This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 85335

2019 Belarus Team Selection Test, 4.1

Tags: geometry
A circle $\omega$ with radius $1$ is given. A collection $T$ of triangles is called [i]good[/i], if the following conditions hold: [list=1] [*] each triangle from $T$ is inscribed in $\omega$; [*] no two triangles from $T$ have a common interior point. [/list] Determine all positive real numbers $t$ such that, for each positive integer $n$, there exists a good collection of $n$ triangles, each of perimeter greater than $t$.

VMEO I 2004, 5

Find all the functions $f:R \to R$ satisfying $$(x + y)(f (x)-f (y)) = f (x^2) - f (y^2),\, \forall x, y \in R$$

Kyiv City MO Juniors Round2 2010+ geometry, 2015.9.4

Circles ${{w} _ {1}}$ and ${{w} _ {2}}$ with centers ${{O} _ {1}}$ and ${{O} _ {2}}$ intersect at points $A$ and $B$, respectively. The line ${{O} _ {1}} {{O} _ {2}}$ intersects ${{w} _ {1}}$ at the point $Q$, which does not lie inside the circle ${{w} _ {2}}$, and ${{w} _ {2}}$ at the point $X$ lying inside the circle ${{w} _ {1} }$. Around the triangle ${{O} _ {1}} AX$ circumscribe a circle ${{w} _ {3}}$ intersecting the circle ${{w} _ {1}}$ for the second time in point $T$. The line $QT$ intersects the circle ${{w} _ {3}}$ at the point $K$, and the line $QB$ intersects ${{w} _ {2}}$ the second time at the point $H$. Prove that a) points $T, \, \, X, \, \, B$ lie on one line; b) points $K, \, \, X, \, \, H$ lie on one line. (Vadym Mitrofanov)

Novosibirsk Oral Geo Oly VII, 2023.4

Inside the convex pentagon $ABCDE$, a point $O$ was chosen, and it turned out that all five triangles $AOB$, $BOC$, $COD$, $DOE$ and $EOA$ are congrunet to each other. Prove that these triangles are isosceles or right-angled.

PEN M Problems, 23

Define \[\begin{cases}d(n, 0)=d(n, n)=1&(n \ge 0),\\ md(n, m)=md(n-1, m)+(2n-m)d(n-1,m-1)&(0<m<n).\end{cases}\] Prove that $d(n, m)$ are integers for all $m, n \in \mathbb{N}$.

1987 IMO Longlists, 45

Let us consider a variable polygon with $2n$ sides ($n \in N$) in a fixed circle such that $2n - 1$ of its sides pass through $2n - 1$ fixed points lying on a straight line $\Delta$. Prove that the last side also passes through a fixed point lying on $\Delta .$

1998 Italy TST, 1

A real number $\alpha$ is given. Find all functions $f : R^+ \to R^+$ satisfying $\alpha x^2f\left(\frac{1}{x}\right) +f(x) =\frac{x}{x+1}$ for all $x > 0$.

2024 Sharygin Geometry Olympiad, 18

Tags: geometry
Let $AA_1, BB_1, CC_1$ be the altitudes of an acute-angled triangle $ABC$; $I_a$ be its excenter corresponding to $A$; $I_a'$ be the reflection of $I_a$ about the line $AA_1$. Points $I_b', I_c'$ are defined similarily. Prove that lines $A_1I_a', B_1I_b', C_1I_c'$ concur.

1971 All Soviet Union Mathematical Olympiad, 146

a) A game for two. The first player writes two rows of ten numbers each, the second under the first. He should provide the following property: if number $b$ is written under $a$, and $d$ -- under $c$, then $a + d = b + c$. The second player has to determine all the numbers. He is allowed to ask the questions like "What number is written in the $x$ place in the $y$ row?" What is the minimal number of the questions asked by the second player before he founds out all the numbers? b) There was a table $m\times n$ on the blackboard with the property: if You chose two rows and two columns, then the sum of the numbers in the two opposite vertices of the rectangles formed by those lines equals the sum of the numbers in two another vertices. Some of the numbers are cleaned but it is still possible to restore all the table. What is the minimal possible number of the remaining numbers?

