Olympiad problems

2013 Bosnia And Herzegovina - Regional Olympiad, 2

In circle with radius $10$, point $M$ is on chord $PQ$ such that $PM=5$ and $MQ=10$. Through point $M$ we draw chords $AB$ and $CD$, and points $X$ and $Y$ are intersection points of chords $AD$ and $BC$ with chord $PQ$ (see picture), respectively. If $XM=3$ find $MY$ [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvYy9kLzBiMmFmM2ViOGVmOTlmZDA5NGY2ZWY4MjM1YWI0ZDZjNjJlNzA1LnBuZw==&rn=Z2VvbWV0cmlqYS5wbmc=[/img]

2010 China Team Selection Test, 3

Tags: number theory , modular arithmetic

Fine all positive integers $m,n\geq 2$, such that (1) $m+1$ is a prime number of type $4k-1$; (2) there is a (positive) prime number $p$ and nonnegative integer $a$, such that \[\frac{m^{2^n-1}-1}{m-1}=m^n+p^a.\]

JOM 2015 Shortlist, A4

Tags: algebra , number theory

Suppose $ 2015= a_1 <a_2 < a_3<\cdots <a_k $ be a finite sequence of positive integers, and for all $ m, n \in \mathbb{N} $ and $1\le m,n \le k $, $$ a_m+a_n\ge a_{m+n}+|m-n| $$ Determine the largest possible value $ k $ can obtain.

2004 USAMTS Problems, 4

Tags: arithmetic sequence

The interior angles of a convex polygon form an arithmetic progression with a common difference of $4^\circ$. Determine the number of sides of the polygon if its largest interior angle is $172^\circ.$

2007 Bosnia Herzegovina Team Selection Test, 6

Tags: combinatorics , induction

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.

2017 Romanian Masters In Mathematics, 6

Tags: cyclic quadrilateral , geometry

Let $ABCD$ be any convex quadrilateral and let $P, Q, R, S$ be points on the segments $AB, BC, CD$, and $DA$, respectively. It is given that the segments $PR$ and $QS$ dissect $ABCD$ into four quadrilaterals, each of which has perpendicular diagonals. Show that the points $P, Q, R, S$ are concyclic.

2017 Sharygin Geometry Olympiad, 1

Tags: geometry , ratio , golden ratio

Let $ABC$ be a regular triangle. The line passing through the midpoint of $AB$ and parallel to $AC$ meets the minor arc $AB$ of the circumcircle at point $K$. Prove that the ratio $AK:BK$ is equal to the ratio of the side and the diagonal of a regular pentagon.

2006 AMC 12/AHSME, 6

Tags:

Francesca uses 100 grams of lemon juice, 100 grams of sugar, and 400 grams of water to make lemonade. There are 25 calories in 100 grams of lemon juice and 386 calories in 100 grams of sugar. Water contains no calories. How many calories are in 200 grams of her lemonade? $ \textbf{(A) } 129 \qquad \textbf{(B) } 137 \qquad \textbf{(C) } 174 \qquad \textbf{(D) } 223 \qquad \textbf{(E) } 411$

2019 BMT Spring, 6

Tags: homothety , geometric transformation , geometry

Let $ \triangle ABE $ be a triangle with $ \frac{AB}{3} = \frac{BE}{4} = \frac{EA}{5} $. Let $ D \neq A $ be on line $ \overline{AE} $ such that $ AE = ED $ and $ D $ is closer to $ E $ than to $ A $. Moreover, let $ C $ be a point such that $ BCDE $ is a parallelogram. Furthermore, let $ M $ be on line $ \overline{CD} $ such that $ \overline{AM} $ bisects $ \angle BAE $, and let $ P $ be the intersection of $ \overline{AM} $ and $ \overline{BE} $. Compute the ratio of $ PM $ to the perimeter of $ \triangle ABE $.

1994 IMO Shortlist, 5

Tags: equation , number theory , binary representation , function

For any positive integer $ k$, let $ f_k$ be the number of elements in the set $ \{ k \plus{} 1, k \plus{} 2, \ldots, 2k\}$ whose base 2 representation contains exactly three 1s. (a) Prove that for any positive integer $ m$, there exists at least one positive integer $ k$ such that $ f(k) \equal{} m$. (b) Determine all positive integers $ m$ for which there exists [i]exactly one[/i] $ k$ with $ f(k) \equal{} m$.

1998 Israel National Olympiad, 6

Tags: polynomial , integer polynomial

Find all pairs $(m,n)$ of integers with $m > n > 7$ for which there exists a polynomial $p(x)$ with integer coefficients such that $p(7) = 77, p(m) = 0$, and $p(n) = 85$.

2010 Canadian Mathematical Olympiad Qualification Repechage, 4

Tags: prime number , number theory

Determine the smallest positive integer $m$ with the property that $m^3-3m^2+2m$ is divisible by both $79$ and $83$.

2012 AIME Problems, 6

Tags: geometry , triangle inequality , inequalities , reflection , geometric transformation

Let $z = a + bi$ be the complex number with $|z| = 5$ and $b > 0$ such that the distance between $(1 + 2i)z^3$ and $z^5$ is maximized, and let $z^4 = c + di$. Find $c+d$.

