Olympiad problems

2015 Math Prize for Girls Problems, 19

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Sabrina has a fair tetrahedral die whose faces are numbered 1, 2, 3, and 4, respectively. She creates a sequence by rolling the die and recording the number on its bottom face. However, she discards (without recording) any roll such that appending its number to the sequence would result in two consecutive terms that sum to 5. Sabrina stops the moment that all four numbers appear in the sequence. Find the expected (average) number of terms in Sabrina's sequence.

2011 Tournament of Towns, 4

The vertices of a $33$-gon are labelled with the integers from $1$ to $33$. Each edge is then labelled with the sum of the labels of its two vertices. Is it possible for the edge labels to consist of $33$ consecutive numbers?

2024 APMO, 1

Tags: geometry

Let $ABC$ be an acute triangle. Let $D$ be a point on side $AB$ and $E$ be a point on side $AC$ such that lines $BC$ and $DE$ are parallel. Let $X$ be an interior point of $BCED$. Suppose rays $DX$ and $EX$ meet side $BC$ at points $P$ and $Q$, respectively, such that both $P$ and $Q$ lie between $B$ and $C$. Suppose that the circumcircles of triangles $BQX$ and $CPX$ intersect at a point $Y \neq X$. Prove that the points $A, X$, and $Y$ are collinear.

2009 Jozsef Wildt International Math Competition, W. 29

Tags: geometry , trigonometry , inequalities

Prove that for all triangle $\triangle ABC$ holds the following inequality $$\sum \limits_{cyc} \left (1-\sqrt{\sqrt{3}\tan \frac{A}{2}}+\sqrt{3}\tan \frac{A}{2}\right )\left (1-\sqrt{\sqrt{3}\tan \frac{B}{2}}+\sqrt{3}\tan \frac{B}{2}\right )\geq 3$$

2022 AIME Problems, 6

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Find the number of ordered pairs of integers $(a, b)$ such that the sequence $$3, 4, 5, a, b, 30, 40, 50$$ is strictly increasing and no set of four (not necessarily consecutive) terms forms an arithmetic progression.

the 13th XMO, P3

Tags: orthocenter and circumcenter , concyclic , geometry

Let O be the circumcenter of triangle ABC. Let H be the orthocenter of triangle ABC. the perpendicular bisector of AB meet AC,BC at D,E. the circumcircle of triangle DEH meet AC,BC,OH again at F,G,L. CH meet FG at T,and ABCT is concyclic. Prove that LHBC is concyclic. graph: https://cdn.luogu.com.cn/upload/image_hosting/w6z6mvm4.png

2025 Bangladesh Mathematical Olympiad, P6

Tags: power of a point , homothety , geometry , radical axis , circles

Suppose $X$ and $Y$ are the common points of two circles $\omega_1$ and $\omega_2$. The third circle $\omega$ is internally tangent to $\omega_1$ and $\omega_2$ in $P$ and $Q$, respectively. Segment $XY$ intersects $\omega$ in points $M$ and $N$. Rays $PM$ and $PN$ intersect $\omega_1$ in points $A$ and $D$; rays $QM$ and $QN$ intersect $\omega_2$ in points $B$ and $C$, respectively. Prove that $AB = CD$.

2019 Online Math Open Problems, 6

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Let $A,B,C,...,Z$ be $26$ nonzero real numbers. Suppose that $T=TNYWR$. Compute the smallest possible value of \[ \left\lceil A^2+B^2+\cdots+Z^2\right\rceil . \] (The notation $\left\lceil x\right\rceil$ denotes the least integer $n$ such that $n\geq x$.) [i]Proposed by Luke Robitaille[/i]

2020 CHMMC Winter (2020-21), 8

Tags: algebra

Define \[ S = \tan^{-1}(2020) + \sum_{j = 0}^{2020} \tan^{-1}(j^2 - j + 1). \] Then $S$ can be written as $\frac{m \pi}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

2005 Czech-Polish-Slovak Match, 1

Tags: function , derivative , analytic geometry , algebra , graphing lines , calculus , polynomial

Let $n$ be a given positive integer. Solve the system \[x_1 + x_2^2 + x_3^3 + \cdots + x_n^n = n,\] \[x_1 + 2x_2 + 3x_3 + \cdots + nx_n = \frac{n(n+1)}{2}\] in the set of nonnegative real numbers.

1995 National High School Mathematics League, 6

Tags: pyramid , 3d geometry , geometry

$O$ is the center of the bottom surface of regular triangular pyramid $P-ABC$. A plane passes $O$ intersects line segment $PC$ at $S$, intersects the extended line of $PA,PB$ at $Q,R$, then $\frac{1}{|PQ|}+\frac{1}{|PR|}+\frac{1}{|PS|}$ $\text{(A)}$ has a maximum value, but no minumum value $\text{(B)}$ has a minumum value, but no maximum value $\text{(C)}$ has both minumum value and maximum value (different) $\text{(D)}$ is a fixed value

2000 Moldova National Olympiad, Problem 5

Tags: fe , polynomial , functional equation

Prove that there is no polynomial $P(x)$ with real coefficients that satisfies $$P'(x)P''(x)>P(x)P'''(x)\qquad\text{for all }x\in\mathbb R.$$Is this statement true for all of the thrice differentiable real functions?

