Olympiad problems

2012 Puerto Rico Team Selection Test, 3

Tags: geometry

$ABC$ is a triangle that is inscribed in a circle. The angle bisectors of $A, B, C$ meet the circle at $D, E, F$, respectively. Show that $AD$ is perpendicular to $EF$.

2016 USA Team Selection Test, 2

Tags: algebra , functional equation

Let $n \ge 4$ be an integer. Find all functions $W : \{1, \dots, n\}^2 \to \mathbb R$ such that for every partition $[n] = A \cup B \cup C$ into disjoint sets, \[ \sum_{a \in A} \sum_{b \in B} \sum_{c \in C} W(a,b) W(b,c) = |A| |B| |C|. \]

2012 CHMMC Fall, 1

Tags: number theory

Find the remainder when $5^{2012}$ is divided by $3$.

2014 Singapore Senior Math Olympiad, 11

Tags: logarithm

Suppose that $x$ is real number such that $\frac{27\times 9^x}{4^x}=\frac{3^x}{8^x}$. Find the value of $2^{-(1+\log_23)x}$

2008 Bulgarian Autumn Math Competition, Problem 10.1

Tags: real root , parameter , algebra

For which values of the parameter $a$ does the equation \[(2x-a)\sqrt{ax^2-(a^2+a+2)x+2(a+1)}=0\] has three different real roots.

2004 AMC 8, 16

Tags:

Two $600$ ml pitchers contain orange juice. One pitcher is $\frac{1}{3}$ full and the other pitcher is $\frac{2}{5}$ full. Water is added to fill each pitcher completely, then both pitchers are poured into one large container. What fraction of the mixture in the large container is orange juice? $\textbf{(A)}\ \frac{1}{8}\qquad \textbf{(B)}\ \frac{3}{16}\qquad \textbf{(C)}\ \frac{11}{30}\qquad \textbf{(D)}\ \frac{11}{19}\qquad \textbf{(E)}\ \frac{11}{15}$

1986 China National Olympiad, 2

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

In $\triangle ABC$, the length of altitude $AD$ is $12$, and the bisector $AE$ of $\angle A$ is $13$. Denote by $m$ the length of median $AF$. Find the range of $m$ when $\angle A$ is acute, orthogonal and obtuse respectively.

2018 Saudi Arabia JBMO TST, 3

Tags: inequalities

Prove that in every triangle there are two sides with lengths $x$ and $y$ such that $$\frac{\sqrt{5}-1}{2}\leq\frac{x}{y}\leq\frac{\sqrt{5}+1}{2}$$

2024 ELMO Shortlist, A4

Tags: algebra

The number $2024$ is written on a blackboard. Each second, if there exist positive integers $a,b,k$ such that $a^k+b^k$ is written on the blackboard, you may write $a^{k'}+b^{k'}$ on the blackboard for any positive integer $k'.$ Find all positive integers that you can eventually write on the blackboard. [i]Srinivas Arun[/i]

2010 Tuymaada Olympiad, 1

Tags: limit , calculus , integration , algebra

We have a set $M$ of real numbers with $|M|>1$ such that for any $x\in M$ we have either $3x-2\in M$ or $-4x+5\in M$. Show that $M$ is infinite.

2009 ISI B.Math Entrance Exam, 6

Tags: number theory

Let $a,b,c,d$ be integers such that $ad-bc$ is non zero. Suppose $b_1,b_2$ are integers both of which are multiples of $ad-bc$. Prove that there exist integers simultaneously satisfying both the equalities $ax+by=b_1, cx+dy=b_2$.

Kvant 2024, M2783

Tags: number theory , sum of digits

The sum of the digits of a natural number is $k{}.$ What is the largest possible sum of digits for[list=a] [*] the square of this number; [*]the fourth power of this number, [/list] given that $k\geqslant 4.$ [i]From the folklore[/i]

1996 Romania National Olympiad, 2

Tags: inequalities

$ a,b,c,d \in [0,1]$ and $ x,y,z,t \in [0, \frac{1}{2}]$ and $ a+b+c+d=x+y+z+t=1$.prove that: $ (i)$ $ ax+by+cz+dt$ $ \geq$ $ min( {\frac{a+b}{2} , \frac{b+c}{2} , \frac{c+d}{2} , \frac{d+a}{2} , \frac{a+c}{2} , \frac{b+d}{2} )}$ $ (ii)$ $ ax+by+cz+dt$ $ \geq$ $ 54abcd$

