Olympiad problems

2021 AMC 12/AHSME Fall, 7

Tags:

A school has $100$ students and $5$ teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The enrollments in the classes are $50, 20, 20, 5, $ and $5$. Let $t$ be the average value obtained if a teacher is picked at random and the number of students in their class is noted. Let $s$ be the average value obtained if a student was picked at random and the number of students in their class, including the student, is noted. What is $t-s$? $\textbf{(A)}\ {-}18.5 \qquad\textbf{(B)}\ {-}13.5 \qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\ 13.5 \qquad\textbf{(E)}\ 18.5$

2014 Contests, 2

Tags: geometry

Let $\triangle{ABC}$ be a non-equilateral, acute triangle with $\angle A=60^\circ$, and let $O$ and $H$ denote the circumcenter and orthocenter of $\triangle{ABC}$, respectively. (a) Prove that line $OH$ intersects both segments $AB$ and $AC$. (b) Line $OH$ intersects segments $AB$ and $AC$ at $P$ and $Q$, respectively. Denote by $s$ and $t$ the respective areas of triangle $APQ$ and quadrilateral $BPQC$. Determine the range of possible values for $s/t$.

2011 Estonia Team Selection Test, 2

Tags: number theory

Let $n$ be a positive integer. Prove that for each factor $m$ of the number $1+2+\cdots+n$ such that $m\ge n$, the set $\{1,2,\ldots,n\}$ can be partitioned into disjoint subsets, the sum of the elements of each being equal to $m$. [b]Edit[/b]:Typographical error fixed.

2010 Contests, 1

Tags: pigeonhole principle , ceiling function , combinatorics

The integer number $n > 1$ is given and a set $S \subset \{0, 1, 2, \ldots, n-1\}$ with $|S| > \frac{3}{4} n$. Prove that there exist integer numbers $a, b, c$ such that the remainders after the division by $n$ of the numbers: \[a, b, c, a+b, b+c, c+a, a+b+c\] belong to $S$.

1998 AMC 8, 2

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If $ \begin{tabular}{r|l}a&b\\ \hline c&d\end{tabular}=\text{a}\cdot\text{d}-\text{b}\cdot\text{c} $, what is the value of $ \begin{tabular}{r|l}3&4\\ \hline 1&2\end{tabular} $ $ \text{(A)}\ -2\qquad\text{(B)}\ -1\qquad\text{(C)}\ 0\qquad\text{(D)}\ 1\qquad\text{(E)}\ 2 $

1952 Putnam, B6

Tags: ellipse , geometry , conic

Prove the necessary and sufficient condition that a triangle inscribed in an ellipse shall have maximum area is that its centroid coincides with the center of the ellipse.

2000 Croatia National Olympiad, Problem 4

Tags: geometry , combinatorial geometry , combinatorics , area of a triangle

Let $ABCD$ be a square with side $20$ and $T_1, T_2, ..., T_{2000}$ are points in $ABCD$ such that no $3$ points in the set $S = \{A, B, C, D, T_1, T_2, ..., T_{2000}\}$ are collinear. Prove that there exists a triangle with vertices in $S$, such that the area is less than $1/10$.

2012 Regional Olympiad of Mexico Center Zone, 6

Tags: invariant , algorithm , rotation , combinatorics

A board of $2n$ x $2n$ is colored chess style, a movement is the changing of colors of a $2$ x $2$ square. For what integers $n$ is possible to complete the board with one color using a finite number of movements?

2020 AMC 10, 11

Tags: median

What is the median of the following list of $4040$ numbers$?$ $$1, 2, 3, ..., 2020, 1^2, 2^2, 3^2, ..., 2020^2$$ $\textbf{(A) } 1974.5 \qquad \textbf{(B) } 1975.5 \qquad \textbf{(C) } 1976.5 \qquad \textbf{(D) } 1977.5 \qquad \textbf{(E) } 1978.5$

2019 Online Math Open Problems, 2

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Let $A$, $B$, $C$, and $P$ be points in the plane such that no three of them are collinear. Suppose that the areas of triangles $BPC$, $CPA$, and $APB$ are 13, 14, and 15, respectively. Compute the sum of all possible values for the area of triangle $ABC$. [i]Proposed by Ankan Bhattacharya[/i]

2006 Iran MO (3rd Round), 2

Tags: modular arithmetic , vector , function , linear algebra , combinatorics

Let $B$ be a subset of $\mathbb{Z}_{3}^{n}$ with the property that for every two distinct members $(a_{1},\ldots,a_{n})$ and $(b_{1},\ldots,b_{n})$ of $B$ there exist $1\leq i\leq n$ such that $a_{i}\equiv{b_{i}+1}\pmod{3}$. Prove that $|B| \leq 2^{n}$.

1998 All-Russian Olympiad, 8

Tags: algorithm , least common multiple , euclidean algorithm , number theory

Two distinct positive integers $a,b$ are written on the board. The smaller of them is erased and replaced with the number $\frac{ab}{|a-b|}$. This process is repeated as long as the two numbers are not equal. Prove that eventually the two numbers on the board will be equal.

