Olympiad problems

1993 Irish Math Olympiad, 4

Tags: integration , trigonometry

Let $x$ be a real number with $0<x<\pi $.Prove that, for all natural number $n$ ,\[sinx+\frac{sin3x}{3}+\frac{sin5x}{5}+\cdots+\frac{sin(2n-1)x}{2n-1}>0.\]

2019 Tournament Of Towns, 4

Tags: angle bisector , geometry , bisects segment , perpendicular

Let $OP$ and $OQ$ be the perpendiculars from the circumcenter $O$ of a triangle $ABC$ to the internal and external bisectors of the angle $B$. Prove that the line$ PQ$ divides the segment connecting midpoints of $CB$ and $AB$ into two equal parts. (Artemiy Sokolov)

2017 F = ma, 9

Tags: fluids

Flasks A, B, and C each have a circular base with a radius of 2 cm. An equal volume of water is poured into each flask, and none overflow. Rank the force of water F on the base of the flask from greatest to least. A) $F_A > F_B > F_C$ B) $F_A > F_C > F_B$ C) $F_B > F_C > F_A$ D) $F_C > F_A > F_B$ E) $F_A = F_B = F_C$

2001 AIME Problems, 4

Tags: geometry , trigonometry , trig identities , law of sines

In triangle $ABC$, angles $A$ and $B$ measure 60 degrees and 45 degrees, respectively. The bisector of angle $A$ intersects $\overline{BC}$ at $T$, and $AT=24.$ The area of triangle $ABC$ can be written in the form $a+b\sqrt{c},$ where $a$, $b$, and $c$ are positive integers, and $c$ is not divisible by the square of any prime. Find $a+b+c.$

2015 Iran Team Selection Test, 2

Tags: geometry

$I_b$ is the $B$-excenter of the triangle $ABC$ and $\omega$ is the circumcircle of this triangle. $M$ is the middle of arc $BC$ of $\omega$ which doesn't contain $A$. $MI_b$ meets $\omega$ at $T\not =M$. Prove that $$ TB\cdot TC=TI_b^2.$$

2024 Germany Team Selection Test, 2

Tags: number theory

Let $a_1<a_2<a_3<\dots$ be positive integers such that $a_{k+1}$ divides $2(a_1+a_2+\dots+a_k)$ for every $k\geqslant 1$. Suppose that for infinitely many primes $p$, there exists $k$ such that $p$ divides $a_k$. Prove that for every positive integer $n$, there exists $k$ such that $n$ divides $a_k$.

KoMaL A Problems 2022/2023, A. 845

Tags: geometry

The incircle of triangle $ABC$ is tangent to sides $BC$, $AC$, and $AB$ at points $D$, $E$ and $F$, respectively. Let $A'$ denote the point of the incircle for which circle $(A'BC)$ is tangent to the incircle. Define points $B'$ and $C'$ similarly. Prove that lines $A'D$, $BE'$, and $CF'$ are concurrent. [i]Proposed by Áron Bán-Szabó, Budapest[/i]

2023 CCA Math Bonanza, L3.3

Tags: logarithm

Given that $\log_{10}(4) = 0.6021$ to the nearest ten-thousandth, find $\log_{10}(5)$ to the nearest thousandth. [i]Lightning 3.3[/i]

2010 Saudi Arabia BMO TST, 4

Tags: system of equations , system , diophantine , number theory

Find all triples $(x,y, z)$ of integers such that $$\begin{cases} x^2y + y^2z + z^2x= 2010^2 \\ xy^2 + yz^2 + zx^2= -2010 \end{cases}$$

2018 Iran MO (3rd Round), 4

Tags: number theory

Prove that for any natural numbers$a,b$ there exist infinity many prime numbers $p$ so that $Ord_p(a)=Ord_p(b)$(Proving that there exist infinity prime numbers $p$ so that $Ord_p(a) \ge Ord_p(b)$ will get a partial mark)

2021 AMC 10 Fall, 4

Tags:

Mr. Lopez has a choice of two routes to get to work. Route A is $6$ miles long, and his average speed along this route is $30$ miles per hour. Route B is $5$ miles long, and his average speed along this route is $40$ miles per hour, except for a $\frac{1}{2}$-mile stretch in a school zone where his average speed is $20$ miles per hour. By how many minutes is Route B quicker than Route A? $\textbf{(A)}\ 2 \frac{3}{4} \qquad\textbf{(B)}\ 3 \frac{3}{4} \qquad\textbf{(C)}\ 4 \frac{1}{2} \qquad\textbf{(D)}\ 5 \frac{1}{2} \qquad\textbf{(E)}\ 6 \frac{3}{4}$

2021 Abels Math Contest (Norwegian MO) Final, 3a

Tags: number theory , perfect square

For which integers $0 \le k \le 9$ do there exist positive integers $m$ and $n$ so that the number $3^m + 3^n + k$ is a perfect square?