2000 JBMO ShortLists, 10

Prove that there are no integers $x,y,z$ such that \[x^4+y^4+z^4-2x^2y^2-2y^2z^2-2z^2x^2=2000 \]

2017 India PRMO, 12

In a class, the total numbers of boys and girls are in the ratio $4 : 3$. On one day it was found that $8$ boys and $14$ girls were absent from the class, and that the number of boys was the square of the number of girls. What is the total number of students in the class?

2010 National Olympiad First Round, 10

How many integers $n$ with $0\leq n < 840$ are there such that $840$ divides $n^8-n^4+n-1$? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8 $

BIMO 2022, 1

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all real numbers $x,y$, we have $$f(xf(x)+2y)=f(x)^2+x+2f(y)$$

1994 Moldova Team Selection Test, 9

Let $O{}$ be the center of the circumscribed sphere of the tetrahedron $ABCD$. Let $L,M,N$ respectively be the midpoints of the segments $BC,CA,AB$. It is known that $AB+BC=AD+CD$, $BC+CA=BD+AD$, $CA+AB=CD+BD$. Prove that $\angle LOM=\angle MON=\angle NOL$. Find their value.

2019 MOAA, Accuracy

[b]p1.[/b] Farmer John wants to bring some cows to a pasture with grass that grows at a constant rate. Initially, the pasture has some nonzero amount of grass and it will stop growing if there is no grass left. The pasture sustains $100$ cows for ten days. The pasture can also sustain $100$ cows for five days, and then $120$ cows for three more days. If cows eat at a constant rate, fund the maximum number of cows Farmer John can bring to the pasture so that they can be sustained indefinitely. [b]p2.[/b] Sam is learning basic arithmetic. He may place either the operation $+$ or $-$ in each of the blank spots between the numbers below: $$5\,\, \_ \,\, 8\,\, \_ \,\,9\,\, \_ \,\,7\,\,\_ \,\,2\,\,\_ \,\,3$$ In how many ways can he place the operations so the result is divisible by $3$? [b]p3.[/b] Will loves the color blue, but he despises the color red. In the $5\times 6$ rectangular grid below, how many rectangles are there containing at most one red square and with sides contained in the gridlines? [img]https://cdn.artofproblemsolving.com/attachments/1/7/7ce55bdc9e05c7c514dddc7f8194f3031b93c4.png[/img] [b]p4.[/b] Let $r_1, r_2, r_3$ be the three roots of a cubic polynomial $P(x)$. Suppose that $$\frac{P(2) + P(-2)}{P(0)}= 200.$$ If $\frac{1}{r_1r_2}+ \frac{1}{r_2r_3}+\frac{1}{r_3r_1}= \frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $m + n$. [b]p5.[/b] Consider a rectangle $ABCD$ with $AB = 3$ and $BC = 1$. Let $O$ be the intersection of diagonals $AC$ and $BD$. Suppose that the circumcircle of $ \vartriangle ADO$ intersects line $AB$ again at $E \ne A$. Then, the length $BE$ can be written as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $m + n$. [b]p6.[/b] Let $ABCD$ be a square with side length $100$ and $M$ be the midpoint of side $AB$. The circle with center $M$ and radius $50$ intersects the circle with center $D$ and radius $100$ at point $E$. $CE$ intersects $AB$ at $F$. If $AF = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$, find $m + n$. [b]p7.[/b] How many pairs of real numbers $(x, y)$, with $0 < x, y < 1$ satisfy the property that both $3x + 5y$ and $5x + 2y$ are integers? [b]p8.[/b] Sebastian is coloring a circular spinner with $4$ congruent sections. He randomly chooses one of four colors for each of the sections. If two or more adjacent sections have the same color, he fuses them and considers them as one section. (Sections meeting at only one point are not adjacent.) Suppose that the expected number of sections in the final colored spinner is equal to $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]p9.[/b] Let $ABC$ be a triangle and $D$ be a point on the extension of segment $BC$ past $C$. Let the line through $A$ perpendicular to $BC$ be $\ell$. The line through $B$ perpendicular to $AD$ and the line through $C$ perpendicular to $AD$ intersect $\ell$ at $H_1$ and $H_2$, respectively. If $AB = 13$, $BC = 14$, $CA = 15$, and $H_1H_2 = 1001$, find $CD$. [b]p10.[/b] Find the sum of all positive integers $k$ such that $$\frac21 -\frac{3}{2 \times 1}+\frac{4}{3\times 2\times 1} + ...+ (-1)^{k+1} \frac{k+1}{k\times (k - 1)\times ... \times 2\times 1} \ge 1 + \frac{1}{700^3}$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

PEN O Problems, 8

Tags: induction
Let $a$ and $b$ be positive integers greater than $2$. Prove that there exists a positive integer $k$ and a finite sequence $n_1$, $\cdots$, $n_k$ of positive integers such that $n_1 =a$, $n_k =b$, and $n_i n_{i+1}$ is divisible by $n_{i}+n_{i+1}$ for each $i$ $(1 \le i \le k)$.