2014 USA Team Selection Test, 2

Tags: analytic geometry , area , geometry , orthocenter , cyclic quadrilateral

Let $ABCD$ be a cyclic quadrilateral, and let $E$, $F$, $G$, and $H$ be the midpoints of $AB$, $BC$, $CD$, and $DA$ respectively. Let $W$, $X$, $Y$ and $Z$ be the orthocenters of triangles $AHE$, $BEF$, $CFG$ and $DGH$, respectively. Prove that the quadrilaterals $ABCD$ and $WXYZ$ have the same area.

2012 Today's Calculation Of Integral, 813

Tags: integration , calculus , geometry , quadratic , inequalities , calculus computations

Let $a$ be a real number. Find the minimum value of $\int_0^1 |ax-x^3|dx$. How many solutions (including University Mathematics )are there for the problem? Any advice would be appreciated.

2024 Korea National Olympiad, 2

Tags: algebra

For a sequence of positive integers $\{x_n\}$ where $x_1 = 2$ and $x_{n + 1} - x_n \in \{0, 3\}$ for all positve integers $n$, then $\{x_n\}$ is called a "frog sequence". Find all real numbers $d$ that satisfy the following condition. [b](Condition)[/b] For two frog sequence $\{a_n\}, \{b_n\}$, if there exists a positive integer $n$ such that $a_n = 1000b_n$, then there exists a positive integer $m$ such that $a_m = d\cdot b_m$.

1945 Moscow Mathematical Olympiad, 100

Tags: geometry , polygon , minimum , combinatorial geometry

Suppose we have two identical cardboard polygons. We placed one polygon upon the other one and aligned. Then we pierced polygons with a pin at a point. Then we turned one of the polygons around this pin by $25^o 30'$. It turned out that the polygons coincided (aligned again). What is the minimal possible number of sides of the polygons?

2014 Harvard-MIT Mathematics Tournament, 25

Tags: geometry

Let $ABC$ be an equilateral triangle of side length $6$ inscribed in a circle $\omega$. Let $A_1,A_2$ be the points (distinct from $A$) where the lines through $A$ passing through the two trisection points of $BC$ meet $\omega$. Define $B_1,B_2,C_1,C_2$ similarly. Given that $A_1,A_2,B_1,B_2,C_1,C_2$ appear on $\omega$ in that order, find the area of hexagon $A_1A_2B_1B_2C_1C_2$.

2012 China Team Selection Test, 1

Tags: circumcircle , inversion , geometry

Given two circles ${\omega _1},{\omega _2}$, $S$ denotes all $\Delta ABC$ satisfies that ${\omega _1}$ is the circumcircle of $\Delta ABC$, ${\omega _2}$ is the $A$- excircle of $\Delta ABC$ , ${\omega _2}$ touches $BC,CA,AB$ at $D,E,F$. $S$ is not empty, prove that the centroid of $\Delta DEF$ is a fixed point.

2001 USA Team Selection Test, 7

Tags: rhombus , geometry , inequalities

Let $ABCD$ be a convex quadrilateral such that $\angle ABC = \angle ADC = 135^{\circ}$ and \[AC^2\cdot BD^2 = 2\cdot AB\cdot BC\cdot CD\cdot DA.\] Prove that the diagonals of the quadrilateral $ABCD$ are perpendicular.

2023 CMIMC Algebra/NT, 2

Tags: number theory , algebra

Find the largest possible value of $a$ such that there exist real numbers $b,c>1$ such that \[a^{\log_b c}\cdot b^{\log_c a}=2023.\] [i]Proposed by Howard Halim[/i]

2022 Durer Math Competition (First Round), 4

Tags: geometry , collinear

Let $ABC$ be an acute triangle, and let $F_A$ and $F_B$ be the midpoints of sides $BC$ and $CA$, respectively. Let $E$ and $F$ be the intersection points of the circle centered at $F_A$ and passing through $A$ and the circle centered at $F_B$ and passing through $B$. Prove that if segments $CE$ and $CF$ have midpoints $N$ and $M$, respectively, then the intersection points of the circle centered at $M$ and passing through $E$ and the circle centered at $N$ and passing through $F$ lie on the line $AB$.

2015 Miklos Schweitzer, 10

Tags: functional analysis , spectral analysis

Let $f:\mathbb{R}\to \mathbb{R}$ be a continuously differentiable,strictly convex function.Let $H$ be a Hilbert space and $A,B$ be bounded,self adjoint linear operators on $H$.Prove that,if $f(A)-f(B)=f'(B)(A-B)$ then $A=B$.

2018 Stars of Mathematics, 3

Tags: algebra , inequalities , function , floor function

Given a positive integer $n$, determine the largest integer $M$ satisfying $$\lfloor \sqrt{a_1}\rfloor + ... + \lfloor \sqrt{a_n} \rfloor \ge \lfloor\sqrt{ a_1 + ... + a_n +M \cdot min(a_1,..., a_n)}\rfloor $$ for all non-negative integers $a_1,...., a_n$. S. Berlov, A. Khrabrov

2023 Belarus Team Selection Test, 2.3

Tags: hyperbola , conic , projective geometry , geometry

Let $ABC$ be an acute triangle with altitude $\overline{AH}$, and let $P$ be a variable point such that the angle bisectors $k$ and $\ell$ of $\angle PBC$ and $\angle PCB$, respectively, meet on $\overline{AH}$. Let $k$ meet $\overline{AC}$ at $E$, $\ell$ meet $\overline{AB}$ at $F$, and $\overline{EF}$ meet $\overline{AH}$ at $Q$. Prove that as $P$ varies, line $PQ$ passes through a fixed point.