2019 Novosibirsk Oral Olympiad in Geometry, 7

Tags: geometry , geometric inequality , convex polygon

Denote $X,Y$ two convex polygons, such that $X$ is contained inside $Y$. Denote $S (X)$, $P (X)$, $S (Y)$, $P (Y)$ the area and perimeter of the first and second polygons, respectively. Prove that $$ \frac{S(X)}{P(X)}<2 \frac{S(Y)}{P(Y)}.$$

2005 IMO Shortlist, 4

Tags: geometry , spiral similarity

Let $ABCD$ be a fixed convex quadrilateral with $BC=DA$ and $BC$ not parallel with $DA$. Let two variable points $E$ and $F$ lie of the sides $BC$ and $DA$, respectively and satisfy $BE=DF$. The lines $AC$ and $BD$ meet at $P$, the lines $BD$ and $EF$ meet at $Q$, the lines $EF$ and $AC$ meet at $R$. Prove that the circumcircles of the triangles $PQR$, as $E$ and $F$ vary, have a common point other than $P$.

2011 Tournament of Towns, 6

Tags: combinatorics

Two ants crawl along the sides of the $49$ squares of a $7 * 7$ board. Each ant passes through all $64$ vertices exactly once and returns to its starting point. What is the smallest possible number of sides covered by both ants?

1981 Bulgaria National Olympiad, Problem 1

Tags: combinatorics , geometry , combinatorial geometry , graph theory

Five points are given in space, no four of which are coplanar. Each of the segments connecting two of them is painted in white, green or red, so that all the colors are used and no three segments of the same color form a triangle. Prove that among these five points there is one at which segments of all the three colors meet.

2024-IMOC, G4

Tags: geometry

Given triangle $ABC$ with $AB<AC$ and its circumcircle $\Omega$. Let $I$ be the incenter of $ABC$, and the feet from $I$ to $BC$ is $D$. The circle with center $A$ and radius $AI$ intersects $\Omega$ at $E$ and $F$. $P$ is a point on $EF$ such that $DP$ is parallel to $AI$. Prove that $AP$ and $MI$ intersects on $\Omega$ where $M$ is the midpoint of arc $BAC$. [hide = Remark] In the test, the incenter called $O$ and the circumcircle called $Luna$ $Cabrera$ You have to prove $AP \cap MO \in Luna$ $Cabrera$ [/hide] [i]Proposed by BlessingOfHeaven[/i]

2003 Germany Team Selection Test, 1

Tags: algebra , functional equation

Find all functions $f$ from the reals to the reals such that \[f\left(f(x)+y\right)=2x+f\left(f(y)-x\right)\] for all real $x,y$.

2012 Today's Calculation Of Integral, 823

Tags: calculus , integration , trigonometry , geometry , geometric transformation , rotation , calculus computations

Let $C$ be the curve expressed by $x=\sin t,\ y=\sin 2t\ \left(0\leq t\leq \frac{\pi}{2}\right).$ (1) Express $y$ in terms of $x$. (2) Find the area of the figure $D$ enclosed by the $x$-axis and $C$. (3) Find the volume of the solid generated by a rotation of $D$ about the $y$-axis.

2016 Singapore MO Open, 4

Tags: coefficient , polynomial , positive coefficients , algebra

Let $b$ be a number with $-2 < b < 0$. Prove that there exists a positive integer $n$ such that all the coefficients of the polynomial $(x + 1)^n(x^2 + bx + 1)$ are positive.

Brazil L2 Finals (OBM) - geometry, 2004.5

Tags: circumcenter , right triangle , right angle , tangent , geometry

Let $D$ be the midpoint of the hypotenuse $AB$ of a right triangle $ABC$. Let $O_1$ and $O_2$ be the circumcenters of the $ADC$ and $DBC$ triangles, respectively. a) Prove that $\angle O_1DO_2$ is right. b) Prove that $AB$ is tangent to the circle of diameter $O_1O_2$ .

1970 Yugoslav Team Selection Test, Problem 2

Tags: 3d geometry , geometry

Describe how to place the vertices of a triangle in the faces of a cube in such a way that the shortest side of the triangle is the biggest possible.

1969 IMO, 5

Tags: combinatorics , combinatorial geometry , counting , convex quadrilateral

Given $n>4$ points in the plane, no three collinear. Prove that there are at least $\frac{(n-3)(n-4)}{2}$ convex quadrilaterals with vertices amongst the $n$ points.

2007 Harvard-MIT Mathematics Tournament, 12

Tags: arithmetic sequence

Let $A_{11}$ denote the answer to problem $11$. Determine the smallest prime $p$ such that the arithmetic sequence $p,p+A_{11},p+2A_{11},\cdots$ begins with the largest number of primes. There is just one triple of possible $(A_{10},A_{11},A_{12})$ of answers to these three problems. Your team will receive credit only for answers matching these. (So, for example, submitting a wrong answer for problem $11$ will not alter the correctness of your answer to problem $12$.)

May Olympiad L1 - geometry, 2017.3

Tags: geometry , rhombus , area

Let $ABCD$ be a rhombus of sides $AB = BC = CD= DA = 13$. On the side $AB$ construct the rhombus $BAFE$ outside $ABCD$ and such that the side $AF$ is parallel to the diagonal $BD$ of $ABCD$. If the area of $BAFE$ is equal to $65$, calculate the area of $ABCD$.