2009 Federal Competition For Advanced Students, P1, 2

For a positive integers $n,k$ we define k-multifactorial of n as $Fk(n)$ = $(n)$ . $(n-k)$ $(n-2k)$...$(r)$, where $r$ is the reminder when $n$ is divided by $k$ that satisfy $1<=r<=k$ Determine all non-negative integers $n$ such that $F20(n)+2009$ is a perfect square.

2017 MIG, 2

Tags:
If a shrub grows at the rate of $6$ inches per $5$ days, how many feet would it grow in a non-leap year? $\textbf{(A) } 2\dfrac12\text{ ft}\qquad\textbf{(B) } 36\dfrac12\text{ ft}\qquad\textbf{(C) } 182\dfrac12\text{ ft}\qquad\textbf{(D) } 73\text{ ft}\qquad\textbf{(E) } 365\text{ ft}$

1994 Brazil National Olympiad, 5

Call a super-integer an infinite sequence of decimal digits: $\ldots d_n \ldots d_2d_1$. (Formally speaking, it is the sequence $(d_1,d_2d_1,d_3d_2d_1,\ldots)$ ) Given two such super-integers $\ldots c_n \ldots c_2c_1$ and $\ldots d_n \ldots d_2d_1$, their product $\ldots p_n \ldots p_2p_1$ is formed by taking $p_n \ldots p_2p_1$ to be the last n digits of the product $c_n \ldots c_2c_1$ and $d_n \ldots d_2d_1$. Can we find two non-zero super-integers with zero product? (a zero super-integer has all its digits zero)

1998 USAMO, 4

A computer screen shows a $98 \times 98$ chessboard, colored in the usual way. One can select with a mouse any rectangle with sides on the lines of the chessboard and click the mouse button: as a result, the colors in the selected rectangle switch (black becomes white, white becomes black). Find, with proof, the minimum number of mouse clicks needed to make the chessboard all one color.

1997 Moldova Team Selection Test, 3

Prove that every integer $ k$ greater than 1 has a multiple that is less than $ k^4$ and can be written in the decimal system with at most four different digits.

2024 Yasinsky Geometry Olympiad, 3

Inside triangle \( ABC \), points \( D \) and \( E \) are chosen such that \( \angle ABD = \angle CBE \) and \( \angle ACD = \angle BCE \). Point \( F \) on side \( AB \) is such that \( DF \parallel AC \), and point \( G \) on side \( AC \) is such that \( EG \parallel AB \). Prove that \( \angle BFG = \angle BDC \). [i]Proposed by Anton Trygub[/i]

2019 Baltic Way, 5

Tags: algebra
The $2m$ numbers $$1\cdot 2, 2\cdot 3, 3\cdot 4,\hdots,2m(2m+1)$$ are written on a blackboard, where $m\geq 2$ is an integer. A [i]move[/i] consists of choosing three numbers $a, b, c$, erasing them from the board and writing the single number $$\frac{abc}{ab+bc+ca}$$ After $m-1$ such moves, only two numbers will remain on the blackboard. Supposing one of these is $\tfrac{4}{3}$, show that the other is larger than $4$.

2021 ELMO Problems, 3

Each cell of a $100\times 100$ grid is colored with one of $101$ colors. A cell is [i]diverse[/i] if, among the $199$ cells in its row or column, every color appears at least once. Determine the maximum possible number of diverse cells.

2015 Switzerland - Final Round, 9

Let$ p$ be an odd prime number. Determine the number of tuples $(a_1, a_2, . . . , a_p)$ of natural numbers with the following properties: 1) $1 \le ai \le p$ for all $i = 1, . . . , p$. 2) $a_1 + a_2 + · · · + a_p$ is not divisible by $p$. 3) $a_1a_2 + a_2a_3 + . . . +a_{p-1}a_p + a_pa_1$ is divisible by $p$.