Olympiad problems

2009 Stanford Mathematics Tournament, 6

Tags: geometry

Equilateral triangle $ABC$ has side lengths of $24$. Points $D$, $E$, and $F$ lies on sides $BC$, $CA$, $AB$ such that ${AD}\perp{BC}$, ${DE}\perp{AC}$, and ${EF}\perp{AB}$. $G$ is the intersection of $AD$ and $EF$. Find the area of quadrilateral $BFGD$

PEN H Problems, 74

Tags: diophantine equation

Find all pairs $(a,b)$ of positive integers that satisfy the equation \[a^{a^{a}}= b^{b}.\]

1956 AMC 12/AHSME, 18

Tags: logarithm

If $ 10^{2y} \equal{} 25$, then $ 10^{ \minus{} y}$ equals: $ \textbf{(A)}\ \minus{} \frac {1}{5} \qquad\textbf{(B)}\ \frac {1}{625} \qquad\textbf{(C)}\ \frac {1}{50} \qquad\textbf{(D)}\ \frac {1}{25} \qquad\textbf{(E)}\ \frac {1}{5}$

ICMC 5, 6

Tags: geometry , combinatorics , covering

Is it possible to cover a circle of area $1$ with finitely many equilateral triangles whose areas sum to $1.01$, all pointing in the same direction? [i]Proposed by Ethan Tan[/i]

1987 Tournament Of Towns, (137) 2

Tags: octagon , tangential , combinatorial geometry , combinatorics

Quadrilaterals may be obtained from an octagon by cutting along its diagonals (in $8$ different ways) . Can it happen that among these $8$ quadrilaterals (a) four (b ) five possess an inscribed circle? (P. M . Sedrakyan , Yerevan)

2015 Postal Coaching, Problem 5

Tags: analytic geometry , combinatorics

Let $S$ be a set of in $3-$ space such that each of the points in $S$ has integer coordinates $(x,y,z)$ with $1 \le x,y,z \le n $. Suppose the pairwise distances between these points are all distinct. Prove that $$|S| \le min \{(n+2)\sqrt{\frac{n}{3}},n\sqrt{6} \}$$

2015 China Team Selection Test, 6

Tags: number theory , analytic number theory , squarefree

Prove that there exist infinitely many integers $n$ such that $n^2+1$ is squarefree.

1994 India Regional Mathematical Olympiad, 8

Tags: inequalities

If $a,b,c$ are positive real numbers such that $a+b+c = 1$, prove that \[ (1+a)(1+b)(1+c) \geq 8 (1-a)(1-b)(1-c) . \]

2024 AIME, 12

Tags:

Let $O(0,0)$, $A(\tfrac{1}{2},0)$, and $B(0, \tfrac{\sqrt{3}}{2})$ be points in the coordinate plane. Let $\mathcal{F}$ be the family of segments $\overline{PQ}$ of unit length lying in the first quadrant with $P$ on the $x$-axis and $Q$ on the $y$-axis. There is a unique point $C$ on $\overline{AB}$, distinct from $A$ and $B$, that does not belong to any segment from $\mathcal{F}$ other than $\overline{AB}$. Then $OC^2 = \tfrac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Find $p + q$.

2020 LMT Fall, B8

Tags: geometry

In rectangle $ABCD$, $AB = 3$ and $BC = 4$. If the feet of the perpendiculars from $B$ and $D$ to $AC$ are $X$ and $Y$ , the length of $X Y$ can be expressed in the form m/n , where m and n are relatively prime positive integers. Find $m +n$.

1984 IMO Shortlist, 10

Tags: number theory , equation , product , perfect square

Prove that the product of five consecutive positive integers cannot be the square of an integer.

2015 China National Olympiad, 2

Tags: ratio , trigonometry , geometry

Let $ A, B, D, E, F, C $ be six points lie on a circle (in order) satisfy $ AB=AC $ . Let $ P=AD \cap BE, R=AF \cap CE, Q=BF \cap CD, S=AD \cap BF, T=AF \cap CD $ . Let $ K $ be a point lie on $ ST $ satisfy $ \angle QKS=\angle ECA $ . Prove that $ \frac{SK}{KT}=\frac{PQ}{QR} $

1961 Czech and Slovak Olympiad III A, 2

Tags: geometry , construction , square , isosceles right triangle , geometric construction

Let a right isosceles triangle $APQ$ with the hypotenuse $AP$ be given in plane. Construct such a square $ABCD$ that the lines $BC, CD$ contain points $P, Q,$ respectively. Compute the length of side $AB = b$ in terms of $AQ=a$.

