of the intersection points of $C$ and $L(\alpha)$. Find $n(\alpha)$. (3) For $0<\alpha <1$, let $S(\alpha)$ be the area of the region bounded by $C,\ L(\alpha)$ and the $x$-axis. Find $S(\alpha)$. 2010 Tokyo Institute of Technology entrance exam, Second Exam.

1988 All Soviet Union Mathematical Olympiad, 473

Tags: combinatorics , game , game strategy

Form $10A$ has $29$ students who are listed in order on its duty roster. Form $10B$ has $32$ students who are listed in order on its duty roster. Every day two students are on duty, one from form $10A$ and one from form $10B$. Each day just one of the students on duty changes and is replaced by the following student on the relevant roster (when the last student on a roster is replaced he is replaced by the first). On two particular days the same two students were on duty. Is it possible that starting on the first of these days and ending the day before the second, every pair of students (one from $10A$ and one from $10B$) shared duty exactly once?

2003 Tournament Of Towns, 5

Tags: inequalities , geometry unsolved , geometry

A point $O$ lies inside of the square $ABCD$. Prove that the difference between the sum of angles $OAB, OBC, OCD , ODA$ and $180^{\circ}$ does not exceed $45^{\circ}$.

PEN P Problems, 11

Tags: inequalities , Additive Number Theory

For each positive integer $n$, let $f(n)$ denote the number of ways of representing $n$ as a sum of powers of 2 with nonnegative integer exponents. Representations which differ only in the ordering of their summands are considered to be the same. For instance, $f(4)=4$, because the number $4$ can be represented in the following four ways: \[4, 2+2, 2+1+1, 1+1+1+1.\] Prove that, for any integer $n \geq 3$, \[2^{n^{2}/4}< f(2^{n}) < 2^{n^{2}/2}.\]

1987 USAMO, 3

Tags: algebra , polynomial , induction , algebra unsolved

Construct a set $S$ of polynomials inductively by the rules: (i) $x\in S$; (ii) if $f(x)\in S$, then $xf(x)\in S$ and $x+(1-x)f(x)\in S$. Prove that there are no two distinct polynomials in $S$ whose graphs intersect within the region $\{0 < x < 1\}$.

2018 Azerbaijan JBMO TST, 4

Tags: combinatorics , TST , AZE JBMO TST

An $n\times n$ square table is divided into $n^2$ unit cells. Some unit segments of the obtained grid (i.e. the side of any unit cell) is colored black so that any unit cell of the given square has exactly one black side. Find [b]a)[/b] the smallest [b]b)[/b] the greatest possible number of black unit segments.

1993 India Regional Mathematical Olympiad, 7

Tags: vector

In the group of ten persons, each person is asked to write the sum of the ages of all the other nine persons. Of all ten sums form the nine-element set $\{ 82, 83,84,85,87,89,90,91,92 \}$, find the individual ages of the persons, assuming them to be whole numbers.

2017 BMT Spring, 9

Tags: geometry , right triangle

Let $AB = 10$ be a diameter of circle $P$. Pick point $C$ on the circle such that $AC = 8$. Let the circle with center $O$ be the incircle of $\vartriangle ABC$. Extend line $AO$ to intersect circle $P$ again at $D$. Find the length of $BD$.

2005 Baltic Way, 16

Tags: number theory , greatest common divisor , number theory proposed

Let $n$ be a positive integer, let $p$ be prime and let $q$ be a divisor of $(n + 1)^p - n^p$. Show that $p$ divides $q - 1$.

2021 Romanian Master of Mathematics Shortlist, A2

Tags: interpolation , RMM Shortlist , algebra , polynomial

Let $n$ be a positive integer and let $x_1,\ldots,x_n,y_1,\ldots,y_n$ be integers satisfying the following condition: the numbers $x_1,\ldots,x_n$ are pairwise distinct and for every positive integer $m$ there exists a polynomial $P_m$ with integer coefficients such that $P_m(x_i) - y_i$, $i=1,\ldots,n$, are all divisible by $m$. Prove that there exists a polynomial $P$ with integer coefficients such that $P(x_i) = y_i$ for all $i=1,\ldots,n$.

2000 Harvard-MIT Mathematics Tournament, 37

Tags: geometry , 3D geometry

A cone with semivertical angle $30^{\circ}$ is half filled with water. What is the angle it must be tilted by so that water starts spilling?

1983 IMO Longlists, 40

Tags: ratio , geometry , 3D geometry , tetrahedron , sphere , function , congruent triangles

Four faces of tetrahedron $ABCD$ are congruent triangles whose angles form an arithmetic progression. If the lengths of the sides of the triangles are $a < b < c$, determine the radius of the sphere circumscribed about the tetrahedron as a function on $a, b$, and $c$. What is the ratio $c/a$ if $R = a \ ?$

2003 China Team Selection Test, 2

Tags: geometry , circumcircle , symmetry , projective geometry , power of a point , radical axis , geometry proposed

Denote by $\left(ABC\right)$ the circumcircle of a triangle $ABC$. Let $ABC$ be an isosceles right-angled triangle with $AB=AC=1$ and $\measuredangle CAB=90^{\circ}$. Let $D$ be the midpoint of the side $BC$, and let $E$ and $F$ be two points on the side $BC$. Let $M$ be the point of intersection of the circles $\left(ADE\right)$ and $\left(ABF\right)$ (apart from $A$). Let $N$ be the point of intersection of the line $AF$ and the circle $\left(ACE\right)$ (apart from $A$). Let $P$ be the point of intersection of the line $AD$ and the circle $\left(AMN\right)$. Find the length of $AP$.

