Olympiad problems

2023 Sharygin Geometry Olympiad, 10.5

The incircle of a triangle $ABC$ touches $BC$ at point $D$. Let $M$ be the midpoint of arc $\widehat{BAC}$ of the circumcircle, and $P$, $Q$ be the projections of $M$ to the external bisectors of angles $B$ and $C$ respectively. Prove that the line $PQ$ bisects $AD$.

1946 Moscow Mathematical Olympiad, 115

Tags: trigonometry , inequalities , acute

Prove that if $\alpha$ and $\beta$ are acute angles and $\alpha$ < $\beta$ , then $\frac{tan \alpha}{\alpha} < \frac{tan \beta}{\beta} $

2014 BMT Spring, 2

Tags: geometry , area of a triangle , areas

Suppose $ \vartriangle ABC$ is similar to $\vartriangle DEF$, with $ A$, $ B$, and $C$ corresponding to $D, E$, and $F$ respectively. If $\overline{AB} = \overline{EF}$, $\overline{BC} = \overline{FD}$, and $\overline{CA} = \overline{DE} = 2$, determine the area of $ \vartriangle ABC$.

2022 JHMT HS, 7

Tags: geometry , optimization , slope , 2022

Two rays emanate from the origin $O$ and form a $45^\circ$ angle in the first quadrant of the Cartesian coordinate plane. For some positive numbers $X$, $Y$, and $S$, the ray with the larger slope passes through point $A = (X, S)$, and the ray with the smaller slope passes through point $B = (S, Y)$. If $6X + 6Y + 5S = 600$, then determine the maximum possible area of $\triangle OAB$.

1995 IMC, 4

Tags: real analysis

Let $F:(1,\infty) \rightarrow \mathbb{R}$ be the function defined by $$F(x)=\int_{x}^{x^{2}} \frac{dt}{\ln(t)}.$$ Show that $F$ is injective and find the set of values of $F$.

2009 AIME Problems, 1

Tags: ratio , AMC , AIME

Call a $ 3$-digit number [i]geometric[/i] if it has $ 3$ distinct digits which, when read from left to right, form a geometric sequence. Find the difference between the largest and smallest geometric numbers.

2007 Today's Calculation Of Integral, 203

Tags: calculus , integration , trigonometry , calculus computations

Let $\alpha ,\ \beta$ be the distinct positive roots of the equation of $2x=\tan x$. Evaluate the following definite integral. \[\int_{0}^{1}\sin \alpha x\sin \beta x\ dx \]

2015 Turkey EGMO TST, 5

Tags: geometry , analytic geometry

Let $a \ge b \ge 0$ be real numbers. Find the area of the region defined as; $K=\{(x,y): x\ge y\ge0$ and $\forall n$ positive integers satisfy $a^n+b^n\ge x^n+y^n\}$ in the cordinate plane.

2010 CHMMC Fall, 11

Tags: CHMMC , Fall CHMMC 2010 , combinatorics , probability

Darryl has a six-sided die with faces $1, 2, 3, 4, 5, 6$. He knows the die is weighted so that one face comes up with probability $1/2$ and the other five faces have equal probability of coming up. He unfortunately does not know which side is weighted, but he knows each face is equally likely to be the weighted one. He rolls the die $5$ times and gets a $1, 2, 3, 4$ and $5$ in some unspecified order. Compute the probability that his next roll is a $6$.

2021 Thailand TST, 2

Tags: inequalities , geometric inequality , geometry , disks , IMO Shortlist , IMO Shortlist 2020 , imo shortlist g4

In the plane, there are $n \geqslant 6$ pairwise disjoint disks $D_{1}, D_{2}, \ldots, D_{n}$ with radii $R_{1} \geqslant R_{2} \geqslant \ldots \geqslant R_{n}$. For every $i=1,2, \ldots, n$, a point $P_{i}$ is chosen in disk $D_{i}$. Let $O$ be an arbitrary point in the plane. Prove that \[O P_{1}+O P_{2}+\ldots+O P_{n} \geqslant R_{6}+R_{7}+\ldots+R_{n}.\] (A disk is assumed to contain its boundary.)

2004 Korea National Olympiad, 3

Tags: geometry , rectangle , geometry unsolved

Positive real numbers, $a_1, .. ,a_6$ satisfy $a_1^2+..+a_6^2=2$. Think six squares that has side length of $a_i$ ($i=1,2,\ldots,6$). Show that the squares can be packed inside a square of length $2$, without overlapping.

2017 CCA Math Bonanza, I8

Tags:

Let $a_1,a_2,\ldots,a_{18}$ be a list of prime numbers such that $\frac{1}{64}\times a_1\times a_2\times\cdots\times a_{18}$ is one million. Determine the sum of all positive integers $n$ such that $$\sum_{i=1}^{18}\frac{1}{\log_{a_i}n}$$ is a positive integer. [i]2017 CCA Math Bonanza Individual Round #8[/i]

2005 JHMT, 4

Tags: geometry

Given an isosceles trapezoid $ABCD$ with $AB = 6$, $CD = 12$, and area $36$, find $BC$.