2003 Iran MO (3rd Round), 1

Tags: gauss , relatively prime , number theory

suppose this equation: x 2 +y 2 +z 2 =w 2 . show that the solution of this equation ( if w,z have same parity) are in this form: x=2d(XZ-YW), y=2d(XW+YZ),z=d(X 2 +Y 2 -Z 2 -W 2 ),w=d(X 2 +Y 2 +Z 2 +W 2 )

Today's calculation of integrals, 854

Tags: calculus , integration , analytic geometry , geometry , limit , ellipse , conic

Given a figure $F: x^2+\frac{y^2}{3}=1$ on the coordinate plane. Denote by $S_n$ the area of the common part of the $n+1' s$ figures formed by rotating $F$ of $\frac{k}{2n}\pi\ (k=0,\ 1,\ 2,\ \cdots,\ n)$ radians counterclockwise about the origin. Find $\lim_{n\to\infty} S_n$.

XMO (China) 2-15 - geometry, 15.1

Tags: tangent circle , geometry

As shown in the figure, in the quadrilateral $ABCD$, $AB\perp BC$, $AD\perp CD$, let $E$ be a point on line $BD$ such that $EC = CA$. The line perpendicular on line$ AC$ passing through $E$, intersects line $AB$ at point $F$, and line $AD$ at point $G$. Let $X$ and $Y$ the midpoints of line segments $AF$ and $AG$ respectively. Let $Z$ and $W$ be the midpoints of line segments $BE$ and $DE$ respectively. Prove that the circumscribed circle of $\vartriangle WBX$ is tangent to the circumscribed circle of $\vartriangle ZDY$. [img]https://cdn.artofproblemsolving.com/attachments/0/3/1f6fca7509e6fd6cad662b42abd236fd4858ca.jpg[/img]

PEN P Problems, 12

Tags: function , floor function , calculus , integration , logarithm , inequalities

The positive function $p(n)$ is defined as the number of ways that the positive integer $n$ can be written as a sum of positive integers. Show that, for all positive integers $n \ge 2$, \[2^{\lfloor \sqrt{n}\rfloor}< p(n) < n^{3 \lfloor\sqrt{n}\rfloor }.\]

2013 Czech-Polish-Slovak Junior Match, 2

Tags: number theory , sum of divisors

Find all natural numbers $n$ such that the sum of the three largest divisors of $n$ is $1457$.

2007 Spain Mathematical Olympiad, Problem 6

Tags: geometry , circles

Given a halfcircle of diameter $AB = 2R$, consider a chord $CD$ of length $c$. Let $E$ be the intersection of $AC$ with $BD$ and $F$ the inersection of $AD$ with $BC$. Prove that the segment $EF$ has a constant length and direction when varying the chord $CD$ about the halfcircle.

2024 China National Olympiad, 5

Tags: geometry , circumcircle

In acute $\triangle {ABC}$, ${K}$ is on the extention of segment $BC$. $P, Q$ are two points such that $KP \parallel AB, BK=BP$ and $KQ\parallel AC, CK=CQ$. The circumcircle of $\triangle KPQ$ intersects $AK$ again at ${T}$. Prove that: (1) $\angle BTC+\angle APB=\angle CQA$. (2) $AP \cdot BT \cdot CQ=AQ \cdot CT \cdot BP$. Proposed by [i]Yijie He[/i] and [i]Yijuan Yao[/i]

2010 Postal Coaching, 2

Tags: geometry , inequalities

Let $M$ be an interior point of a $\triangle ABC$ such that $\angle AM B = 150^{\circ} , \angle BM C = 120^{\circ}$. Let $P, Q, R$ be the circumcentres of the $\triangle AM B, \triangle BM C, \triangle CM A$ respectively. Prove that $[P QR] \ge [ABC]$.

2013-2014 SDML (High School), 7

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Two flag poles of height $11$ and $13$ are planted vertically in level ground, and an equilateral triangle is hung as shown in the figure so that [the] lowest vertex just touches the ground. What is the length of the side of the equilateral triangle? [asy] pair A, B, C, D, E; A = (0,11); B = origin; C = (13.8564064606,0); D = (13.8564064606,13); E = (8.66025403784,0); draw(A--B--C--D--cycle); draw(A--E--D); label("$11$",A--B); label("$13$",C--D); [/asy]

2006 Turkey Team Selection Test, 2

Tags: euler , angle bisector , power of a point , radical axis , geometry

From a point $Q$ on a circle with diameter $AB$ different from $A$ and $B$, we draw a perpendicular to $AB$, $QH$, where $H$ lies on $AB$. The intersection points of the circle of diameter $AB$ and the circle of center $Q$ and radius $QH$ are $C$ and $D$. Prove that $CD$ bisects $QH$.

2018 Azerbaijan BMO TST, 2

Tags: perfect square , functional equation , functional equation in n , number theory

Find all functions $f :Z_{>0} \to Z_{>0}$ such that the number $xf(x) + f ^2(y) + 2xf(y)$ is a perfect square for all positive integers $x,y$.

1974 Canada National Olympiad, 5

Tags: projective geometry

Given a circle with diameter $AB$ and a point $X$ on the circle different from $A$ and $B$, let $t_{a}$, $t_{b}$ and $t_{x}$ be the tangents to the circle at $A$, $B$ and $X$ respectively. Let $Z$ be the point where line $AX$ meets $t_{b}$ and $Y$ the point where line $BX$ meets $t_{a}$. Show that the three lines $YZ$, $t_{x}$ and $AB$ are either concurrent (i.e., all pass through the same point) or parallel. [img]6762[/img]

2021 Estonia Team Selection Test, 1

Tags: combinatorics , recursion

Let $n$ be a positive integer. Find the number of permutations $a_1$, $a_2$, $\dots a_n$ of the sequence $1$, $2$, $\dots$ , $n$ satisfying $$a_1 \le 2a_2\le 3a_3 \le \dots \le na_n$$. Proposed by United Kingdom