2007 Italy TST, 2

Tags: modular arithmetic , combinatorics proposed , combinatorics

In a competition, there were $2n+1$ teams. Every team plays exatly once against every other team. Every match finishes with the victory of one of the teams. We call cyclical a 3-subset of team ${ A,B,C }$ if $A$ won against $B$, $B$ won against $C$ , $C$ won against $A$. (a) Find the minimum of cyclical 3-subset (depending on $n$); (b) Find the maximum of cyclical 3-subset (depending on $n$).

2005 District Olympiad, 3

Tags: geometry , 3D geometry , tetrahedron , circumcircle , trigonometry , trig identities , Law of Sines

Prove that if the circumcircles of the faces of a tetrahedron $ABCD$ have equal radii, then $AB=CD$, $AC=BD$ and $AD=BC$.

2005 Belarusian National Olympiad, 4

Tags: combinatorics

An $n \times n$ table is called good if one can paint its cells with three colors so that, for any two different rows and two different columns, the four cells at their intersections are not all of the same color. a)Show, that exists good $9 \times 9$ good table. b)Prove, that fif $n \times n$ table is good, then $n<11$

2012 NIMO Summer Contest, 2

Tags: inequalities

Compute the number of positive integers $n$ satisfying the inequalities \[ 2^{n-1} < 5^{n-3} < 3^n. \][i]Proposed by Isabella Grabski[/i]

1967 German National Olympiad, 1

Tags: geometry , Locus , Equilateral , square

In a plane, a square $ABCD$ and a point $P$ located inside it are given. Let a point $ Q$ pass through all sides of the square. Describe the set of all those points $R$ in for which the triangle $PQR$ is equilateral.

2022 All-Russian Olympiad, 2

Tags: geometry

On side $BC$ of an acute triangle $ABC$ are marked points $D$ and $E$ so that $BD = CE$. On the arc $DE$ of the circumscribed circle of triangle $ADE$ that does not contain the point $A$, there are points $P$ and $Q$ such that $AB = PC$ and $AC = BQ$. Prove that $AP=AQ$.

2020 HMNT (HMMO), 2

Tags: geometry

In the future, MIT has attracted so many students that its buildings have become skyscrapers. Ben and Jerry decide to go ziplining together. Ben starts at the top of the Green Building, and ziplines to the bottom of the Stata Center. After waiting $a$ seconds, Jerry starts at the top of the Stata Center, and ziplines to the bottom of the Green Building. The Green Building is $160$ meters tall, the Stata Center is $90$ meters tall, and the two buildings are $120$ meters apart. Furthermore, both zipline at $10$ meters per second. Given that Ben and Jerry meet at the point where the two ziplines cross, compute $100a$.

2013 Bangladesh Mathematical Olympiad, 8

Tags: geometry

$\triangle ABC$ is an acute angled triangle. Perpendiculars drawn from its vertices on the opposite sides are $AD$, $BE$ and $CF$. The line parallel to $ DF$ through $E$ meets $BC$ at $Y$ and $BA$ at $X$. $DF$ and $CA$ meet at $Z$. Circumcircle of $XYZ$ meets $AC$ at $S$. Given, $\angle B=33 ^\circ.$ find the angle $\angle FSD $ with proof.

1979 Czech And Slovak Olympiad IIIA, 3

Tags: geometry , geometric inequality , square , cyclic quadrilateral

If in a quadrilateral $ABCD$ whose vertices lie on a circle of radius $1$, holds $$|AB| \cdot |BC| \cdot |CD|\cdot |DA| \ge 4$$, then $ABCD$ is a square. Prove it. [hide=Hint given in contest] You can use Ptolemy's formula $|AB| \cdot |CD| + |BC|\cdot |AD|= |AC| \cdot|BD|$[/hide]

2000 National Olympiad First Round, 29

Tags: geometry , geometric transformation , homothety

One of the external common tangent lines of the two externally tangent circles with center $O_1$ and $O_2$ touches the circles at $B$ and $C$, respectively. Let $A$ be the common point of the circles. The line $BA$ meets the circle with center $O_2$ at $A$ and $D$. If $|BA|=5$ and $|AD|=4$, then what is $|CD|$? $ \textbf{(A)}\ \sqrt{20} \qquad\textbf{(B)}\ \sqrt{27} \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ \frac{15}2 \qquad\textbf{(E)}\ 4\sqrt5 $

2005 QEDMO 1st, 2 (G2)

Tags: geometry , geometric transformation , reflection , geometry proposed

Let $ABC$ be a triangle. Let $C^{\prime}$ and $A^{\prime}$ be the reflections of its vertices $C$ and $A$, respectively, in the altitude of triangle $ABC$ issuing from $B$. The perpendicular to the line $BA^{\prime}$ through the point $C^{\prime}$ intersects the line $BC$ at $U$; the perpendicular to the line $BC^{\prime}$ through the point $A^{\prime}$ intersects the line $BA$ at $V$. Prove that $UV \parallel CA$. Darij

2018 Mexico National Olympiad, 6

Tags: geometry , incenter , arc midpoint , angle bisector , geometric transformation , reflection , circumcircle

Let $ABC$ be an acute-angled triangle with circumference $\Omega$. Let the angle bisectors of $\angle B$ and $\angle C$ intersect $\Omega$ again at $M$ and $N$. Let $I$ be the intersection point of these angle bisectors. Let $M'$ and $N'$ be the respective reflections of $M$ and $N$ in $AC$ and $AB$. Prove that the center of the circle passing through $I$, $M'$, $N'$ lies on the altitude of triangle $ABC$ from $A$. [i]Proposed by Victor Domínguez and Ariel García[/i]