1998 China National Olympiad, 2

Tags: inequalities , geometry , parallelogram , geometry unsolved

Let $D$ be a point inside acute triangle $ABC$ satisfying the condition \[DA\cdot DB\cdot AB+DB\cdot DC\cdot BC+DC\cdot DA\cdot CA=AB\cdot BC\cdot CA.\] Determine (with proof) the geometric position of point $D$.

2002 AMC 12/AHSME, 24

Tags: trigonometry , geometry , AMC , AIME , calculus , integration , modular arithmetic

Find the number of ordered pairs of real numbers $ (a,b)$ such that $ (a \plus{} bi)^{2002} \equal{} a \minus{} bi$. $ \textbf{(A)}\ 1001\qquad \textbf{(B)}\ 1002\qquad \textbf{(C)}\ 2001\qquad \textbf{(D)}\ 2002\qquad \textbf{(E)}\ 2004$

2022 Purple Comet Problems, 24

Tags: combinatorics

Find the number of permutations of the letters $AAABBBCCC$ where no letter appears in a position that originally contained that letter. For example, count the permutations $BBBCCCAAA$ and $CBCAACBBA$ but not the permutation $CABCACBAB$.

2010 Tournament Of Towns, 3

Tags: geometry , 3D geometry , combinatorics unsolved , combinatorics

A $1\times 1\times 1$ cube is placed on an $8\times 8$ chessboard so that its bottom face coincides with a square of the chessboard. The cube rolls over a bottom edge so that the adjacent face now lands on the chessboard. In this way, the cube rolls around the chessboard, landing on each square at least once. Is it possible that a particular face of the cube never lands on the chessboard?

2005 India National Olympiad, 2

Tags: number theory unsolved , number theory

Let $\alpha$ and $\beta$ be positive integers such that $\dfrac{43}{197} < \dfrac{ \alpha }{ \beta } < \dfrac{17}{77}$. Find the minimum possible value of $\beta$.

2007 Bosnia Herzegovina Team Selection Test, 6

Tags: induction , combinatorics unsolved , combinatorics

The set $A$ has exactly $n>4$ elements. Ann chooses $n+1$ distinct subsets of $A$, such that every subset has exactly $3$ elements. Prove that there exist two subsets chosen by Ann which have exactly one common element.

2010 Brazil Team Selection Test, 2

Tags: function , algebra , modular arithmetic , number theory , Divisibility , IMO Shortlist

Let $f$ be a non-constant function from the set of positive integers into the set of positive integer, such that $a-b$ divides $f(a)-f(b)$ for all distinct positive integers $a$, $b$. Prove that there exist infinitely many primes $p$ such that $p$ divides $f(c)$ for some positive integer $c$. [i]Proposed by Juhan Aru, Estonia[/i]

2019 CMIMC, 9

Tags: geometry , circumcircle

Let $ABCD$ be a square of side length $1$, and let $P_1, P_2$ and $P_3$ be points on the perimeter such that $\angle P_1P_2P_3 = 90^\circ$ and $P_1, P_2, P_3$ lie on different sides of the square. As these points vary, the locus of the circumcenter of $\triangle P_1P_2P_3$ is a region $\mathcal{R}$; what is the area of $\mathcal{R}$?

PEN A Problems, 82

Tags: quadratics , function , inequalities , Vieta , Divisibility Theory , Vieta Jumping

Which integers can be represented as \[\frac{(x+y+z)^{2}}{xyz}\] where $x$, $y$, and $z$ are positive integers?

1958 AMC 12/AHSME, 38

Tags: trigonometry , analytic geometry , ratio , function

Let $ r$ be the distance from the origion to a point $ P$ with coordinates $ x$ and $ y$. Designate the ratio $ \frac{y}{r}$ by $ s$ and the ratio $ \frac{x}{r}$ by $ c$. Then the values of $ s^2 \minus{} c^2$ are limited to the numbers: $ \textbf{(A)}\ \text{less than }{\minus{}1}\text{ are greater than }{\plus{}1}\text{, both excluded}\qquad\\ \textbf{(B)}\ \text{less than }{\minus{}1}\text{ are greater than }{\plus{}1}\text{, both included}\qquad \\ \textbf{(C)}\ \text{between }{\minus{}1}\text{ and }{\plus{}1}\text{, both excluded}\qquad \\ \textbf{(D)}\ \text{between }{\minus{}1}\text{ and }{\plus{}1}\text{, both included}\qquad \\ \textbf{(E)}\ {\minus{}1}\text{ and }{\plus{}1}\text{ only}$

2013 Dutch IMO TST, 2

Tags: Perfect Squares , number theory , rational

Determine all integers $n$ for which $\frac{4n-2}{n+5}$ is the square of a rational number.

1979 IMO Longlists, 51

Tags: geometry unsolved , geometry

Let $ABC$ be an arbitrary triangle and let $S_1, S_2,\cdots, S_7$ be circles satisfying the following conditions: $S_1$ is tangent to $CA$ and $AB$, $S_2$ is tangent to $S_1, AB$, and $BC,$ $S_3$ is tangent to $S_2, BC$, and $CA,$ .............................. $S_7$ is tangent to $S_6, CA$ and $AB.$ Prove that the circles $S_1$ and $S_7$ coincide.