2016 Moldova Team Selection Test, 5

Tags: number theory

The sequence of polynomials $\left( P_{n}(X)\right)_{n\in Z_{>0}}$ is defined as follows: $P_{1}(X)=2X$ $P_{2}(X)=2(X^2+1)$ $P_{n+2}(X)=2X\cdot P_{n+1}(X)-(X^2-1)P_{n}(X)$, for all positive integers $n$. Find all $n$ for which $X^2+1\mid P_{n}(X)$

2021 AMC 10 Spring, 20

Tags: prob

In how many ways can the sequence $1,2,3,4,5$ be arranged so that no three consecutive terms are increasing and no three consecutive terms are decreasing? $\textbf{(A) } 10 \qquad \textbf{(B) } 18 \qquad \textbf{(C) } 24 \qquad \textbf{(D) } 32 \qquad \textbf{(E) } 44$

2002 Belarusian National Olympiad, 3

Tags: combinatorics

There are $20$ cities in Wonderland. The company Wonderland Airways (WA) established $18$ air routes between them. Any of the routes is closed and passes (with landing) through some $5$ different cities. Each city belongs to at least three different routes, for no two cities there exist more than one routes, which allow to fly from one to another without landing. Prove that one can fly from any city of Wonderland to any other one by airplanes of WA. (V. Kaskevich)

2018 Pan-African Shortlist, C2

Tags: game , combinatorics , polynomial , integer root , algebra

Adamu and Afaafa choose, each in his turn, positive integers as coefficients of a polynomial of degree $n$. Adamu wins if the polynomial obtained has an integer root; otherwise, Afaafa wins. Afaafa plays first if $n$ is odd; otherwise Adamu plays first. Prove that: [list] [*] Adamu has a winning strategy if $n$ is odd. [*] Afaafa has a winning strategy if $n$ is even. [/list]

2005 AMC 12/AHSME, 15

Tags:

The sum of four two-digit numbers is $ 221$. None of the eight digits is $ 0$ and no two of them are same. Which of the following is [b]not[/b] included among the eight digits? $ \textbf{(A)}\ 1\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 3\qquad \textbf{(D)}\ 4\qquad \textbf{(E)}\ 5$

2005 Manhattan Mathematical Olympiad, 4

Tags: geometry , search

Here is a problem given at the mathematical test at some school: [i]The hypotenuse of the right triangle is 12 inches. The height (distance from the opposite vertex to the hypotenuse) is 12 inches. Find the area of the triangle[/i] Everybody in the class got the answer $42$ square inches, except for the two best students. Can you explain why the two best students could not get the same answer as the majority?

2013 Today's Calculation Of Integral, 883

Tags: calculus , integration , trigonometry , calculus computations , integral inequality , logarithm

Prove that for each positive integer $n$ \[\frac{4n^2+1}{4n^2-1}\int_0^{\pi} (e^{x}-e^{-x})\cos 2nx\ dx>\frac{e^{\pi}-e^{-\pi}-2}{4}\ln \frac{(2n+1)^2}{(2n-1)(n+3)}.\]

2014 Singapore Senior Math Olympiad, 32

Tags: inequalities , trigonometry

Determine the maximum value of $\frac{8(x+y)(x^3+y^3)}{(x^2+y^2)^2}$ for all $(x,y)\neq (0,0)$

2020 Kazakhstan National Olympiad, 2

Tags: algebra , inequalities

Let $x_1, x_2, ... , x_n$ be a real numbers such that\\ 1) $1 \le x_1, x_2, ... , x_n \le 160$ 2) $x^{2}_{i} + x^{2}_{j} + x^{2}_{k} \ge 2(x_ix_j + x_jx_k + x_kx_i)$ for all $1\le i < j < k \le n$ Find the largest possible $n$.

2007 Harvard-MIT Mathematics Tournament, 32

Tags: circumcircle , geometry

Triangle $ABC$ has $AB=4$, $BC=6$, and $AC=5$. Let $O$ denote the circumcenter of $ABC$. The circle $\Gamma$ is tangent to and surrounds the circumcircles of triangle $AOB$, $BOC$, and $AOC$. Determine the diameter of $\Gamma$.

2024 CMI B.Sc. Entrance Exam, 1

Tags: geometry , integral calculus , calculus

(a) Sketch qualitativly the region with maximum area such that it lies in the first quadrant and is bound by $y=x^2-x^3$ and $y=kx$ where $k$ is a constent. The region must not have any other lines closing it. Note: $kx$ lies above $x^2-x^3$ (b) Find an expression for the volume of the solid obtained by spinning this region about the $y$ axis.

2009 Pan African, 2

Tags: geometric transformation , reflection , geometry

Point $P$ lies inside a triangle $ABC$. Let $D,E$ and $F$ be reflections of the point $P$ in the lines $BC,CA$ and $AB$, respectively. Prove that if the triangle $DEF$ is equilateral, then the lines $AD,BE$ and $CF$ intersect in a common point.