2014 China Northern MO, 1

Tags: geometry , perpendicular , isosceles

As shown in the figure, given $\vartriangle ABC$ with $\angle B$, $\angle C$ acute angles, $AD \perp BC$, $DE \perp AC$, $M$ midpoint of $DE$, $AM \perp BE$. Prove that $\vartriangle ABC$ is isosceles. [img]https://cdn.artofproblemsolving.com/attachments/a/8/f553c33557979f6f7b799935c3bde743edcc3c.png[/img]

2005 Today's Calculation Of Integral, 75

Tags: calculus , integration , function , trigonometry , induction , calculus computations

A function $f(\theta)$ satisfies the following conditions $(a),(b)$. $(a)\ f(\theta)\geq 0$ $(b)\ \int_0^{\pi} f(\theta)\sin \theta d\theta =1$ Prove the following inequality. \[\int_0^{\pi} f(\theta)\sin n\theta \ d\theta \leq n\ (n=1,2,\cdots)\]

2017 Indonesia MO, 1

Tags: parallelogram , geometry

$ABCD$ is a parallelogram. $g$ is a line passing $A$. Prove that the distance from $C$ to $g$ is either the sum or the difference of the distance from $B$ to $g$, and the distance from $D$ to $g$.

2024/2025 TOURNAMENT OF TOWNS, P5

Tags: geometry

Given a circle ${\omega }_{1}$ , and a circle ${\omega }_{2}$ inside it. An arbitrary circle ${\omega }_{3}$ is chosen which is tangent to the two latter circles and both tangencies are internal. The tangency points are linked by a segment. A tangent line to ${\omega }_{2}$ is drawn through the meet point of this segment and the circle ${\omega }_{2}$ . Thus a chord of the circle ${\omega }_{3}$ is obtained. Prove that the ends of all such chords (obtained by all possible choices of ${\omega }_{3}$ ) belong to a fixed circle. Pavel Kozhevnikov

2021 Purple Comet Problems, 16

Tags:

Paula rolls three standard fair dice. The probability that the three numbers rolled on the dice are the side lengths of a triangle with positive area is $\tfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

2000 Romania Team Selection Test, 2

Tags: polynomial , algebra

Let $P,Q$ be two monic polynomials with complex coefficients such that $P(P(x))=Q(Q(x))$ for all $x$. Prove that $P=Q$. [i]Marius Cavachi[/i]

2013 IPhOO, 6

Tags: trigonometry , mechanics

A fancy bathroom scale is calibrated in Newtons. This scale is put on a ramp, which is at a $40^\circ$ angle to the horizontal. A box is then put on the scale and the box-scale system is then pushed up the ramp by a horizontal force $F$. The system slides up the ramp at a constant speed. If the bathroom scale reads $R$ and the coefficient of static friction between the system and the ramp is $0.40$, what is $\frac{F}{R}$? Round to the nearest thousandth. [i](Proposed by Ahaan Rungta)[/i]

1997 Baltic Way, 9

Tags: geometry , 3d geometry , sphere , number theory

The worlds in the Worlds’ Sphere are numbered $1,2,3,\ldots $ and connected so that for any integer $n\ge 1$, Gandalf the Wizard can move in both directions between any worlds with numbers $n,2n$ and $3n+1$. Starting his travel from an arbitrary world, can Gandalf reach every other world?

1996 Cono Sur Olympiad, 5

Tags: geometry , combinatorics

We want to cover totally a square(side is equal to $k$ integer and $k>1$) with this rectangles: $1$ rectangle ($1\times 1$), $2$ rectangles ($2\times 1$), $4$ rectangles ($3\times 1$),...., $2^n$ rectangles ($n + 1 \times 1$), such that the rectangles can't overlap and don't exceed the limits of square. Find all $k$, such that this is possible and for each $k$ found you have to draw a solution

2003 Bulgaria Team Selection Test, 5

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

Let $ABCD$ be a circumscribed quadrilateral and let $P$ be the orthogonal projection of its in center on $AC$. Prove that $\angle {APB}=\angle {APD}$

1976 Miklós Schweitzer, 9

Tags: advanced fields , combinatorial geometry

Let $ D$ be a convex subset of the $ n$-dimensional space, and suppose that $ D'$ is obtained from $ D$ by applying a positive central dilatation and then a translation. Suppose also that the sum of the volumes of $ D$ and $ D'$ is $ 1$, and $ D \cap D'\not\equal{} \emptyset .$ Determine the supremum of the volume of the convex hull of $ D \cup D'$ taken for all such pairs of sets $ D,D'$. [i]L. Fejes-Toth, E. Makai[/i]

1998 Czech and Slovak Match, 2

Tags: integer root , integer polynomial , algebra

A polynomial $P(x)$ of degree $n \ge 5$ with integer coefficients has $n$ distinct integer roots, one of which is $0$. Find all integer roots of the polynomial $P(P(x))$.