2018 Junior Balkan Team Selection Tests - Romania, 3

Tags: geometric transformation , reflection , geometry , equal angles , circumcircle

Let $ABC$ be a triangle with $AB > AC$. Point $P \in (AB)$ is such that $\angle ACP = \angle ABC$. Let $D$ be the reflection of $P$ into the line $AC$ and let $E$ be the point in which the circumcircle of $BCD$ meets again the line $AC$. Prove that $AE = AC$.

2005 Moldova Team Selection Test, 3

Tags: extremal combinatorics , linear algebra , matrix , combinatorics

For an ${n\times n}$ matrix $A$, let $X_{i}$ be the set of entries in row $i$, and $Y_{j}$ the set of entries in column $j$, ${1\leq i,j\leq n}$. We say that $A$ is [i]golden[/i] if ${X_{1},\dots ,X_{n},Y_{1},\dots ,Y_{n}}$ are distinct sets. Find the least integer $n$ such that there exists a ${2004\times 2004}$ golden matrix with entries in the set ${\{1,2,\dots ,n\}}$.

1993 India National Olympiad, 2

Tags: polynomial , quadratic , algebra

Let $p(x) = x^2 +ax +b$ be a quadratic polynomial with $a,b \in \mathbb{Z}$. Given any integer $n$ , show that there is an integer $M$ such that $p(n) p(n+1) = p(M)$.

2015 HMMT Geometry, 9

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Let $ABCD$ be a regular tetrahedron with side length $1$. Let $X$ be the point in the triangle $BCD$ such that $[XBC]=2[XBD]=4[XCD]$, where $[\overline{\omega}]$ denotes the area of figure $\overline{\omega}$. Let $Y$ lie on segment $AX$ such that $2AY=YX$. Let $M$ be the midpoint of $BD$. Let $Z$ be a point on segment $AM$ such that the lines $YZ$ and $BC$ intersect at some point. Find $\frac{AZ}{ZM}$.

2020 USMCA, 14

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Call a real number [i]amiable[/i] if it can be expressed in the form $a - b\sqrt{2}$, where $1 \le a, b \le 100$ are integers. Find the amiable number $x$ that minimizes $\left|x - \frac{1}{3}\right|$.

2018 HMNT, 1

Tags: number , hmmt

What is the largest factor of $130000$ that does not contain the digit $0$ or $5$?

1994 Polish MO Finals, 2

Tags: projective geometry , geometry , trapezoid , conic

Let be given two parallel lines $k$ and $l$, and a circle not intersecting $k$. Consider a variable point $A$ on the line $k$. The two tangents from this point $A$ to the circle intersect the line $l$ at $B$ and $C$. Let $m$ be the line through the point $A$ and the midpoint of the segment $BC$. Prove that all the lines $m$ (as $A$ varies) have a common point.

1998 May Olympiad, 3

Tags: combinatorics

There are four boats on one of the river banks; their names are Eight, Four, Two and One, because that is the number of hours it takes each of them to cross the river. One boat can be tied to another, but not more than one, and then the time it takes to cross is equal to that of the slower of the two boats. A single sailor must take all the boats to the other shore. What is the least amount of time you need to complete the move?

2014 Saint Petersburg Mathematical Olympiad, 7

Tags: angle bisector , geometry , circumcircle , incenter

$I$ - incenter , $M$- midpoint of arc $BAC$ of circumcircle, $AL$ - angle bisector of triangle $ABC$. $MI$ intersect circumcircle in $K$. Circumcircle of $AKL$ intersect $BC$ at $L$ and $P$. Prove that $\angle AIP=90$

2007 Mexico National Olympiad, 3

Tags: inequalities

Given $a$, $b$, and $c$ be positive real numbers with $a+b+c=1$, prove that \[\sqrt{a+bc}+\sqrt{b+ca}+\sqrt{c+ab}\le2\]

2003 National High School Mathematics League, 14

Tags: complex number , geometry , perpendicular bisector

$A,B,C$ are points that three complex numbers $z_0=a\text{i},z_1=\frac{1}{2}+b\text{i},z_2=1+c\text{i}(a,b,c\in\mathbb{R})$ refer to on complex plane (not collinear). Prove that curve $Z=Z_0\cos^4t+2Z_1\cos^2t\sin^2t+Z_2\sin^4t(t\in\mathbb{R})$ has only one common point with the perpendicular bisector of $AC$, and find the point.

2019 Pan-African Shortlist, G1

Tags: geometry , circumcircle

The tangents to the circumcircle of $\triangle ABC$ at $B$ and $C$ meet at $D$. The circumcircle of $\triangle BCD$ meets sides $AC$ and $AB$ again at $E$ and $F$ respectively. Let $O$ be the circumcentre of $\triangle ABC$. Show that $AO$ is perpendicular to $EF$.

2025 Kosovo National Mathematical Olympiad`, P4

Tags: remainder , number theory

When a number is divided by $2$ it has quotient $x$ and remainder $1$. Whereas, when the same number is divided by $3$ it has quotient $y$ and remainder $2$. What is the remainder when $x+y$ is divided by $5$?